I. Irreversible Thermodynamics of Systems Containing Radiation. Application to Photochemical Reactions. II. Mean Values in Ensembles of Systems of Particles with Long-Range Potentials. III. Approximations to Time-Dependent Ensemble Averages Citation Mortimer, Robert George (1963) I. Irreversible Thermodynamics of Systems Containing Radiation. Application to Photochemical Reactions. II. Mean Values in Ensembles of Systems of Particles with Long-Range Potentials. III. Approximations to Time-Dependent Ensemble Averages. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6XPY-Y044. https://resolver.caltech.edu/CaltechTHESIS:04302012-143409544 Abstract I. The general theory of the irreversible thermodynamics of chemically reacting systems interacting with radiation is developed. From this, a limit to the efficiency of photochemical processes is derived, and this is compared with approximate experimental efficiencies of published researches, which are found to be considerably lower. II. A general cluster method of obtaining approximations to ensemble averages of functions depending on coordinates and/or momenta at a single time is derived, starting with a method, due to Mazo and Zemach, valid only for short-range forces, and proceeding to a method capable of treating systems of particles with long and short-range potentials, such as ionic solutions. III. Approaches to the time evolution of distribution functions and to time-correlation functions are discussed. A general perturbation series approach to time-correlation functions through the exponential Liouville operator is derived, and this is applied to the calculation of the first-order correction to a Fokker-Planck friction constant and to first and second-order corrections to a momentum autocorrelation function. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry and Physics) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Mazo, Robert M. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1963 Additional Information: Title varies in the 1963 Caltech commencement program: I. Irreversible Thermodynamics of Photochemistry. II. Approximations to Ensemble Averages. Record Number: CaltechTHESIS:04302012-143409544 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04302012-143409544 DOI: 10.7907/6XPY-Y044 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6992 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 02 May 2012 14:37 Last Modified: 08 Jan 2024 23:38 Thesis Files Preview PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

I. IRREVERSIBLE THERMODYNAMICS OF SYSTEMS CONTAINING RADIATION. APPLICATION TO PHOTOCHEMICAL REACTIONS II. MEAN VALUES IN ENSEMBLES OF SYSTEMS OF PARTICLES WITH LONG-RANGE POTENTIALS III. APPROXIMATIONS TO TIME-DEPENDENT ENSEMBLE AVERAGES Thesis by Robert George Mortimer In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1963 ACKNOWLEDGAENTS It is a pleasure for we to acknowledge the nelp of Dr. Robert M. Mazo, who nas cuided the research which has resulted in this thesis. In addition, I am erateful for the help and encourageuent of others, both fellow students and teachers, too numerous to mention, from whom I have learned much. Thanks are also due to my parents, Dr. and Mrs. Willian E. Mortimer, who have always encouraged ay progress. More especially, deep gratitude is due my wife, Dottie, who has helped me in more ways than she realizes. I am grateful to the National Science Foundation for fellowships which have made this thesis financially possible, to F. I. duPont de Nemours and Company and to the Shell Companies Foundation for research grants, and to the California Institute of Technology for a tuition scholarship during sy last term of residence. — ti ( ‘ ABSTRACT I The general theory of the irreversible thermodynamics of chemically reacting systems interacting with radiation is developed. From this, a limit to the efficiency of photochemical processes is derived, and this is compared with approximate experimental efficiencies of published researches, which are found to be considerably lower. II A general cluster method of obtaining approximations to ensemble averages of functions depending on coordinates and/or momenta at a single time is derived, starting with a method, due to Mazo and Zemach, valid only for short- range forces, and proceeding to a method capable of treating systems of particles with long and short-range potentials, such as ionic solutions. It Approaches to the time evolution of distribution functions and to time-correlation functions are discussed, A general perturbation series approach to time-correlation functions through the exponential Liouville operator is derived, and this is applied to the calculation of the first-order correction to a Fokker-Planck friction constant and to first and second-order corrections to a momentum autocorrelation function. Eis TABLE OF CONTENTS LITLE Trreversible Thersodynanics of Systems Containing Sediation. Application to Photochemical Reaction A. Introduction R. General Theory C. Photochesiical Efficiency D. Calculations Ee. Discussion Glossary of Symbols References “ean Values in Fnsembles of Systens of Particles with Long-Range Potentials A. Introduction 1. Explanation of the Problem 2. Survey of the Literature B. Preparatory Development C. Treatment of Two-30dy Functions D. tiscussion Appendixes 1. Proof of Mayerts “Chain-Snrinking" Theorex ?. Proof of Equation 40 3. Fxtension of Salpeter’s “etnod to the Yulticomponent Case Slossery of Symbols 2eferences we oe ia] III. Approximations to Time-Tependent Frsemble Averares 89 \. Introdiuction &9 1, Explanation of the Problem RS 2. Survey of the Literature 91 3. Approsches to the Time Evolution Probvlea 95 C. Tineslenendent Fnsemble Averages 105 1. Theory 105 2. Calculations 113 a. FokxerePlanck Friction Coefficient 113 De “omentum Autocerrelation Punction 125 re. DPiscussion 140 References 143 Propositions 145 I. IRREVERSIBLE THERMODYNAMICS OF SYSTEMS CONTAINING RADIATION. APPLICATION TO PHOTOCHEMICAL REACTIONS.* (1) A. INTRODUCTION The behavior of a radiation field in the presence of absorbing matter has been extensively studied, largely from the point of view of astrophysical applications. (2) The chemical study of the interaction of radiation and matter has been largely concerned with such topics as quantum yields and the mechanisms of photochemical reactions. Of particular interest are those photochemical reactions which proceed with a positive free-energy increment (i.e., which will not go "in the dark"). Typical important examples are photosynthesis and ozonization of oxygen in the upper atmosphere. In this section of the thesis we give a general treatment of the irreversible thermodyna- mics of systems containing radiation and apply our results to the determination of the efficiency of photochemical reactions which will not go ‘in the dark."" Although most of the complication inherent in the general treatment will be unnecessary for the particular application which we make, we nevertheless present it here, for it is not without intrin- sic interest and possible applicability. * This part of the thesis is a revision of a published article, reference 1. B. GENERAL THEORY If unpolarized radiation is in thermal equilibrium with matter at a temperature |, Planck (3) has shown that the specific intensity of the radiation is given by Ty) = Zhy? , <" exp(hyyq) — | where L,) is the specific intensity in units of energy (1) per unit area per unit time per unit solid angle per unit frequency interval, and 4 denotes a unit vector giving the direction. The constant h is Planck's constant, V is the frequency, ¢c is the velocity of light, R is Boltzmann's constant, and | is the temperature of the matter in equilibrium with the radiation. In the case of equilibrium, one can speak of T as being the temperature of the radiation, T, . However, Planck (3) has also shown that it is often convenient to ascribe a temperature [,. to nonequilibrium vadiation, defined by equation 1, which can be solved for |, : hv kn [(Zhvez)+l] ' Although Ty is now a function of position, direction, and (2) frequency, it is a useful concept when the radiation can be considered to be in local thermodynamic equilibrium, which is assumed to be the case in the following development. ‘ 3 The system under consideration is assumed to be come- posed of a fluid and radiation from an outside source. Discontinuities in the fluid are supposed to be negligible. Symbols will be introduced as needed, generally following the notation of Kirkwood and Crawford (4), who give a discussion of the macroscopic equations of transport for systems not containing radiation. The Stokes' operator will be written as a total derivative, while a derivative taken at a fixed point in the coordinate system will be written as a partial derivative. Thus, ad _ 9 The system is described by the following equations: a) the energy-balance equation, which will be developed in detail below; b) the equation of motion of matter, Vv 5 se = eX + VS (4) dt ~ ~ ? where 0~ is the total stress tensor, X is the external "body" or "volume" force, VY is the macroscopic local velocity, and - is the density of the fluid; c) the equa- tions of continuity of the chemical species present , de; t hek: +V-(¥ +FeXJ=—e 7» d Ne (5) where ©, is the density of the ith component, 7, is the ~ diffusion flux of this component relative to the local 4 center of mass, an is the gtoichiometric coefficient in mass units for the ith component and the kth reaction, and A, is the progress variable of the kth reaction, also in mass units; and the sum of these equations, the equation of continuity of matter, dP _ J+ + ¥ Cex) =O ; (6) and d) the equation of transfer of radiation, of which only a consequence will be needed: Ee = - V-F _— J r y ~ dt where pe is the rate of energy absorption per unit volume (7) by the matter from the radiation field, E, is the space density of radiant energy, and EF is the radiant energy flux, defined by 6 = 6 ((T0drd +4 ((Lmdod 0 +8((Tncod 0, coy ~~ where the é are wit vectors in the x, y, and z directions and L, wm, andn are direction cosines taken from the same directions. An energy-balance equation is obtained by equating the sum of the rate of increase of energy in an arbitrary region of the system, (2GkriEl , @ and the rate of energy efflux through the surface of the region, S[ie+ Gehrke v+E]-dd ao A ~~ to the rate at which work is done on the matter in the region, [YEEMvfeldTs[ueds ay e=( V Forces due to reflection of radiation at discontinue ities, such as the boundary of the system, are neglected. The surface integrals are transformed by using Gauss’ divergence theorem, after which the integrands may be equated, since the region of integration is arbitrary. me gives 2. e E+. Lov4e)+ Vf de +(PE+39¥ )y +E) = . (12) S =7 4%: + eyvL + vve In these equations, Fy is the internal energy per unit mass, de is the energy flux with respect to the local center of mass, and X- ig the external force on component i. From equation 12, using the equation of motion, equation 4, one obtains after rearranging terms: “GE etre) ) oe + v. “Tes) = 6 where equation 7 has been used. The first and fourth parentheses on the left hand side vanish by the equation of continuity, and the second and third can be simplified by using the Stokes’ operator, After use of the identity, Vey I ViVGT + ZV , (14) Equation 13 becomes ‘ bhi CoA — V4, é os = g: Vv 4 . f = page =o re phi Ve f (15) c= Thies equation is not yet in the most useful form. We sub- stitute for Ie ; ky _ 5) ; _ BbeapeD ph = pt DIME YY, oo —— where ax: is the chemical potential, E- and 5S, are partial specific quantities, and F and I are heat fluxes defined by equation 16, Note that the chemical potential is the partial specific quantity, not the partial molar quantity. We define further the nonradiative entropy flux excluding the convective flux: [— Ici -£ + fee (17) - ke a and the gradient of the chemical potential minus the effects of the external forces and the temperature gradient: Va _ Va -X; of Le Vi (18) 7 After substituting the expressions for the energy flux and the entropy flux into equation 15, we obtain: After some manipulation and use of the identity, v-(24) =-aVA+A-Va , _ the desired non-radiative energy transport equation is obtained: pthiol -T 04, $a PGW F Ws spe. — Equation 7 is already an equation of transport for radiant energy, but needs to be put in a more useful form, such that the divergence of F 4s expressed in terms of the entropy flux, S , which is defined: 2 = 6 fh todd «6 |L,smabrall? + G/L navel _ (22) From the definitions of F and S , we find = | [plano : 3 [fombodih 3 gnc dQ 2) =< if, Abd 4 “afl yadda, ndydll.. (A) We know that (3) hry — (25) aT, i Thus, if conditions for interchanging the order of differ- entiation and integration are met, TVG = V-F (26) One special case for which this is satisfied is that of a monochromatic plane wave. Although a plane wave cannot have a finite intensity (3), many photochemical experiments are performed using nearly monochromatic, nearly unidirection- al radiation, so we shall consider this case from here on. One could treat the general case as a superposition of plane waves, or perhaps by defining a mean temperature. In order to investigate the entropy production, we shall seek two partial entropy equations of the form (PS, 5) 4 9-E Ty. - ae (27) and = which, when added, wiil give an expression for £; = A, + he . the entropy production density. Let us first transform equation 27 into an equation in terms of quantities we have used thus far. Using the Stokes’ operator, equation 27 becomes A Ss SF 7 Sop -v (GS )+ VE - gd = O., (29) Now AS 3 — + ¥Vp=- V-pyJev Up =-pV-¥ | (0) Therefore, poe + VE. -~pev)-f =o. @D But 7, - pPI,¥ = s,. » 80 equation 31 becomes pie +l -f = 0. o Equation 28 and 32 can now be added to obtain a total entropy equation yielding g. ° To obtain an equation of the form of equation 32, we differentiate the Gibbs' equation, AE, =7, dG, - bdV + Jrock ome, (33) with respect to time, substitute for the volume in terms of the meaET sf » and solve for HS, fal o, obtaining: dS, P LE fb ap lk | F dé 7 at pl dt LZ Att 7. We know also (3) os. _ - JF ——e - (35) ee Pot Substituting equation 7 and equation 21 into equation 35 and equation 34, we obtain the two entropy equations, which we combine, obtaining: S ~ - far $e -£-VH.-0S, 10 wo where the entropy source Z< is a sum of d= ae(erha vu , P 7 ad) —— >” DR ke ft he ae wo N BN i. 7 aN “ 7 TIN Se (37) tw Each of these terms can be separated into a product of a "generalized force" with a "flux". We choose fe as a flux and a 7 as a force, and split the other terms in the conventional manner. According to the theorem of Onsager, if certain hypotheses are met (5), the entropy production is a sum of products of forces and fluxes, and the matrix of phenomenological co- efficients expressing the linear dependence of the forces on the fluxes is symmetric. In an isotropic medium if the fluxes depend on the forces linearly, then scalar fluxes can only depend on scalar and tensor forces, but not on vector forces. This situation arises because, to make a scalar © depend linearly on a vector, the coupling coefficient must be a vector. In an isotropic medium, there are no available 11 vectors (in the absence of external fields) which do not depend on the vector forces. But if the coupling constant depends on the vector forces, the hypothesis of linearity is violated. There are, however, isotropic tensors, so that scalars may be isotropically coupled to tensors. This result is known in the literature as a Curie's theorem, and the usual reference is reference 6. However, the referee of the paper by Mortimer and Mazo (1) quite rightly pointed out that reference 6 does not contain the above statement. We restrict our attention to isotropic media and assume that the relation between chemical reaction rates and the stress tensor, temperature gradient and concentration gradients are linear. We do not assume that the reaction rates are linear functions of the various AFs or of the radiation flux. Such assumption would restrict our considerations to physically uninteresting situations and is, in any case, unnecessary. It follows that the rates of chemical reactions will be affected only by the radiation and by the forces de- seribed by the stress tensor; the latter effect hae never been experimentally demonstrated, and would be expected to be very small, if found. C. PHOTOCHEMICAL EFFICIENCY Since we are dealing with reactions that will not pro- ceed in the dark, the Gibbs free energy of the system will Snerease as these reactions proceed photochemically. This 12 inerease of free energy of products over reactants is in principle completely available to do work through electro- chemical or other processes using the reverse reactions. We define the sted as follows: Where / is the sctheteney and ea the free energy change per unit mass of reaction &. Those reactions (call them reactions 4+/, h+2, --. , t ) which do not happen to be coupled through phenomenological coefficients with a photochemical reaction of positive Af are excluded from the sum for reasons we will discuss. We require that the entropy production be non-negative, according to the second law of thermodynamics. Since the forces giving rise to o, and By cannot influence those giving rise to BZ » Poy and Ds ; by the theorem of Curie, and since the effect of the force connected with ZB on the chemical reactions is assumed negligible, we require the sum of 4 and De to be non-negative. aA _f\s £Zaae + pe = = oO. (39) However, if some, say t-h, of the reactions are not photochemical and are not coupled to photochemical reactions, they obey a separate inequality, and equation 39 still holds if these reactions, A+ / »--) Cy are excluded from the sum. The free energy increments of the photochemical reactions 13 we are considering are positive, so if their dA, [dt are to be positive, either pe must be positive, or else some of the coupled reactions must have a necative product Af, AA MAE » in which case our nominally photochemical reactions are being driven by coupled reactions instead of photochemicaily. In this case we can say nothing about photochemical efficiency. It may even turn out that i is negative. Now from equation 39 we conclude h — _ fs dA, < p- he 40 Pre eee ae Sp caf This Limit is the same functionai form as the Carnot effi- ciency of a reversible heat engine operating between heat reservoirs of temperature Tt and i ° Clearly the maxinua efficiency caa be approached only if the (negative) free energy production due to reactions which go spontaneously ("in the dark") can be made negli- gible. For the case of reactions which are photochemically catalyzed, rather than driven, ‘ will, in general, be negative. In this case, equation 40 still holds, but tells us nothing of interest. In a study of the production of Light from therwasalh energy, Weinstein (7) stated in a footnote that arcunents Similar to his could be used to show that equation 40 is the limiting efficiency for the conversion of radiant 14 energy into work. It can also be shown that a reversible heat engine using a black-body radiation field (photon gas) ag a working fluid has the efficiency given by equation 40 if 7, and / are replaced by the upper and lower tempera~ tures with which the radiation is equilibrated. D. CALCULATIONS Let us now compare our theoretical limit with experi- mental average efficiencies, which may be calculated approx- imately from data reported in the literature. The results of such comparisons for three published etudies (8, 9, 10) are given in Table I. Rabinowiteh and Wood (8) studied the dissociation of iodine gas into normal fodine atoms by visible light from a carbon arc. In order to make an approximate calculation, it was necessary to compute first the radiation temperature at their reaction vessel, and then to calculate their exper- imental efficiency. Since only a crude comparison can be made, only two wavelengths of the continuous carbon arc spectrum were chosen at which to calculate the radiation temperature, the wavelength of maximum emission of the arc, and the wavelength of maximum absorption. The radiation tempera- tures were obtained as mllows: The reported number of photons absorbed per unit volume per unit time was TABLE I COMPARISON OF EFFICIENCIES Reaction Io -~> 21 305 > 204 2NH, > No +3Ho Reference Wavelength (2) Approx. T, (°K) Max. Eff. Exp. Eff. 8 9 10 5050 5180 1295 1470 1295 1470 3450 3400 4750 5225 5400 5900 0.91 0.91 0.94 0.94 0.94 0.95 0.45 0.31 0.43 16 multiplied by the energy per photon at the wavelength of maximum absorption as an approximation to the total rate of energy absorption per unit volume. From this, the total incident intensity was calculated, using the reported mean absorption coefficient, <, , of 18.0 en tatm * and the given cell length of 2 em, and the reported iodine pressure of G15 am of mercury. These figures yleld a transmitted intensity of 98.4% of the incident intensity, which was therefore approximated by the rate of energy absorption divided by 0.016. From this, it was possible to get the scale on the ordinate of a graph of the spectral distribution of their carbon are given by Rabinowitch and Wood, by the use of graphical integration. This allowed the intensity per unit wavelength interval at any given wavelength to be read directly from the given curve, and this quantity was con- verted to intensity per wnit frequency interval for the two frequencies of interest. This intensity was then converted to intensity per unit solid angle by assuming that the arc subtended an area of 1 square centimeter and that the incident intensity came equally from all parts of this area, which was located 16 centimeters from the reaction vessel. The radiation tempera~ ture was then calculated at the two wavelengths of interest, 5180 & and 5050 & by substitution into equation 2, giving 3400 and 3450 degrees Kelvin, 17 In order to estimate the experimental efficiency, it was assumed that it would be possible to dissociate enough Lodine molecules te make the iodine atom con- centration equal to the iodine molecule concentration, and it was assumed that these conditions could have been approximated by an experiment of the type done by Rabinowitch and Wood. A standard free energy change for the dissociation reaction of 28.9 kcal/mole was taken, and the standard formula used: Af, =AF,° 4+RT&Q (41) K where AF, "ts the standard free energy change, and where © is the quotient of activities raised to the proper powers (see glossary of symbols). A quantum yield of unity was assumed, and using 55.4 kcal/einstein as the average photon energy, the average efficiency was calcu- lated with equation 38. The theoretical limit was calcu- lated by equation 40 assuming that the radiation temperature did not vary over the reaction vessel. Groth (9) studied the formation of ozone from oxygen, using light in the vacuum ultraviolet from a xenon lamp with a partial pressure of Xe of 0.1 mm and a partial pres- sure of Ne of 20 mm. Oxygen was passed through an illumi- nated vessel at varying rates and the ozone production was studied as a funetion of this rate. The ozone fore mation wag assumed entirely due to the two emission lines at 1295 & and 1470 &. In order to get the intensity per 18 unit frequency interval, it is necessary to know both the width and the shape of these lines. The literature seems to contain no information on the width of these lines in the emission spectrum, but McLennar and Turnbull (11) reported that the extreme width of the absorption line at 1470 & was about 5 & at a pressure of 20 mm, including the wings of the line. Not knowing the shape of the line, we make the very rough approximation that each line may be approximated by a block of radiation 1 AR wide. This 4s a reasonable width, according to the work of Margenau and Watson (12). A more reasonable shape would be gaussian, but this would lead to a different radiation temperature for each frequency over the line, and we would have to take some sort of average temperature. Using a square line shape gives us effectively some sort of average temperature, and simplifies the calculations. The assumption was made that each line was of equal intensity in photons per unit time. This is only very roughly correct, since Groth reported that 73% of the ab- sorbed energy came from the 1470 A line. The approximate incident radiation temperature was calculated as before from equation 2, making reasonable assumptions about the area subtended by the source and the distance from the gource to the reaction vessel. However, in this case, there is another complication, since the absorption coefficient of oxygen at the wavelengths in question is very large. The transmitted intensity was 19 accordingly very small after the light passed through only a few millimeters of the cell. We make the very rough approxi- mation that the mean radiation temperature was about equal to the arithmetic mean of the incident radiation temper- ature and room temperature. The experimental efficiency was estimated from the reported quantum yield of about 2.9. Since no data on actual concentrations were given, the standard free change of the ovization reaction, was taken as an upper limit to the actual free energy change, and equation 33 was used, assuming equal numbers of photons absorbed from each of the two spectral lines. The data of Wijnen and Taylor (10) were even harder to wor with, since they used a reaction vessel surrounding their light source, which was a xenon lamp similar to that of Groth, except that it operated at a xenon pressure of 1.5 mm, with no ether gas present. This would give sharper emission lines than would Groth's lamp, so the radiation temperature was approximated by repeating the calculations on Groth's lamp assuming a block of radiation 0.3 A wide for each line. In order to estimate the experimental efficiency, it was assumed that the only products of the ammonia decom position studied were nitrogen and hydrogen, and that the free energy change could be approximated by the standard free energy change of 8.0 keal per mole of NH3. Quantum yields of from 2 to 11 were reported, so the efficiency was 20 calevlated on the basis of a quantum yield of 11, and it was again assumed that the two spectral lines had equal intensities of photons per unit time. Thus the efficiency was approximated by 11 times the standard free energy change divided by the average energy per einstein of photons at 1295 and 1470 &. E. DISCUSSION The preceding development has assumed local equilibrium in the fluid system, and also in the radiation field. If absolute equilibrium were attained, the radiation would be isotropic, no chemical reactions would occur, and the radiation temperature would be equal to the thermal temper« ature of the fluid. This, however, ie far from the case we are interested in. In most photochemical experiments, one uses nearly unidirectional radiation of rather high radiation temperature; one should ask under what conditions the actual state of such systems can be approximated by assuming local equilibrium. First, the radiation should not be required to be isotropic, or even nearly so, since we have assumed a nearly homogeneous, Lsotropic fluid. In such a fluid, the effect of an individual photon is independent of the direction from which it arrives, and since the photon- photon coupling is very weak, one could think of the radiation field as being composed of essentially 21 independent photons. This allows us to assume anisctropy, such as the nearly unidirectional radiation we discuss. Although the case in which the radiation temperature is only slightly higher than the thermal temperature might be of interest, it should not be necessary to assume this case in order to use local equilibrium arguments. In the discussion of systems without radiation, it is sufficient to assume that gradients of macroscopic properties are small on the scale of intermolecular distances. Therefore, we shall assume for the foregoing development that the radiation temperature does not vary significantly over distances of the order of intermolecular distances. This seems equivalent to requiring local equilibrium separately in the fluid and in the radiation field, but not requiring equilibration of the radiation with the fluid. This cri- terion will probably be met with sufficient exactness in typical photochemical systems. The approximation which was introduced by neglecting forces due to reflection of radiation at discontinuities is not likely to be of importance, since most photochemical reactions are carried out ir systems of sufficient size that the surface of the system is not very important. The two assumptions mentioned above are those of basic importance to sections B and C. However, the erudity of the numerical calculations of section D should probably be emphasized again. The experiments cited were designed 22 to investigate quantum yields or reaction kinetics, and the data necessary to calculate radiation temperatures and experimental efficiencies were incomplete and had to be augmented with reasonable estimates of many numbers. However, the comparisons of Table I are expected to be qualitatively valid. However, it would have been more useful to compare limiting and actual efficiencies of experiments designed to maximize the photochemical effielency, but no such experiments could be found in the literature. Instead of trying to maximize the efficiency by using the longest wavelength that would cause their reactions to proceed, for example, the investigators cited used convenLently available wavelengths. Again, they used relatively high intensities, making the limiting efficlency 30 high as to be experimentally indistingvishable from unity. Still, the fact remains that the actual efficienctes were lower thar: the theoretical limiting efficiency, as must be the case even with experiments designed to approach this limit. The conclusion that radiant energy cannot be fully utilized in a photochemical process is not surprising, since it has been known since the time of Planck (3) that plane waves, which one might think of as sources of pure energy without entropy, cannot have finite intensities. It has also long been realized that parallel beams of 23 wonochromatic photons do not exist, since the womentum and the position of even a Sinzle photon cannot be siml- taneously specified to arbitrary exactuess, Indeed, the result here obtained waight have been anticipated if one thought of radiation as a means of transferring enersy between black bodies, which could be used as heat reservoirs with ordinary reversible heat enzines. ms: ie ye ym mM M ® Ss ¥ X & . nN Bids wwe F HD gigats: LUMA \\ 5 GLOSSARY OF SYMBOLS Element of area times the normal unit vector Speed of light Internal energy per unit mass Partial specific internal energy Space density of radiant energy Radiation energy flux (see equation 8) Free energy change per unit mass of the kth reaction Planck's constant Partial specific enthalpy Intensity of radiation per unit area per unit time per unit solid angle per unlit frequency interval Energy flux wich respect to the local center of mass Diffusion flux of component i Non-rediative entropy flux (see equation 17) Boltzmann's constant Radiant entropy intensity similar to I,, Mass of component i Total pressure Second law heat flux First law heat flux a2 b Gn Fe Activity quotient for reaction a@cak..-- eo Gas constant Entropy per unit mass Partial specific entropy Space density of radiant entropy 3 S ay by ~S NRA ha os = | es N | = eS &% 25 Radiation entropy flux (see equation 22) Time Thermodynamic temperature Radiation temperature Macroscopic local velocity Element of volume Mass fraction of component i Force on component i due to vector fields and radiation absorption 2. X; Total force as above Unit tensor Rate of energy absorption per unit mass Efficiency Progress variable of the kth reaction Chemical potential per unit mass Gradient of chemical potential in an equivalent field-free isothermal system (see equation 18) Frequency of radiation Density of matter Total stress tensor Entropy source strength Element of solid angle 1. 2. 3e 7. 8. 9. 10. 11. 12. REFERENCES R. G. Mortimer and R. M. Mazo - Chem. Phys. 35, 1013-1018 (1961) .* S. Chandrasekhar, ES nsfer, (Dover Publications, New york, M. Planck fan Eo 3 of n, Trans. by M. Masius, over Publications, New Yor : 1959) J. Kirkwood and B. Crawford, Jr. . Phys. Gpew. 56, 10481051 (1952) , a B.D. Coleman and C. Truesdell - Chem. Phys. 33, 28-31 (1960) oa P. Curie p. 127, (Gauthier-Villars Paris, 13085" , , , M. A. Weinstein, J. Optical Soc. Am. 50, 597-602 (1960). E. Rabinowitch and W. C. wood, J. Chem. Phys. 4972504 (1936). Z 4» We Groth, Z. Physik. Chem. 378, 307-314 (1937). M. H. J. Wijmen and H. A. Taylor, J. Chem. Phys. . 233-236 (1953). # ab J. C. McLennar and R. Turnbull, Proc. Roy. Soc. (London) Al39, 683-698 (1933). H. Margenau and i. «. Watson, Rev. Mod. Fhys. 3 22-53 (1936). ‘ ; 27 II. MEAN VALUES IN ENSEMBLES OF SXSTEMS OF PARTICLES WITH LONG-RANGE POTENTIALS. A. INTRODUCTION 1. Explanation of the Provlem The principal business of statistical mechanics is the task of obtaining mean values, either over an ensemble of systems, or over an interval of time. One of the funda- mental postulates of statistical mechanics is that an ensemble average may be identified with an observed macro- scopic variable, which is by ite nature a time average. An ensemble is an assembly of a very large number of systems, which are all described by the same numbers specifying a small number of degrees of freedom, usually macroscopic, while the remaining degrees of freedom are described by parameters distributed in a way postulated for the kind of ensemble considered. The treatise by Hill (1) contains a discussion of various ensembles and their application to statistical thermodynamics. This work will use the canonical ensemble, in which the individual systeme all have the same number of particles and composition, and are in thermal contact with the same heat reservoir. For classicai systems, the canonical distribution function, which is proportional to the density of system phase points in the 6-N dimensional phase space of the Neparticle system 28 ({[ -space), is postulated to be Ww F Cppr-tyK-R) & evp[- BH TKK), where B=lAT, His the Hamiltonian of the system, the p's are momenta, and the K's are coordinates of the particles. 7 is the absolute temperature, and & is Boltzmann's constant. The mean value of any function of positions and coordinates can be obtained by integrating ite product with the distribution function over the entire phase space: alps -pihe Kade Uy afl Ph (> = (2) fed did LK, The denominator of the right hand side of equation 2 is called the partition function, Dy Many ensemble averages, such as equilibrium thermodynamic quantities, can be obtained from OF by manipulation of its definition to give an equation of the form of equation 2. We treat those functions for which the average cannot be obtained directly fron Qys Kirkwood and his students have sometimes used the symbol (xs $Y for a mean value, emphasizing the fact that equation 2 has the form of 4 scalar product on [espace of © and the normalized distribution function, poy. oat” . The fact is not important for this work, but the canonical ensemble is simply connected with 29 thermodynamics by the formula where A ig the Helmholtz free energy, or work function. If the function, C , of which we require an ensemble average has as arguments the coordinates and/or momenta of only a small number of particles, say, ', we can, instead of equation 2, use an expression in terms of reduced dis- tribution functions, which we define as follows: (n) /) Crp Rok =|E Rh Rd pdt ht, & /, where £ w is now the normalized distribution function, Oy (o-f8 - . We can similarly define reduced distribue tion functions, 4 ae ” depending on ” coordinates and ‘™ momenta, where 7 xm , or where either 7“ or 7% is zero. n Using f @”) » equation 2 becomes ; (a) (lp. Ry APB REP ARMA Ah Two very useful distribution functions are the two- (2) particle coordinate distribution function, fe (K, K ) (2,2) (strictly we should write f ), and its correlation fmetion, 30 defined by: (2) id V7, f &R)=9 Th ROE ER (6) In fluid systems g° is radially symmetric, and is called the radial distribution function. In sclutions, if we can designate one of the species present as solvent, a useful distribution function depends on the coordinates and/or momenta of all the solute particles and on none of the solvent particles’ coordinates or momenta. A useful function is the equilibrium potential of mean force, yi, defined as follows: The negative gradient of Ww” with respect to the coordi- nates of a particle can be shown to be equal to the ensemble mean force on that particle as a function of the posittons of the other particles in the set n, averaged over the positions of all other particles (2). This part of the thesis consists of an investigation of mean values in an equilibrium ensemble of systems of particles interacting with long and short range forces, and te restricted to functions depending on coordinates measured at only one time. Another, more difficult case, involving coordinates and momenta measured at more than one time, and therefore involving all EN coordinates and momenta through the equations of motion, will be considered 31 in part III of this thesis. Due to the basic similarity between equation 2 and equation 5, the methods developed will be applicable both to gases of charged particles and to ionic solutions, in which case f w is interpreted as a reduced distribution. £unetion involving only the solute ions. An example of a function for which we might seek an ensemble average is a 2; Z ef Dr, » where © 2- is the charge on ion , , (f is the dielectric constant, and | is the magnitude of 5 » the vector separating. Kj ions ¢ and ; . This is the potential energy of set of ions in a dielectric fluid. Another function of interest might be a fi » Of use in neutron scattering (3). Still another function of interest is d(%- -K.) 1G, -F)s the ensemble average of which is f - ad K, Ke) - We shall consider only classical systems. However, for quantumemechanical cases, ensemble averages may be written in a form formally identical with equation 2 if Wigner distribution functions are used (4) instead of the density matrix formalism. Another, equivalent formalism has been employed by Mazo and Zemach (3), using Slater sums to express the ensemble average. In this part of the thesis, the notation of Mayer (5) will be followed as much as possible. 32 2. Survey of the Literature The modern statistical treatment of ionic solutions began with the important work of Debye and Huckel (6), based on the use of the Poisson-Soltzmann equation, the solution of which however is known not to satisfy certain integrability requirements (7). Debye and Hiickel linearized the equation, obtaining a solution satisfying these re- quirements, and obtained their wealleknown expression for the radial distribution function, DAR] Gi where dre J 6, x7 kK = 2Te C; Z ‘ ik? = ° 2 c=/ PD 18 the dielectric constant of the medium, Z. and C, are respectively the charge number and concentration in particles per unit volume of the fonic species Ly» € is the magnitude of the charge on the electron, and y- ig the sealar distance between the particular ions of species < and , being considered. Equation 8 has had adequate experimental verification as an exact limiting law as the concentration of electrolyte approaches zero. In the following three decades, many attempts were made to improve on the DebyesHuckel theory, including the work of Kramers (8) and many others. Onsager (9) gives 33 an early review, and the treatise by Harned and Owens (10) contains much information about these attempts. Probably the most fundamental contribution since that of Debye and Huekel was made by Mayer (5), who obtained expressions for the ensemble averages of the eeuvtic pressure of an fonic solution and the activity coefficients at that pressure, valid to the next order of approximation beyoad the Debye= Huekel theory. He assumed the potential of mean force between ions at infinite dilution to be a sum of twoebody coulomb forces with hard-core repulsions, and applied cluster techniques developed for noneideal gases (12) and extended to mixtures by McMillan and Mayer (11). Divergences in the cluster integrals were taken care of by selective resummation according to groupings of terms so as to cancel out ine finities of opposite sign and give a finite approximation. Since 1950, others have continued work with Mayer's equations. Poirier (13) has obtained numerical results and various thermodynamic relationships, and Haga (14) has carried the evaluation of contributions to those of order kK » whereas Mayer carried them only to order x~ . Meeron (15) has summed some of the expressions explicitly, and has discussed the use of a single hard-core repulsive potential in a mixture of ions. Meeron has worked on expressions for the radial distri-e bution function and twoebody potential of mean force using extensions of Mayer's technique (16), and has obtained results 34 apparently valid to order XK * | one order better than the Debye-Huckel result. Some of the more recent work on radial distributions for classical fluids interacting with shorterange forces is of interest to this problem, since techniques developed to treat dense systems are very similar to those developed to treat longerange forces in dilute systems. Beginning with the work of Salpeter (17), Van Leeuwen, Groeneveld, and deBoer (18), and also Rushbrooke (19), have developed a “hyperchain" approximation, which also involves selective resummations, equivalent to work of Meeron (20). Morita has also invented a “hyper-netted chain" approximation (21), which Meeron (22) has pointed out, is also equivalent to his work. Friedman (23) has derived density expansions of distribution functions without assuming the potential to be a sum of two-body potentials, but a model for the 3-body and higher potentials ia not furnished. He has also ex- tended the Mayer theory in the same sense (24). B. PREPARATORY DEVELOPMENT In order to begin work on ionic solutions, it will be necessary to lay a groundwork of more general applicability. The formalism of Mazo and Zewach (3), which holds for quantum as well as classical systems, will be generalized to multicomponent systems. 35 Consider first the expression of distribution functions in terms of Urselletype cluster functions. The distri- N~) of particles of chemical species // 2 .-. ,0 will temporarily be called Wy instead of { (“eq follow Mazo and Zemach. bution function of the set f/ = (MM, Since the momentum dependence ig immediately separable for classical systems at equilibrium, Wa) is assumed to be a coordinate distribution function, and is assumed separable. That is, if the set NV is divided into the mutually exclusive subsets 7% and iV —1 , which are then moved apart until their separation exceeds the range of intermolecular forces, then The expression of Wy in terms of cluster functions, Ly proceeds by use of — functionals and functional differentiation. The product of parametric functions 4,/f,)46;,) ---a4f,)-9,-(fy,) £8 represented by a” a, Ww). alo) or by Q () - #e will fre- quently use an underlined symbol with an underlined exponent to represent a product of quantities raised to various powers in this manner. The generating functional for the W functions is defined gi We ae“) (10) =O 36 from which expression one obtains a particular Wy by functional differentiation, as in Mazo and Zemach's onee component case. (4) daly) daly) daly) $2, (ty, ) a 20 (11) ~~ A generating functional for a set of Co, functions is de- fined exactly ag in equation 10, and the U; functions are uniquely defined by _ 7 o Gjla) = expo ., (12) Just as in the one-component case, equation 12 relates the (/ and VU/ gunctions such that if the set WN is separated as before, the separability of the W ¢unetions implies that Ny approaches zero. From equations 11 and 12, one finds that N- rs — /, NM (13) rs TT UG) X= /- where the right band side of equation 13 is 4a sum over all ways of dividing the set v into mutually exclusive clusters (not counting permutations within a given cluster as dis- tinct). This agrees with the result of MeMillaa and Mayer (11), who also inverted equation 13 by combinatorial methods, 37 obtalning N, Gyo) = Wye) -2, We Ce) Wo (c¥) + a, Wo! TT” We Ge) ™ x= = On! UW, Colt) (LM7H) where the sum is again over all distinct ways of sivnaios the set // into clusters. The notation Gita used to represent a division of // into 4 elusters such that MM, My, ; My are the sets of particles in each of these clusters. Equation 14 can also easily be obtained by taking the logarithm of equation 12 and then applying functional differentiation. In the previous work, we have assumed that the U/'s and U's are symmetric functions of all the coordinates of molecules of the same species. We now generalize this te separable functions Wy yk (v* ) ~~ x) » which are symmetric functions of coordinates of molecules of the same species only if they are in the same group of argu- ments, as for example in §& | Any number of groups of arguments can be taken, but three groups will be chosen for illustration. The generating functional becomes G,/ (a,b,c) a) W. (4) me @X15) -2, 20 ala ne b Wry ae opr Meo Nzo R20 wy w 38 where the L's and c's are parametric functions similar to the @'s. we have used the notation al= Tred ’ which will be used from now on. we also use the notation a= Sf; . Similarly with the previous case, Wage = = f dL. _ d na 1G (16) —_—— ~ ala) Sagi) bb.) Tey eg) ae ae al We define also, _ re G G@bo = ep Gu a,b) ; (17) which again defines the U functions uniquely. The integrais of the W and U functions will be of interest, although the integrals of the U functions will diverge formally for potentials of infinite range, due to the impossibility of separating subsets of particles to distances greater than the range of the potential. Equation 17 aan for our purposes be rewritten = C8) (4) () SS ca), (0) Ck) ay ke ee ini fee aes 39 w where the parametric functions have been replaced by constants, %,..d, b,, -- be G,-f—e » and where we define (19) u _ sj fu, (2) cl, A) ; mR lulk mM, alos ads ol In order to use the above for ensemble averages, we define Wioy = Wi, =f (strictly pho) ), the reduced coordinate distribution function of the set 1 (all fons), with all configurations of the solvent averaged. ~ All of the following will apply equally well to a gas of charged particles, in which case fe would represent the total distribution function of the particles of the gas. We consider first functions depending asymmetrically on’ the coordinates of only two particles, defining for the classical case, f_ J If we require that Woo, be the distribution function, then c{ can depend on the coordinates of only the two particles. Note that quantum-mechanical definitions can also be made, as by Mazo and Zemach, using Slater sums. ve Functions Wo a and Wes. 4 aiso must exist, due to ~ 49 the assumed separability, by which they are defined. A special case of separability which is of some interest is that of Wo x =Wo;., *O, corresponding to tds wv Pie “i = Gs (f-Si-| > eo From equation 18 and the above definitions, we fird In order to express the ensemble average in terms of the C/ integrals, we restrict 7 and 72 in equation 18 to © or / . We separate all terms in the exponent independent of the 2's aud 4's and expand the remaining exponential factors. Then we equate coefficients of equal powers of the @'‘e and &'s in the resulting expression, and obtain, for example: = N a1 & Te) ~~ N Je ™ (22) S|, Rte £ — os ve = as a CU), Z.5 Uso x Uer,p [expo C Use 2 . However, by writing equation 12 as we did equation 15, we find: _ £ _ =f Be cS Z c ‘tide Z = £L%X 2 iS fe oO é (23) yee ee mee &. El r me ? and by equating coefficients of cH 4n equation 22 after substituting equation 23, we find 41 s WwW I LN ns 4. aot /~ My atleyne)ahap ay (24) Equation 2% resembles a result of Mazo and Zemach, but has multicomponent (underlined) subscripts. From it and equation 21, we find <x gil: 26 ae ath (25) < (r s) NW. plCsns + Mom Uosam) 4 n=e mM=o C. ; where Z, is the fugacity, in molecular units, of come ponent ¢ . We have used the relation (25) 7 MeeWen = ZB Woow (26) ee Bj Equation 26 holds true only for//>7't , which will oblige us to consider an infinite system. When we allow the system to become infinite, however, we must allow the concentration of each species to remain finite and fixed. Although the fugacity, pe ; Of an Lonic component, cannot be measured thermodynamically, mean fugacities re- lated to lt can, so we will consider equations 25 and 26 to be meaningful for Lonic solutions. Mayer (5) and others 42 (13, 15) have studied the fugacity in tonic solutions under an external pressure equal to the osmotic pressure, 80 we can apply their results and work from equation 25. Mayer found that a relation of McMillan and Mayer (11) can be modified to hold for the Coulomb potential: fi = = € ~ Oe (27) Here S.. is given by §.2 2S = <2, By c* (28) . ac: Poo nen Syke. The Br are cluster integrals having as integrands sums of all products of certain functions connected with diagrams in a specific way. The differentiation is simply a formal way to express the series desired, and although the & integrals diverge for the Coulomb potential, they are considered independent of C, in the differentiation. Mayer obtained a finite approximation by a selective re- suomation, so we shall assume Sy to be known. Equations similar to equation 25 can be derived for functions, -, depending on coordinates of any number of particles. For example, V Kk m)> — ! ; 7 nthe (29) 43 and ee ~ a K Mel My - by XM SiS) . W-lcly Yq x ee 61) Ik, A ¥ 2 [Osea Hoar ¥ W1l=oO (30) = Ay, 4 'k, w- tor -O jsf Fee Page etternt )f 3 ; Now that the expansions of Mazo and Zemach are we HOY Mg I A122, 0 vt extended to the multicomponent case, any result of cluster theory can in principle be obtained. In the following, we shall extend the applicability of this formalism to the ease of the Coulomb potential. C. TREATMENT OF TWO-BODY FUNCTIONS For the case of long-range potentials, the expressions such as equation 25 and 29 are not directly useful, since almost every 2 integral diverges. The intermolecular potential in a gas of charged particles and the interionic potential of mean force at infinite dilution in an Lonic solution are of infinite range, 80 we must seek a selective resummation of the terms in such expressions, to get a 4d finite approximation to KX > . For this, it is almost imperative to connect the 2 integrals or their inte- grands with cluster diagrams. In order to do this using elementary techniques, we cestrict ourselves to classical systems. In the following, we treat explicitly the case of equation 25. The extension to mean values of functions depending on different numbers of coordinates, such as the cases of equation 29 or 30, is straightforward. Consider first the well-known connection of Mayer's reducible cluster diagrams with his cluster integrals, bn (which are \/ ~ times our een ). (12) The potential energy, or potential of mean force in the solution case, is assumed given by a sum of two-body potentiais, Fe (Wiss) s dependent only on | Vy. . The iy | functions f.. are defined by ) 2 Pl tu) a K fy | Pah Ll) -_ er PI” y (31) it) \ eet / 7 — ad % The integrands of the by are gums of products of f functions, and each term corresponds to a cluster diagram having 7” labelled vertices representing the set 7 .- A line between two vertices, /, and a represents an of function, having as argument //,. - For example, *) 45 The integrand cf Z/o0» is known (12) to consist of the sum of all products of £ functions corresponding to "at least singly connected'' (ALSC) diagrams on the set of vertices «x . ALSC diagrams are those in which every vertex is connected to every other vertex either by a line or by a path of lines and vertices. The above statement about on is equivalent to equation 14, as can be seen by writing the V/s as products of factors for any particular set Av ° We now take the logarithm of equation 17, and by functional differentiation we find 4.4. = — ol? od Sian ae ee mar P a si = jy Mealy ’ fo 5 2 a jen We, where et, means a sum over the distinct ways of permuting arguments of the set //-/:/- between the different )/ functions in a product, and where the limits on the other gum mean summation over all possible sets of numbers Mz Ms,.-.,'Ng~ up to NM, Me, My, a) Ay-/, =f Ay; -/, My “ Comparison of equation 33 with equation 14 shows that a similar cluster correspondence would obtain for functions, K=A-aA. 4 that are products of one-body functions. An dt 46 : equation for OU, jr, Wy OF Ah; 0 ¢ analogous with equa- tion 33 can be obtained: Ub yy = We al ; _ 7 —; Cy MG OS Ay eee = Wo jn Wee K-a-y “ “e'= oO Comparison of this with equation 14 shows that ‘6, lj N-/, fe consists of the sum of all products of - functions corresponding to all ALSC diagrams on the set / of vertices, multiplied by a » where: ; => 4, WW; )a. Gp (35) 7 "el, —> gO . « a XK: Y. ) J 4 In the cluster diagrams for Uo». vi-/- » we use 2 J, —= triangle instead of a circle for the particle /; . For example <A aK Saag J eS AN, - £ Gy (36) ‘ { ” U* J VGte Ved ag Nig I a4) yk For the terms in .o -le » Which are similar, we choose 5 to use a square for particle /, ‘ if = asa, » then Ut, ve-lel, is a sum of all products corresponding to ALSC diagrams on the set #2, such as, for “//242 for example: pe & xX i => « Ui, Di, Chin, Elite) tag" ( Va) 77 iF i 4? i x A ALK. a.(%,) for finite Vp: » some of the preducts corresponding to disconnected graphs survive in equation 33. If, for a particular Uo ox » equation 14 is written in diagram form, it is seen that all less than singly connected (LTSC) diagrams cancel. (Note that an ALSC diagram on the set 7- 77? multiplied by an ALSC diagram on 7% is equivalent to one LTSC diagram on 7% .) If equation 33 is written in diagram form for a particular V, it is seen that all LTSC diagrams that would become ALSC by addition of a line between ee and /; survive. We call these diagrams LTSC (1) diagrams. These survive in pairs, one diagram with positive sign multiplied by X , and one with negative sign multiplied by CZ; a. In order to see that this is true in general, let us see how the canceilation of diagrams arises in equation 14. Consider a LTSC diagram which must cancel out in equation 14. It is made up of, say, /V disconnected ALSC pieces. This diagram will occur, with coefficient ey Cu-n! a certain number of times, say X,/ , in a sum in equation 14 having as terms products of (M=1,2,---, Vv) W funetions. The factor xi is not in general unity, since the / pieces can come from ‘4 elusters in a number of ways equal to the number of ways of putting WV distin= guishable balis in /{ identical boxes, which is mM VL 48 Here @_. is the number of balls in box number ~<, L. 4s the number of boxes containing @ balis, and the gum is over all sets of occupation numbers. We know from equation 14, that W/ Vo mel v7, aa) (MDIxX, = © , (39) since the LTSC diagrams cancel in equation 14. In equation 33, exactly the same diagrams occur, except that particles a and iP are represented by a square and a triangle, respectively, and some of the diagrams are multiplied by << while others are multiplied by a, . The following cases occur: (a) Particles iy and a are ir; the same ALSC diagram piece. Since /° and a are arguments of the same “/ function, all these diagrams are multiplied by , and they cancel just as in equation lA. (b) Particles /; and tj are in different ALSC pleces say piece A and piece BB, and M=23, All terms in which A and B arose from the same Y will be multiplied by ~ , and all others will be multiplied by Aas Single out those multiplied by ~< . In these, 4 and 5 ,together can be considered one piece, and we have the same cancellation as in case a) except that we now have V-/ “pieces”. In other words the LTSC diagram is multiplied by s",, mM ~", J At XK 4 /) (M-1)s Xe wes C) =H (40) 49 Now look at the terms multiplied by @, a, : These have a coefficient Vv 7 Wl ayy M >¢ M = b) but both sums here are zero, 30 these cancel also. c) The last case, that of LTSC (1) diagrams is different. Here for each diagram, a nonzero coefficient —%«— 4.4% remains. We have therefore shown that U1. v-icy. Corresponds to the sum of all products of f functions corresponding to ALSC diagrams multiplied by ©“, plus all products corresponding to LTSC (1) diagrams multiplied by K- Alp , If we desire, we can connect diagrams with the 7¢ integrals instead of their integrands, the U functions. If we do so, the diagrams are similar, but we denote only the chemical species of each particle without labelling them as though distinguishable. Of course, aiter inte- gration, many terms in a given () may make equal contri- butions, so each distinct diagram enters with a com- binatorfial coeffieient, which is equal to the number of permutations of vertices of the same chemical species which leave the term unchanged after integration. Finding this coefficient for the general case is a difficult combinatorial problem, which is not yet solved. 50 We now return to equation 25, which expresses the mean value of c as a series, and note that the double sum ig equivalent to a sum of LTSC (1) diagrams. However, the combinatorial coefficients in the integrands here are not the same as in the aa oc Since permutations between the disconnected parts are not allowed in the Wis 0, 2 Ue. -m product diagrams. The ratio of the coefficients of a particular LTSC (1) diagram in 1s ©, 104 Le jm and in 2, 4, wast then be 7~ oe as va): eae (9) LEAf ye fae This is the inverse of the ratio of the factors in front of the integrals, so the double sum cancels exactly the LTSC (1) diagrams in the single sum with coefficient -4,q. . / J Wow if we define 1¢,.,. ,, as being like 7¢,. Thy “1, Ve except for deleting the terms multiplied by -AG, » its integrand contains all products corresponding to ALSC and LTSC (1) diagrams, all multiplied by c<, amd equation 25 becomes h-leh J ri fees =O “_ - We neglect bi, compared with Ay; and define: ty _ -2 TL ¢ = VW Up y, £ (42) Equation 41 becomes, using equation 27: Malet, £L vo = ely Pa x jf. = y . 43 <X(h5,%) es Let “((- ER et -) =o d ~ 51 where we use a a he thy +h-¢ : Equation 43 is now a density expansion, at least formally, but it is not in the most convenient form. However, we shall work first with this form, as do Mazo and Zemach (3). We begin by expanding the f. functions, following Mayer (5), assuming the potential of mean force at infinite dilution to be the Coulowb potential with a dielectric constant plus a repulsion at short range: bey) = Ayla + 2 agg [ABB UC Nelo- /) where | and ik tel Here /D is the dielectric constant of the medium, ig the magnitude of the electron's charge, y 2 is Boltzmann's constant, and / is the temperature. The parameter -< will l.ter be allowed to approach (>. The function Ry similar to a . exeept that only the short-range repulsive potential a fis included: / 4. Vig.) = = hay aH (ot) A ler) / Cf hit 52 If this repulsive potential is that of hard spheres, =_—— 7 _ ke Q ; &, (Ky) _ / SKY (45) y/ —_—~ where Ay is the distance of closest approach of ions of species . and Jj? g Each term in a given Sn is now replaced by an infinite number of terms, each of which we associate with another type of diagram, in which a solid line between two vertices, say k and ts represents ‘ (Yee). A broken line will represent a WA function, two solid lines will represent 79 Xr), one broken and one solid line wiil represent k()9(0, ete. For example, | fe X x =H (i, 7 i 19Ge Le WI 9G 24 WRIh u) (46) 2 7 le fy _, We shall call Lg‘) the term in Ly corresponding to the diagram 7 - The contribution to <> from » L is ag 4 “lq 53 A row of vertices, each connected only by two solid lines, is called a chain. If a diagram contains no chains, except possibly chains composed entirely of the particles /- and fi , it is called a prototype diagram. A diagram, 4» will be said to belong to a prototype, ae Lf it becomes identical to § upon replacing all chains (exeluding those containing particles / and fy ) by lines. Je shall later use a prototype diagram to represent the sum of all diagrams belonging to it, in which case we shall draw it with wiggly lines. A typical diagram and its prototype are: ( 4s Se: 4 Os, ” / rn ay t & yb 4 “3 hc i, 4; Note that / and / vemain, even if contained in chains in the diagram. In the set 2‘, we call the vertices in the chains 7% , and the other vertices aa Expression 47 becomes fg) 6% exp (- mS MI; Su, 3 — | (49) Mayer has shown (5), for the case of the irreducible cluster integrals, that a sum of the form 4, c ~ te replaced by a gum of the original form including only prototype diagrams, and that in these, instead of functions, the solid lines, @ St which we now draw “wiggly following Salpeter, represent (o) - ; TA g@Uuwye= S§ - (59) ut 4-77 Mayer's proof of this (5) is included in Appendix 1. The theorem holds true also in our case for the contribution in equation 49 independent of z HS, : The only differences are that our definition of prototypes must be used and that all integrands are multiplied by x Ny yi) This doesn't affect the "chain-shrinkiag”’ since /- ‘and ris J bution of a prototype, S$ , we write are never in the set /2 . The contri- m o c) ~ a a (s)& 0) The contributions from the further terms in the series in expression 49 are slightly more complicated. For these, we restrict ourselves to the case |Z | = [Zz == (Z-). For this case 5 =5,.= ey Oa in equation 49 now becomes - The sum in the bracket _ Dv > nS, =~ Sn = 8, >We (52) es X=] 2 where we number the chains in yy kxom {/ to D, and Hoe is the mumber of particles in chain - . Consider each of the ) terms in equation 52 separately. The summation over diagrams belonging to the prototype ¢ will lead to 55 the same result as before except for chain number X , which is replaced by (0. Sf — KY Vi 1) = 2 NG —— (33) U/ — S17 - ee The contributions multiplied by S:, = a Vox, 18 give us ) terms in expression 49, Summation over dias grams belonging to a prototype, i » gives a result similar to the previcus _ except that oer >< and chain IS are replaced by ia “) instead of g° a if «<> [SB - lf X=8 , this chain must be replaced by (eo Z If) = 2 2 CA 1% (rv) he Tey) > G4) in summing over the diagrams. The contribution from the Ss? sum is similar. There are Wv-1N Y-2) terms with different combinations of three chains replaced by FO instead of @) (-) , BYOF), terms with one chain replaced by (Y y) and one by # (2) (7) , and terms, each with one chain re- piaced by Ke Hy = a) 5} (r= KE_C-K7743kr-| _) GS) i" 327 - 56 Before we can express all orders ‘e contributions to <K»> » we need an expression for ¢ de » whieh is dee fined: - ao # in = ENG Ke?) a “Za (r) ; (56) =O and which we seek as follows. First, we note that g @ (Ko. (57) jj ~( 2 OK / li (yy ) where G is ~~, by equation 50. Since ” occurs in the ee of rs ey) in the product AV’ we can write Ff “r) conveniently as a polynomial in -KY times eK os: f O) = CKn eC = 1S, Geie = . (58) k 2 By operating on equation 58 with 2 de’ we obtain the recursion relation S 5 S=/ a) ae -—? f = & & ; 96, pt Cp (59) by equating equal powers of Yr, By inspection of a few special cases, we concluded that 57 the following was probably the correct expression for C et / Ss a a cs Cp = SS. ft CY (60) goo £6 (s-b)/ | Equation 60 can be proved by induction, but sinee the proof is slightly tedious, it is relegated to the Appendixes (Appendix 2). (2 Now that we have 7 Y/, we ean Find the contributions of higher order than “a simply by proceeding in an analogous manner, with obvious results. In order to write the contribution of a given proto- type 5 » we introduce left subscripts, as well as left superscripts, on Dn (e)* If §$ has © chaine, there are »Y left aibacetoce, and if the contribution is to he wultiplied by S a we affix a left superscript / . If the xt" left subscript is /2 , this means that after summation the x” chain is replaced by gn). qt is apparent that a given set of left subscripts oe ostee more than once in the Uv "tems multiplied by oc /yl P py Since we have / ams = ots Ss multiplied x~<~=/ [jot C=, together, the coefficients will be the multinomial coe / _ j // y? Xx=/ fe ° . efficients: (61) 38 e ~ Thus the contribution of a given prototype § to KK > fs an =m™S, fo ane feel ~ ‘ 0 7- f/ —_ ce [ don ig) “Sy sepinis) 33 a (62) Hh ao / ar iid _ Se Uf. : Sh where the sume are over all sets of left subscripts con- sistent with = lie =f. Now in order to obtain an approximation to a given tw Ln> to any order in A , we simply need to sum expression 62 over whatever 7 and & necessary. However, it is very difficult to determine, a priori, which prototypes must be considered, and it may be difficult to determine how many terms of the series in square brackets in equation 57 to take. As a matter of fact, consideration of the form of expression 57 leads one to conelude that an infinite subset of prototypes should be taken immediately to cancel the expression Lip Ca 5) ° However, this can in effect be accouplished more easily by returning to equation 41 and deriving a more useful density expansion, as suggested by the work of Meeron (16) and of Salpeter (17). We shall use the method of Salpeter, which is illustrated in Appendix 3, where it is 59 extended to the multicomponent case. We write equation 41 in the form NI: 1 ntkf; —4 by 2 ~ Y Aer D> =. ad = x —— mnM=0O0 and recall that the integrand of Ln contains all products of f functions corresponding to ALSC and LTSC (1) diagrams, including those containing articulation points. (An articulation point is a vertex at which the diagram may be severed into disconnected parts by breaking all the lines drawn to that vertex.) Consider now an ALSC diagram representing a term in the integrand of rT in equation 63, say the diagram E 3 In this diagram, vertices lp and <y are articulation points, , ¥* and the parts of the diagram attached to the set eA = . 1 t th ints will alled Che |, Ie, 2k, b, Xe, Ze) i ya 10se points will be calle appendages. The set A constitute an “at least doubly connected," ALBC, diagram (unless there is only one chain connecting /- and 4 ) or an "irreducible diagram”. 69 An ALDC diagram contains at least two independent paths leading between any two vertices. We call the set of particles in the appendage attached at vertex “, the set 4 bly)» and this set may be of any size (or empty) and connected among its own members in any ado way, but can be attached to neo other member of A except As . When the integration in re is performed, inte- gration in turn over the égovdinares of particles re- presented by appendage vertices gives factors multiplying the contribution obtained by integrating over the coordinates of the set ke, Of course, since [a must contain the vertex set nN, we can write R= Mkt = As > k(n) (64) The LTSC (1) diagrams are treated in exactly the same way, except of course for the fact that #* contains only /° and /-, and there is no irreducible diagram. L The contribution of £€ to equation 63 can be written ~ HITE Jai, %, dF sm dy - “es (red A6e! £y4 “—~ 61 ry The first (/[ + | must correspond to an ALDC diagram, unless there are fewer than two paths from /; to /; : and the others correspond to ALSC appendages. In the ease of the LTSC (1) diagrams, instead of the ALDC cone tribution we have simply SK, Ke) Az 2d Fr. Since we have till now regarded the particles as distinguishable, we must now multiply expression 65 by the number of ways to pick the sets & Lis) o Lh ) a i) DO a from the set “2 5 | ~* ¢ (4a)! / giving os % C£0us)) Lee Pi (rf Jel 3, wees) V= Ril ALDe = ba MALSC on y, (66) ot on k* Si p,) +My ; a” 4] okt ® - If we call Vo Ck | ) times the first integral, Livy where Y -vefers to the ALDC diagram, and call the sum * Ny over all ALDC diagrams, on k » K 2 » and then sum expression 66 over all such “Y and over all ALSC diagrams on each set <(x,)+ Ke , we obtain “ x, [| (Lowe, Bo ey 62 By equation A34, summing this over all £ CL) for each Ss gives & (68) Summing expression 68 over all Ak givea us a convenient density expansion for <x > : MV - -lely’ << 9 A Kalin >= 2 as (59) Yee | =O Now we treat this exactly as we did the term of equation 43 independent of Sy 3 expanding the functions, defining new diagrams and prototypes. The only difference is that only diagrams occur in which all particles lie on direct paths from ij: to /; (irreduc- ible diagrams). The chaineshrinking theorem of Mayer applies directly, and we obtain finally after summing over all diagrams belonging to the same prototype, Vil La y a ee <b, A= 2, SB, E&° , wW=C a Here a is defined as was kon » except that the (2) integrand is a sum of products of 4 functions and #& functions, corresponding to irreducible prototype diagrams. In principle, the problem of obtaining a finite approximation for Kx» is now solved, since all the integrals will converge for functions A of interest. 63 However, we still have an infinite number of prototypes, and there is no systematic way to pick those which must be included to obtain an approximation exact to some order in the concentration. This is due to the fact that the @ funetions are concentration-dependent, due to the dependence otf COP the Debye parameter, < . Thus, one can not obtain an approximation to order a” just by taking all prototypes up to those for whieh 77-7 , since later <2 funetions could be proportional to high inverse powers of concentration, cancelling all or part of the factors of concentration multiplying then. This is a general failing of cluster integral theory, and future work could well be concentrated on seeking a systematic ordering scheme. Nevertheless, one can ine vestigate the smaller prototype diagrams individually, hope that there are no prototypes among the larger ones that behave much differently, and thus hope to order the contributions provisionally. We illustrate the process of ordering the contributions of prototypes by considering the case of A= 3(0-K, dla-- Ge for which : Ka lh i> =<S (nk SRY “Exy) _ RG 64 Here the S's are three-dimensional Dirac delta functions and gr) is the radial distribution funetion or two-body correlation function. However, as previously stated, the radial distribution function in ionic solutions has been studied by diagram techniques by Salpeter (17) and by Meeron (16), and the work of Meeron can be used dixectly to order the pro- totype contributions for the case of equation 71, and hence indirectly for general ~ . Explicit expressions for contributions reducing to prototypes in a way different from simply summing over chains have also been obtained. We shall review now some of the results of Meeron* which are applicable to the case of A= d(n- K.. WO, -Ki,) « This work uses diagrams in which two points are fixed while the others are integrated over. In our scheme, these are the vertex oO and the vertex V » and are ine tegrated over using the delta functions. The diagrams are otherwise equivalent. A few differences in notation are noted. We have followed the notation of Mayer, in which the chain bonds are called -bonds, and the short-range bonds & -bonds. Meeron calls the chain-bonds X bonds, and the short-range bonds £ pends. His K = 4m", but otherwise the notation is similar, except for wie co of U* where we use av* , Where the notation differs, we present the results in our notation. * See section 5 of reference (16). 65 It is possible to sum over prototypes differing only in the number of chains joining pairs of vertices, and such summing over all numbers of chains (excluding unity, since that is the ¢ bond) gives the sw bond," which represents the function —- Sf (2) 7 % Cay) = Ty eg) + As Cig, 2B; ala where x () . wet) / Nete that / differs from F since the contribution of the € bond, —A 220) must be subtracted off. If we use a line -«-e-«e— to represent a YP bond, then symbolically 2, . =2_ Py ve (n chains) Using 7 -bonds, the sum over prototypes is replaced by one of the same form, but the prototypes now have any combination of / -bonds and ¢ ~bonds. The «bonds ean occur in chains, whereas the ¢ ebonds still cannot. No contributions to the radial distribution function up to order K~ have been found, besides those of the 66 following diagrams (for compactness, we use a product representation, as for example gJ-——V X OOO = = fee 3 v_| x tk Lb le [ | + ye ey - aa + iy c78) ‘ke le g (E ke le +2r “tay y 42 This corresponds to the expression, effectively — «+ Oo oy +N factored in the same manner as the diagrammatic expression: @) - - C | s z= / (75) In equation 75, (7 is the contribution of the first three diagrams in the second bracket, for s =H 3 Je ' ig the contribution of the next diagram for s=. anid t= 3 and [ he. is the contribution for the remaining two, for s=K and t=L . These functions are given by: 7 67 = 21327972", )exy[- 2 Kudo r Med) (2s24Z r' hile) KP pr la Kite bl dK Ki, (78) The second term in the integral of equation 76 can be evaluated by the convolution theorem to obtain - KK; — 4 222. e ’ fb a4 (76a) but the first term must be done numerically. Equations 77 and 78 can be evaluated analytically; at least for certain specific forms of 10*, We now have the radial distribution function to order Ks and hence by using the same diagrams, any other two-body function to a similar approximation. The Debye-Huckel radial distribution, equation &, is obtained by ignoring the hae in equation 75, and then linearizing the expression for F with respect to the charges 2- and Zz; ° 68 D. DISCUSSION Although we have treated explicitly only the case of functions depending on the coordinates of two particles, the extension to other cases, such as functions depending on momenta or on other coordinates, symmetrically or un- symmetrically, is straight forward, and will not be presented. In the case of a two-body function depending on coordinates and momenta, the result of equation 70 is a function of the two momenta. Since the momenta are separable at equilibrium, multiplication of this result by the two one-particle momentum distribution functions followed by integration gives the final result. This study is subject to the limitations of cluster treatments in general. First, we assumed that the potentials involved are sums of two-body potentials. This is not strictly true, although it would be expected to hold to a much better approximation for an Lonized gas than for an ionic solution. Friedman (23, 24) has attempted to avoid this limitation, and has formally in- cluded 3-body, 4-body, and higher potentials. However, his work is not of current interest, since no simple model is available for these potentials. Our work could easily be extended in the Friedman sense, if a model for the higher potentials becomes available. Another limitation is the use of a dielectric constant 69 to represent the solvent. Since the solvent is not a continuous fluid, but is very inhomogeneous on a molecular scale, this approximation is probably consider- ably in error for close ion-ion approaches. However, for sufficiently dilute solutions, such approaches are rare, and do not justify a more complicated model. Probably a wore serioug shortcoming is the fact that it is impossible, as yet, to make certain the ordering of the contributions of the prototypes without investi- gating each one of the infinite number explicitly (16b). This occurs because series such as equation 70 are not well-ordered Taylor expansions to begin with, since the coefficients, which themselves depend om the expansion variables, are evaluated to all orders. If may ve possible to remedy this situation, perhaps through a very sophisticated use of topology, and future work in this area might be fruitful. A related shortcoming, is that large classes of diagrams having cross-links between chains are ignored, er at least relegated to belong to more complicates protoe types than can be examined. For example, consider the two diagrams C1) (79) 70 and (2) Ke : ; S (80) Diagram 1 belongs to the prototype Ss | con which is easily included, and in the scheme of Meeron is ‘ even included in the simplest V -bend prototype —--Y (82) whereas diagram < , which has only two more f ~bonds, belongs to the prototype nee (83) which is not likely to be investigated at all. t might be that this is as it should be, but there is no proof that such is the case, nor is there an apparent way at present to analyze the problem. However, in spite of such limitations, the rise of cluster techniques in the past decade involving selective rvesummations constitutes a considerable advance, since previously no way was available to obtain finite approxi-~ mations. The value of this study should be discussed. The development based on ALSC and LTSC diagrams leading to equation 62 is an original contribution, the exact worth 71 o£ which is yet unclear. The development leading to equation 70 is essentlally equivalent to a previous development of the radial distribution function (16), but ts useful in showing the exact relation between this work and that of Mazo and Zemach (3), which formed the starting point of this study. It may be in the future thet applications will arise for which the irreducible diagram development would be inapplicable or awkward, necessitating the use of the development leading to equation 62. An example of such a possibility might be a method of using reducible classical cluster diagrams to order contributions for a quantum mechanical treatment. In addition, the extension of Mazo and Zemach's work to the multicomponent case is of interest in itself, as may be the various summations cf convolutions leading to equations 53, 54, 55, amd 58 through 60. 72 APPENDIX 1 PROOF OF MAYER'S '‘CHAIN-SHRINKING' THEOREM (5) Consider an integral, & » with an integrand cone- sisting of a sum of products “o£ te functions corresponding to diagrams on the set of vertices £k » including a factor (£/) ‘io the definition, and consider a series, of which this integral, multiplied by Cc = is a term. We need the following lemma, which is actually 7 fairly simple convolution (16b), but which was originally proved by Montroll and Mayer (26) in a complicated fashion using bilinear expansions in a wnit cube: If ie ( =f [ge ga) J (ayn) Ha) Fd hm, ay where gr is a radially symmetric function of r~ , and if G(t) is the Fourier transform — ) , 2D — / Fa = ae 2 © (6) =— mar / A satr redr (a2 pe J Zr D , then the transform, Odd), et gn(t) , is given by Qall) = [G (t) } ; ~ 73 and if Ye, i) = &—.(-K* TY, f ane (—« Fae (4A) then the transforn, 2H » of f () ts given by / In our case, —S ff go = © JET 5 8) Ge _. (ox 24 £2). a5) and —KV “is LEY (A8) = - (iY) in the limit of ~«->o0 , which limit we now take. Consider now the contribution of a particular diagram, YA » representing one term in the integrand of Beg ‘ after exparsion of the f£ functions. This diagram can now have any number of solid lines (including zero) and either one or no broken line, joining any two vertices. If there are no vertices in the diagram which are commected by being in chains off -bonds, the diagram is called a prototype. Each diagram, “4 » belongs" to a prototype, 5 » Obtained by replacing all 7 -bonds 74 chains by lines. In general, there will be a number, DVD , of eae chains in 7, and we shall call the vertices in the chains the set 7. We call further the set in chain number & the set -?7_.. The vertices not in chains will be called the set >. The ho of the single term corresponding to the diagram url Thos) yl tH, (9) where Ly) is the integral corresponding to the prototype 5, except that Zn. replaces the «™ bond. The number of chain ends attached to vertices of species S is ™ 6 Note that v So. Y= 2» 3S =f ° ff 74 a = Expression 9 must be wultiplied by Cc, by (£! ) : and by the total number of terms leading to identical con- tributions when we sum over “4 . This last combinatorial coefficient is equal to the cominatorlal coefficient of the prototype, Ams)» multiplied by: 1) The number of ways of selecting the sets 7! 11, --- No from {= m+n , which is; Cm+u)! Ua mu /oru.! a aia @ 75 and 2) The number of ways of ordering the particles in the chains, which is: >: / T} not A=) where = — Collecting the above factors gives (a) [tte a le “Ante Th Tf CAB Cs) c) La (10) WV t Nast rs) This now needs to be summed over all possible chain composi- tions and lengths to give the summed contributions of all diagrams belonging to the prototype Ss , First we gum over all possible compositions of the chains for a certain set of chain lengths, 77, 7, VW y 04) p~ That is, we sum over all 7L,. such that a Vas = Ms ee) 4 This affects only the part in curly brackets, which be- cones ? n ra om fo je «S i ( 4 Ne! {{ As . cs) . | Nas =~ To.. / (ail) fT \ seat - ( Den ) be ; 76 However, by the wultinomfial theorem, this is exactly Vv o~ Nn Th (“AZ BS) = TEx?) (A12) X= S| - <= | - 2 where X is the Debye inverse shielding length. Expression Ald now becomes Oo am /_» la : Cy) [irs |e (C8) Ar) Toy Since the integrand of lay (5) has yn, in it for the bonds «= !2) --- -,> >, summing over all Yl. from © to 2% ro gives from expression Al3 and Mayer's lemma: au 4 ot J f OY [Te |< Aso Tag wo / Here ay m(g) is the integral corresponding to the proto type S$ with eC r) replacing all chains in the inte- grand. But expression Al4 has exactly the form of the con- tribution of the prototype to the original series a Be ot except for the fact that bonds have been replaced by ebonds, so we have reduced the contributions of all diagrams to a sum of the original form, but involving only we prototype diagrams having F -bonds instead of -bonds. In such prototypes we choose to use “wiggly” lines for the 77 &- «bonds, retaining broken lines for the R -borig, following Salpeter (17). APPENDIX 2 PROOF OF EQUATION 60 We want to show that equation 60 is the correct expression for the coefficients in equation 58. We assume that equation 60 holds for all values of 4 <~7L, and substitute into the right hand side of equation 58; obtaining: 4 t~S Sai AG), ere s* S fe = SY, : t+ 2 (A15) ei wu, E10s-t)h og, EL Cs-t-0! Ss 6S S-t -s+/ _ al 2 we = BRS =s5= Se. +> EC? G-8) ar) =! t!i€s-tl! an ei fset Di so n+! Ne al + / fo} — 2 E” =f: a (-!) | (Al17) =f i J (s -t)! S li f / = CiaetiD- ~t-S -_- > z : C1) - (60) shows ¢ i gsi . 78 Thus equation 60 ig established for the case S</. The case S=/ remains, since we are unable to use the recursion relation, because © , is not defined. Howe ever, inspection of equation 57 shows that Cy must always equal unity, so we need only show that equation 60 gives wity for S = P ‘ r+l rm We assume Crs! and show Cy, = Cy, . From equation 60, we have r+! Hh ter] ro .Lr Cosa = 2, E € a! “. t CV) (A18) t=/ t Sepia t=, ZJcr-t)f yal vele gee! rE oy t-rl = >, ~° (= _ + Dt (-)) Or-t+)) (419) ta ELOrt+))! tay th dr-tti)! y+! , tr! —_— > -}) (r+! ) — ge wae ee ee 2 s+) ey Cre ) So sl Oresddtti“‘“ ;‘ S Gt Ey or. (a2) 79 which we show is zero. Consider the quantity: We can cbtain expression A22 from A23 by the follow ing recipe: f y—| 7 ° <3 BS. yr-/ < 3 \ — - / ; (a a } (x £) [x6 | > tO) Ge) x") ~ CA L X =| Sr-o sl fryr=g i x= (A24) is rS, \¥-] won r/ (-1) (stl) 5=0 =F Cv-s)! 7 which is the same as expression A22. 7 But if we operate directly on (x-1)" 5 Y-) or fewer times with X AY, » we note immediately that every teca in the result must contain (x-1) at least to the first power, causing the result to vanish when X=/ is substituted. Therefore / - = O Cy, Cy, = , (A25) and equation 60 is established. 80 APPENDIX 3 EXTENSION OF SALPETER'S METHOD (17) TO THE MULTICOMPONENT CASE For this appendix, we shall consider only the deriva- tion of the activity coefficient expression in terms of a fugacity series. Equation 26 is He “eeyn = JF UWbowv (29) The integral %4% o » is the configuration integral, but Salpeter uses a different definition of the configuration integral, which we adopt for this appendix and extend to the multicomponent case: (A26) _ Wi —=— Se Woo N 1W \V 5 Equation 26 becomes, for nx a n £. 2X = Z yon (427) LIN 81 We now want an expression for / no / ye gt (A28 Ti ee ed Consider hae . Assume the separability of the potential energy, U ,» in equation A26 into pair forces, and write a as a sum of eluster terms, as is usually done. A general term in this expansion is repre- sented by a diagram with the entire set 4 as vertices, and an arbitrary set of lines connecting some or all of them in one or wore groups. Consider a term with a diagram consisting of one ALSC part containing particle /; and the set £ » plus other disconnected parts. Let . / 2B 3_ (L) C be the integral over the set oats £ the product of f functions belonging to the ALSC part. Summing all terms in Sy with the particular ALSC part and all other disconnected parts must give 7 ‘TZ ~Chti]y ONL - (A390) This we multiply by the number of ways to select the set £ from the set M- If. : CM-le}{ oe —_~ 82 The cluster integral, by, » is defined as usual, (brl:) es 13, Pee (hE (Vv) < 2, Whe av Tu Og where 2s means a sum over all ALSC diagrams on the set kei. The total contribution to By from all diagramsa in which /- oceurs in a cluster with a set 4 is then (433) Using equation A27 and summing over all sets £ > we find . 7 —t S f . a L=o : ~ oe This is the desired result. Extensions of other expressions of Salpeter are similar. Bi faz ) a 4 Am () bn ey 83 GLOSSARY OF SYMBOLS Parametric function in generating functionals Distance of closest approach of fons of type < and / Helmholtz free energy Combinatorial coefficient Mayer's reducible cluster integral Parametric function similar to Q; (fx. ) Mayer's irreducible cluster integral Concentration of component 4 in particles/unit volume Parametric function similar to @, (%, ) Coefficient of (Ky in B Dielectric constant Magnitude of the electron charge Mayer's cluster expansion function Distribution function of 4 particles in phase space Reduced distribution function depending on 72 coordinates and >, momenta Reduced distribution funetion depending on 72 coordinates and 77 momenta ~ Funetion in equation 72 Two-body correlation function or radial distribution function Function in equation 44 Fourier transform of Go) vv - 34 Generating fimectional for () g£unetions Generating fmmetional for VU/ funetions System hamiltonian Cluster integral (see equation 42) Contribution to re from diagram /7 Zexvo order contribution of protetype £ to equation 62 A pe order contribution of prototype to equation 62 Integral in equation A9 Integral in equation Al4 Cluster integral in equation 69 Contribution to ey from diagram 5 Prototype cluster integral in equation 790 Boltzmann's constant Funetion in equation 44 ‘i fC The set ee (4 La,--) £,- ) plus particles fe and y The set in She protatres part of a diagram in “chaineshrinking’ proof The set in the chains of a diagram Total number of particles in set 7% Set of particles in chain oC Set of all charged particles Momentum of particle « (Lia Polynomial in 7 Cr) General chain-replacing function Function in Appendix 1 (see equation AL) 85 Partition function The scalar separation of particles /- Temperature General cluster integral Cluster functions Volume of system Potential of mean force Fourier transform of £n Cr) J Coordinates of particle x. Sum in Mayer expression for fugacity Special cluster integral (see equation 41) General cluster functions General separable function integrals Short-range repulsive potential General separable function Charge number of ion of type Fugacity of component Configuration integral in Appendix 3 (AT) Activity Function Function Function coefficient of in equation 75 in equation 75 in equation 75 Kronecker delta f component C (see equation 76) (see equation 77) (see equation 78) and i; / Distribution function in MazoeZemach formalism d(r) f5) Ye) 86 Dirac delta function Debye inverse length (see equation 8) Wee® ff LT" Number of chains in a dlagram Potential function Funetion represented by Meeron's “ Y -bond"’ 10. ll. 12. 4. 15. 87 REFERENCES Hill, Terrell L. tatistical Mechanics, p. 59-80 McGraw-Hill Book.” Ga Nos York, CEI ° , Mazo, R. M., and Zemach, A. C., Phys. Rev., 109, 1564-1572 (1958). Irv J. He, and Zwanzig, Robert ¥., J. Chem. Phys. 19, ‘13 921180" (1951). ° ° ° Mayer, Joseph E., J. Ghem. Phys., 18, 1426-1436 (1950). Kirkwood, John G., and Poirier, Jacques C., J. Phys. Chem., 58, 591-598 (1954). » J. Phys Kramers, H. A-, Proc. Roy. Acad. (Amsterdam) wx, 145-158 (1927). oe ° Onsager, Lars, Chem. Revs., 13, 73-89 (1933). Harned, H. S., and Owens, B. B., The Physical ee Se Electrol tic Solutions, 3rd ed. Reinhold Pub. es aw ¥ 3 ost. McMillan, iim. G., Jv., and Mayer, Joseph E., J. Chem. Phys., 13, 276-305 (1945). an Mayer, Joseph Edward, and Ma er, Maria Goppert, Seabie cies. Meshes p. 277-294, John Wiley and ms, New XYOrk, a , a) Poirier, Jacques C., J. Chem. Phys., 21, 965-972 (1953). b) Poirier, Jacques C., J. Chem. Phys., 21, 972-985 (1953). Haga, Eijiro, J. Phys. Soc. Japan, 8, 714-723 (1953). 16. 17. 18. 19. 20. 21, 22: 23. 24. 256 26. 88 a) Meeron, Emmanuel, J. Chem. Phys., 27, 1238-1246 (1957). b) Meeron, Emmanuel, J. Chem. Phys., 28, 630-643 (1958). Salpeter, Edwin E., Ann. Phys., 5, 183-223 (1958). Van Leeuwen, J. M. J., Groeneveld, J., and Deboer, J., Physica, 25, 792-808 (1959). a) Rushbrooke, G. S., Physica, 26, 259-265 (1960). b) Rushbrooke, G. S. aa Hutchinson, P., Physica, 27, 647-656 (1961). a) Meeron, Emmanuel, Phys. Fluids, 1, 137-149 (1958). b) Meeron, Emmanuel, and Rodemich, Eugene R., Phys. Fluids, 1, 246-250 (1958). c) Meeron, Emmanuel, J. aeet Phys., 1, 192-201 (1960). d) Meeron, Emmanuel, Physica, 26, LA 5-448 (1960). See, for example, Morita, Tohru, Progr. Theor. Phys., 23, 829-845 (1960) . Meeron, Emmanuel, Progr. Theor. Phys., 24, 588-592 (1960). Friedman, Harold L., J. Chem. Phys., 34, 73-78 (1961). Friedman, Harold L., Mol. Phys., 2, 23-38 (1959). Hirschfelder, Joseph 0., Curtiss, Charles F., and Bird, R. Byron, Molecular Theory. of Gases and Liguids, p. 141-147 John Wiley and Sons, New York, 1954. Montroll, Elliot W., and Mayer, Joseph E., J. Chem. Phys., 9 626-637 (1941). III. APPROXIMATIONS TO TIME-DEPENDENT ENSEMBLE AVERAGES Ae INTRODUCTION 1. Explanation of the Problem In equilibrium statistical thermodynamics, one is generally interested in ensemble averages of various functions which are not explicitly dependent on the time, and which have time-independent enseuble averages. The arguments of these functions are the coordinates and moments of particles in the system, measured at a single time. If we, however, desire the enseable averaze of a function having as argzunents positions and rowenta measured at two or more different tines, we find this average to depend on the times at whicn the coordinates and gomenta are to be measured, even in an equilibrius enseuble. For example, consider the ensemble average Cale) ltrs)» =/F RB) BRR aR OARAR , (a) where fe and &% are arbitrary functions of coordinates and momenta and do not depend explicitly on time. We weite Alt) ror S(RLtel, PLt.]) and x(t, +s) for x (R(tors),PLtot5) ) ‘ where the time arguments of the coordinates and nonenta 90 indicate the times at which they are to be measured, and “& and f represent the coordinates and momenta of all particles in the system. We will usually employ the shorthand notation, R= Rl) 8: Ft.) R- RO , f= FE) where t= totS . In equation 1, £OYCR, £2.) is the distribution function of the entire systen. Since the classical systems in our ensemble are governed by deterministic equations of motion, R=-RlRES) ; C= FCRRSD- (2) The ensemble average in equation 1 is a function of S (as well as being a functional of f¥ Band & )s even in an equilibrium ensemble. In the case that - and << are the same functional form, expression 1 is called an autocorrelation function, or loosely, a time-correlation function, and may be of considerable interest. If the functions are complex, the complex conjugate of %X is used in equation 1. Such functions are of interest in the Kirkwood theory of Brownian motion (1), and in the theory of nuclear spin relaxation (2). Time-correlation functions and other functions of the type of equation 1 can be defined for equilibrium and for 91 non-equilibrium systems, if we assume these to be re- presentable by a Gibbsian ensemble (3). Im this case, ©) depends explicitly on 7% . In addition to studying ensemble averages of the form of equation 1, non-equilibrium statistical mechanics concerns itself directly with the related study of the time evolution of £) and its reduced distribution functions, £@) » which are also explicitly time- dependent. In this study, we shall consider first the evolution of ion distribution functions in a non-equilibrium electro- lytic solution, and shall then consider time-dependent ensemble averages in ionic solutions and gases of charged particles. 2. ‘Survey of the Literature There is a considerable body of literature concern- ing the evolution of distribution functions in non- equilibrium ensembles, and this survey will not attempt to be exhaustive, or even completely representative. In addition to this survey, some references will be introduced in the development to follow. In 1946, Kirkwood (1) began a long series of papers on irreversible statistical mechanics by deriving a differential equation governing the evolution of a one- particle distribution function under assumptions appearing reasonable for a heavy particle immersed in a fluid of 92 lighter particles, but perhaps also giving a reasonable approximation for other cases. This equation is of the form of the Fokker-Planck equation in phase space (4). Kirkwood's derivation was later refined by Ross (5), and Helfand (6) studied the friction constant in the equation under a linear trajectory approximation. Brittin (7) has extended the original Kirkwood formalism to apply specifically to a fully ionized gas. Rice and his collaborators (8) have begun an in- vestigation of moderately dense fluids, beginning where the original Kirkwood articles ended (9), with rigid spheres, After considering rigid spheres, they worked with rigid spheres having weak attractive potentials, and essentially combined the hard-sphere and Brownian motion treatments. There are many other treatments of the time evolution of distribution functions, including the elegant methods of Prigogine (10a) which are not understood by the present author. Weinstock (10b) has published an expansion of the Green's function solution of Liouville’s equation for a system of hard spheres, giving the time evolution of the distribution function, and Andrews (11) has also treated this evolution in dense fluids. Further survey of work on fluids in general will not be attempted, but some relevant references from work on plasmas and electron gases will be cited. No © ~ 1 a references were fourd to work on the evolution or dise tribution fPunctions in ionic solutions, but there are aany articles which deal with plasias. Thompson’ ec review of plasma dynasiecs covers up to about tne end of 195° (12), and way be consulted for references. Since then, many articles have Leen writter. One of the most widely-qucted is that of lfostoker and RosenbdLluth (13). They treat a plaswe im which one particle ls singled out as 2 test particle, and by expanding the hierarchy of equations obtained by integration of the Liouville equations, obtain an equation governing the time evolution of reduced distribution functions. Tere are extensions and refinenents of the work of Restoker avd Rosenbluth (14) snd there are several other treatuents, for exauple references (15) to (18). In most treatnents, orly the coulomb part of the potential is retained, and close approaches are usually vieclected. Zon and Kaluwan (19a, 195, 19¢e) have also studied the interaction of a test particle with a plasza, using an approach based on the Fokker-Planck equation approsca of Tastiorewiez, “Neuman, ard tiddell (19d), which te based on the oricinal approach of Kirkwood {i}. Vou Koos has solved the Viasov equation (208), using a diasrapn techrique wnica ae develoved for tue solution of the Liouville equation (28>), and which is of nore general snterest. Any ether articles of Interest to us will be © 94 mentioned in the development to follow. tw Tt does not appear that there are many article in the Literature especially concerned with classical time correlation functions. ‘lowever, Oppenhiein and Bloom (2) apply them to nuclear spin relaxation and derive sone approximate expressions for tine-displaced reduced distribution functions, which can be used to obtain such averazes. Vineyard (21) also presents sone forual results with such distribution functions. Taylor (22) nas obtained the tine correlation function of the alectric field in a plasma, to a zeroth approxinsation, ana has used it to express the dynasical friction on a test particle. 95 B. APPROACHES TO THE TIME EVOLUTION PROBLEM In any dynamical system, the behavior in time of any function, FP , of coordinates and momenta of particles in the system is given by ag. og + TMel, (3) ‘i where d?/2t 1s the time derivative of the arbitrary function P taken along the natural trajectory in the 6-N-dimensional phase space, Ip fat is the partial derivative of the functional form with all the generalized coordinates, fe » and all the generalized momenta, fe " fixed, and [he] a 2. +r op.) (4) is the Poisson bracket of (the Hamiltonian of the complete system) and Va Liouville's theorem (23) states that if P is the phase space density function, rally then expression 3 vanishes, so that w) @yyousets 9) as =~ [HFN] = cLF (5) 96 The second equality defines the hermitian Liouville operator, 24. . Equation 5 is the starting point of most time-dependent problems in statistical mechanics, but it is also possible to try to work with the equations of motion of the individual particles. This is what Pines and Bohm (24) did in their collective coordinate treatment of a classical plasma, represented by a one- component charged gas neutralized by a uniform background. Their success suggests a trial of collective coordinates for ionic solutions, in an attempt to avoid treating the solvent as a molecular fluid. If we begin by taking an electrically neutral, multi-component gas of particles interacting only through Coulomb force, we can go through the Pines-Bohm random- phase approximation procedure, and obtain the result that collective oscillations occur. Presumably, all of their work with dispersion relations, etc., could be so extended. Next, we can try representing the solvent as a structureless fluid with a dielectric constant, through which the ions move, experiencing a viscous drag. In this system, we might hope to find suitable collective coordinates that will show damped oscillations. Attempts were made to find such coordinates, but the presence of the friction terms made the problem intractable. Even if this had been successful, we would have 97 veer left with an unphysical usymptotic picture of motionless iors. 4 stochastic term could have been gdded, but although tnere are mwechanisas to excite collective oecillations in plisuas, to szssume such in an tonic solution would be unworranted, since the bombard- ment of individual ions in © solution by solute molecules 4s certainly not collective, o even collectively stochistic. Tf we abindon attespts to work with collective cooriinates, taere is still anotner possibility to try before we rive up tryinz to lenore the wolecular nature of the solvent. It its known (25) that non-conservative systecs can be treated with Lazringiun sethois if the dissipetive forces sare proportional to tne velocity of the nurticles on which they act. If we sssume this for our ions in their solvent, we c:n derive a sodified Liouville equation for the distribution function of the ions. If we use the dissivation function of ‘ayleicn, A/ = > ) { > af Ne ; me . Fae eur Me) (6) c= where A: 4s the friction constint of particle <¢ . snen thit the friction:il force is ———> ss ote: ina the V‘S re velocity consponents, tnen Lugranze's o @ equitions becowe (275): « ~y = uf —— (s > ee i oe — Vainy equation 7 inste vd of the ordinary Larrunire's ecuctions in Jerivines Liouville's theore. vives, for cartesian ae med Jee -_ m ~ where i= 5. R- i+ cain be shown thot density functions setisfying eguz tion »-etain their norazsliz.tion for all tize, but if we write © :s IO then Z. 43g not aeruitian. | 0 “hag is not a serious Jr. uxless we were to try elecint technidcues such ony <A Andrews', wnicn require real eirenval ues = tne fact tnat we could not include oe o9 stocnastic teras in the Lagranzcian equutions, since these are neither derivible fron « potential nor propor- tional to 3 velocity, reaulting ag.in in the unphysicsl notionless state 26 ~>eo , We conclude trat there ig nothing to be gained oy i1omorine the aolecular nature of the solvent, ond turn aue attention to the entire sysetes of particles. The dynarieal behavior of the system is governed by Liouville's equition, eauation 5, which describes the bine evolution of the phase density, or distribution function, for all N particles. Tnis equation is valld whetner the ensexble represents ur equilioriun or 4 non- equilibrium systes. Rerinmning with eguxtion 5 ond integrating over the coorlinates ard sonenta of «11 of the particles except one, it is possible to obtain an equation of the Pokker- "launek type in tne phase space of one particle (1, 5)\s The result of iess is (5): MU) > _ si) Sx >) Fil) > xt) > 5) wnere the friction constant, s » is defined: S ar [[Feorite 698) ) “Eel re) oD » dPde ds ds 200 2 and where the other sysbols -.re the sae .s those of reference (1): OK /y=7vo , Kk a Pee KES + Ohh & + FF (12) ase ere K<FD is tne eguilibriua ensenble sen force on particle l with tne position .nd nowentuxy of this pirticle fixed. “ne cuantity 4s a correction tera presuned Beall and not further eviluated. 4l1so (2 “ = aS Vote Grjnla dp > a ©, where LN “ere We is the equilibriun potertisil of mean force between ourcticles 1 und L, and ty & is 2 tine larrve enourn to .llow the interral to :ttiin 4 "pluteau value" waich “irkwood «ssumes to exist. tote that the equation 4nvolves the tiue-snootnei distribution function ~ Cl) of 3 WO f Crp,t) = =| f Cy, Aj t+) ds , (15) [e) This tise-smootsines appears necessucy to introduce Lrrevergibility inte the dynzrics of the systex, sovernes by the tise-syusetrical Liouville equation. o) in contrust to seue of the results of equilibriua 189) s$itisticsl nechanics, there is nothing in che Jjerivation of eaguxtion 16 thit requires che scotenviuls involved te 4Zave «a finite range, ond -elfeni nas usel eciuition 11 to e.leulate © friction constant for a plusaes (6). ‘owever, the devivecions of both toss and Kirkwood «are sufficiently clovijei by uneroxiuations that 1% is iif- ficult to tell wnetner ecuatior 11, waich correeponis to the zero-order tery of om «.utocorrel2tion funetion {the effects of intersoleculsr forces are nezlected), 4g. proper cpproxis:tion to the friction constant. Kirekwood’s orizingal expression for 3,18 is o / a a AEG — ° of 2. 5, ret) < FGF GH) ds ai <t D) whien pees tne entire 2utocorrelsatior function of waich enusttion 11 uses only the ceroeorder tera. in tite crse ef Suonuleive poberntielse (suel: as yard spheres), for axiuple, tue Linear trajectory «pproxziaition asy tell very Little sbeut actuel particle trajectories, ond it would seem sove sensible to use equation 16 vratser than oe ustion 11. “iowever, in the cace of isnpulsive poter- tials, the entire derivation is net valid, and the FokverePlancs eguntion approaca decoges questionable. 3Ace and Allmatt (286, Ge) treat systens naving jawd-core vrapulsive potentials and sort attraetive poter= tials. Tnielr netsod essentially separates the effects oF ’ A. - 4 me oe pe + “hn n- 2 a” a the two parts of sche potential. “he fard-core part Gi 102 the potential mrives rise to a “collision” integral of the space type as that of fice et al. (9), while the soft part of the potential «ives rise to Fokker-Planck terms siniler to those in equation 10. The result is DE? -7 +g. + dat” aa) where 20) _ Ct) aka ce pFV% = e+ F OF Up FG, and QEV HV, esl -% [fF FU aT ines (19) Here ova is the macroscopic local velocity of the fluid. The friction coefficient, £ » is the same se in equation 11, except that only the soft part of the potential is fneluded. The "collision" integrals, J, and J, are given oS =§. co (VLF On, RAP OE CpArit) F Cn, ptt Cn ast bdbdedy, (20) Te = Pel) ((TF Cae AG: élaak- Gf ~%r, pant) a fC, pytok- aa Ciypit)| 2 t bolbdedp, , #4) wnere g- is the hard-core Gdiageter, is the reduced wass, b is the impsct paraceter, C is an azimuthal angle, 4 48 a unit vector along the line of centers at impact, and Fir 4g the relative somentun. 105 Rice and Allnatt have also solved equation 17 by a Chapuan-Enskeg procedure, amd obtained a result re- C2) quiring only g a (7) for 1ts evaluatiou.* A siuilar solution for equation 11 was derived by Lebowitz, Prise, and elfand (25). As in tne case of Kirkwood's Fokker-Planck equation, tnere is netoine in the derivation ef equation 17 require ine the forces te be of short range, 50 that tne prodlea of the time evolution of one-body distribution functions 4n an tonic solution is as well in hand as that of ordinary fluids if the friction constant S can be eval- usted. Of course, we have the additional couplicaticn 4n coeputiryur < that more then one kind of potential eceurs, 28 the solvent-solvent ana eclvent ion inter- setions are of short ranmze while the lon-ion interactions ave of lon: ranze. It would, at first glance, appear that we were faced in this Fokker-Planck equation approach with a task akin to raising ourselves by our vootstrans o We obtain information about the distribution function throug) an autocorrelation function, but for this we recuire the distribution function. fowever, in a srearized theory such ss the FokkerePlanck gpproack of Kirkwood, tne zero-order or eguillorius distribution funetioyr would seen to suffice for the evaluation of the ® See equation 22" of reference 3b. p= O4 force autocorrelation, and thus in a second-order theory the fireteorder distribution should suffice for whate ever averazes oust be taken, and so forth. ve now turn our attention to means of obtaining approxiuations to autocorrelation functions and similar time-dependent ensemble mean values. 105 C. TIME-DEPENDENT ENSEMSLE AVERA‘ES 1. Theory we shall consider the calculation of ensesble averages auch as that of euuation 1. We use again the notation K, © for the coordinates and monenta of all NV particles; 4 / for the coordinates and aowenta of yw particles; and RP for tne coordinates and nomenta of W-n particles. We also use 2 =2o+5, but will later sonetizes set a =. © ° Since the classical systeus in our ensemble are rovermea by deterministic equations of motion, RK = OC) = Ue CR, te, s) (22) f= CD = (CR FS) (23) | In addition, of course, via and a are functionals of the fumetional forn oF the equations of motion. Faustion 3 becomes, using the definition of Ly (equation 5 ): te ses Li we de (20) since ax /dt =O by assumption. A formal selution of ecuation 24 is: 106 x(K #) = Lapp ish) x (Ra, be) (25) We can now write equation 1:3 i ¢ <A.) (t.15)> -[Fte, fate )/H he Re LOR, RIACRAK (26) The ensemble averace is row explicitly a funotior of S, and we note that the interrand of the right hand side of equation 26 ie explicitly a function of “zeroed" vari- ables only. Tne theory of the exponential Liouville operator, Lxuy-esh ) » has been the subject of considerable research.® However, the work of Prigogine and others in his school is very abstract, involving sophisticated mathematical and disgram techniaues, and seems to be of a nature not well suited to use in equation 26. All previous work with the exponential Liouville operator seers to be directed towards the time evelution of dis= tribution functions directly, through the Liouville equation, equation 5, waich has the solutions FR, tt) = pO CRE). (an ® See, for exauple, references 1, 10, arid 11. 107 If we were interested only in the time evolution of the distribution function, it would probably be wise to work directly with equation 22, but there are tinee correlation functions, such as the one-particle souwentun time-correlation function, that are of interest in their own right, so we berin with equation 25. It is interesting to note that knowledge of the tine- dependent distribution function at time t is not enough to calculate a time-correlation function if we do not use the exponential Liouville operator. Distribution functions involving two times are necessary, such as those of Vineyard (21). An alternative foraalisa can be developed using such functions (see proposition III). We now return to an approximation schene using the exponential Licuville operator. Sy the use of Hamilton's att - equations and F >= [& » we obtain J — es; : a J &€ 29 J =! ™m; J D a. where F am UR) 4s the force on particle wr A and PH; 48 its mass. At this point, we restrict oure selves to cases in which the kinetic energy is auch ereater than the potential energy, so that the second tern under the sum in equation 29 has auch leas effect then the first in deterninine trajectories of particles over short periods of time. Unfortunately, this case 103 4s not Likely te apply to ionic solutions, so we defer treatnent of these until later. We write A= hk +h (30) where YF /° = L =) eV, = oS (31) and ‘ NM — 7 e = - [= = = HS , & . (32) Thus, 8s _es(U4+L) Qtij=ee” =e (33) This exponential expression can be written using Feynaan's operator calculus (25), in which tige becomes an operator ordering a C2 ét,) = epi) me ole’ opel (34) ’ The argunents of the scperators go not show any explicit time dependence, but are sicply labels interpreted such that . Ee Pada) means ; ; re a a and . Uf fe == 7 109 We expand the second exponential factor, rerenbering that the presence of the order paraneters takes care of. all comautation difficulties. (if Lr ana Li were comauting operators, the probles would be auch sinpler.) Equation 34 becomes ¢ ¢ n fey, , =f) VAD a 2 (Et) = apf} to [Loe | (35) i ts The first, or zero-order, tern on the rightehand side of equation 35 is the only one that would occur if there were no forces between the particles, and use of it alone results in the linear trajectory epproximation (6), in which + _ rb) = &)+ SF (36) : = . We now proceed to disentangle the operators ir equation 35, writing each tern so that the operators act fron right to left in the usual manner. The first-order term becones t 2% to) = - UN ti) VG, t)Aé ) (37) 1. where we write é. 110 © Tne second-order tera is bt é ee (Dy. \_ enn ery [@UG dedé. @ Yet), psf ctear fer (C6) a) (39) te lo When this is disentanyvled, we wust have two terus, corres= ponding to the two possibilities, f,< Ca and t, 4 ar ° We introduce a step function, S 3 S(x) =} ( 3 BE (40) > 6 (@, > AS Equation 39 nov _— 1e8 a ee) = Lup Fo eae Le MENS Etuede. | to +e apf veuel). ia VA e-tele dee? (ay This expression can be disentangled to yield the followings: O%t)= 1 f ‘ (SEA) Men Plea letdeat. ae é¢ _ ) i a) {a (E46) Mele ner. et) SE -2 Alte, ?) fo bo Since there is no reason to require a particular order of interration, tim two tergs in equation 42 differ only in a pernutation of integration variables, and so are identical. lll We write | t ty @y fonts i 4) c 4 to to where we change the limits of integration instead of using the step function. Ag we continue with the higher-order terus in equation 35, the same pattern occurs. In the nth order teria, n! terns occur, differing only in verautations of interration variables, giving C £ Gusi 8 ; ») c c ° Q “i t,) A) \ fe fee eee) LEM Eb) (ue) to b& &e Weinstock (10b) has presented, without derivation, an expression fer the Green's function for the Liouville equation whicn is equivalent to equation 4&. towever, equation 44. was derived independently, and it appears worthwhile to include this sizple derivation. The application of Weinstock is to systeus of hard spheres, end is thus wore or less unrelated to the present applicate ions. Equation 44 can be given an interpretation sinilar to that of Feynuan’s work in quantun electrodynanics and to diagran éwohniigves in irreversible statistical mechanics % CHU«EZE) describes the propagation of the systen fron tine z to tise fe as though no 112 forces acted. In equation 44, the right-hand side is interpreted as free propazation frou Le te ty . action of the potential at fn, free propagation froa tn to Tres » etc, The integrations vary the times of each free progation fron O to t-te , keep~ ing their sun equal to 7 Equation 25 now becomes yo? ARP) =2 (Lb) (KR) = = Duck C2 Et) MEK). (5) Equation 26 becones Oe 3, a 1,64 Hes) ZEs jal, feed (46) < pu dalOs” In order to use — a few terms of equation 46, we have restricted cur discussion to systeus in which the potential energy is much smaller than the kinetic energy (with the zero of potential enersy taken with all particles infinitely separated). However, if we can devise a way to include large numbers, or perhaps infinite subsets, of teras in ecuation 46, 1t can be applicd to any systen, as no approxizatione have been uced in ite derivation. Of course, the implicit assumption of convergence has been unde. Purther investigation would be necessary to justify this, but we can obtain convergence by avoiding racions of phase space in which L has singularities or 115 other troubles arise. I:spulsive potentials wignt need special attention. RB, Caled lons "se wor i e peced! sects S11 now 96 (a 1@ eslcul: Sion of sO royimations to ensenble ‘OB v Fo Sfe 2 F 5 ‘ Coz es oLe 2 e consider, ead of + Fol erePis friction coerricient of @ ty LL or of envatic eco-el coefficiers (4); Te _ _ _ £ = 5 = Xe <6@ Fede “oO elfend (45) crleul¢ete AY us i j7e Linear ec TPS Lam, e717 corres sc? » pee of only rvero=-order ber _— /— JS Ly co _ ral £< FEA At j eS rd A ~ Bet gt iy ; T > — (| Se R)R@O> alt =[at ede te enh afhcen enn TIhobE). 6 114 In equation 49, (©) —s ands £2. ,0F = F RrtB) , (50) where V4 neons © eo nore “FA, eons Ey fe ¢ By hy, Safin 4. - ( aay MA, -—l aw A + “ax fx + Suyfin + Eve fx (the velocity vector U in 3Nedimensional space). Inserting equation 50 into equation 49 Fives: To. ly ~ tn) { <E@r@rddt- (At [dt Rae €°(@ fro) (51) (oy, .\a = PEO LER #E ) Since the Gs occur only with the Ks 9 we can write instesd o% Ve e@ operator t, Ve Ly by whic “at Ne, By. fare) (52) j 5 = a Equation 51 becores CaP ete (EL, Soll) [ekersdes (4€ fat, (let hs0) x (53) KER )E Ch t-E) We sean. 6 (RrtVe, 115 — We write F= YU » and now consider the case of CN Yelfand, in which F CRB, 0) can be replaced by the equilibriug distribution furction, oe 2xp (— 4 K) OO - [%,,0@) Jo Mn. au, [e PO"), pives Now so an integration vy parts ( Bokwsdt- ~ 5 hee, HEAR e-PKe -PY x Ves Y ay lisp, Stott) Ye eee Yu, fs ht) € ~ = le dean BK, AU, r YVvU-vvU + VOUVTU f ess ve assune UR) = bpansforn of 7 Fae differentiation is equivalent to multiplication by — _(R or oR » so equstion 55 becomes: ra oh; (%;) and use the Fourier J LLé T |. Wy w ££ K U [atcFeakny = 5), (ltfae (dae er?” x o ( V lo & J robe dts exp ch. ([t-tJve- (t-tJy,. - ch: (ei ty, pr can)? lane z 3 ON +(e ac gore (Lette Len, chy (Ole tin)?» tj 4 os ¢ Vv v dh dhe >, ?, ey (e-40%,-Leem) cke(ty-t v3) x Qn) Qn} 42> 72 Lt; — = a 7 ; cap {chet hd 1, BARR) | (67) L147 We know that for equilibrium: Loup thkv)\ = oxp(-#*¢ Japon.) (58) w~ where Gy 3s the Pourier transfor: of di | and di 4g the radial distribution function. In order to proceed, we must make some approximation for the terms involving two field particles (particles other then nuaber 1). If we nerlect correlaticn between the field rarticles, then % I, k) 5 ae : 4 (is) 4 (%3) (59) 4f 2 ar 1d Ler Ri lic, he WS, = 1G, &) GC, bd ~ Bx SMG, b) rar G i A)§ (é.) 3 are field particles. Under this approximation (60) _ (an) SB) MA ; eke Ge keV = v2 [GAG (4k) + Can) Cpl heh, 40 SJGy har &,) _ (41) t an SALAD - 118 Mote that we carn leave out isunediately all of the terns ss proportional to SC, Jor d Ch, ) since every integrand nas the factor Ab. é ‘ ee 57-61 yield: c= [ot Fiokio>’- =— vi feel ‘dé, tb / (2n}3 (an)? [2 Sit ik atadoyf [eed 2d]? Ab hehe bd) X AG iss G,&) Z / ay £2 7 A)G G,, Ch) tie ith, boy Fin," f +) «End sat pekeh pond perf C8 » b hha /. bh+ bitsy ) YY AA - Moos ote ee 5p ONE) Lt; = ni V , exe ot Sim —! (¢lhrh)-th] y) J 3) is tr duced TasB, 119 The three double time integrals in equation 62 are of the sane form: “C~ om a Azye T= [dt [ole é gt (Ant, ~2B.00) (63) wa th are functions of b - wiere v € Ax, Bu -_ . Lay 2) “n,n, > ME and 7, » but independent of tf and ba Comcleting the scuscre vives: L, =f een oet BEAL Ae i ste Bt) _ (64) a Call the t intecral ae and let At _& .— = x — €(A,- Bi/Ay ) . ao —X | Ta = ae [+ Be Ax Oo et /A, ole fe“ Ac BY, the Aa 7 2A. 4 + £8. anf [tA Bey] af] (46) T 2 Zz Lot Bt \S! (tu) T= [Atop (CEP EE VT fe At eo BAL] + LEla feng tlacbey)-of F 89]? (67) 126 The first two terus can ve integrated to zive error functions, but the last two are likes =. Co Zz by T Et" £! fa) Jo ps where tip i and rx are functions of Bx and A. r Equation %8 can ve siuwplified by a partial integration to yield Thus J. is riven by } f=] | fi Sq Poe Soya T Dy; Cte 26. ) ates - i —_ ery) fA - La 7 121 In the foregoing, Ax, b. and CC. are given bys Am = _- a 6, = z heh, ) C, ~ (bet de )* RBM , (Baw. = = Ss, Chirk d, C= br) A LAS 2ay amy Bam, aie — 2// 4! ‘ As= 4 (2, yan © som } 4h bth, 2) safe be Rey am i= & th) , An, # Rhy Aomy Bene and ae Cy- Mey 4) 122 Our lst crder correction to the friction coefficient is “ss proportional tos enon —! (Foe - a i, dhe BG, (hb) x (an) (2m) ’ Hd) ab) de Ae (hb 4A) T, £ = +722 an Gy (a) ahi lh) » Ker j= > VW ~/ 7 a Now that the tise interrals have oeen evaluated, we aave proceeded as far as we can without choosing a particular potential and substitut aR, actual fonctions inte equation 72 for AEA) oe ate (4) « in other words, equation 72 is our result for s zeveral potential. Sinee the wy are fairly cowplicated, we sake the sane approximation used by lielfand in nis zero-order tern, that of lettin: CT —->od - It all orders were li- 123 cluded, this should force is to vanish, but it does not pace ¥ °°) and ¢” venieh, so the lower orders taken by themselves may be meaningful, and at least it is of interest to _ Len (Em EwS Sdt r-3 20 with ¢ Tt . 7 ’) ait KFOFaD dt , which is Helfand's linear trajectory approximation. 0 aed In this limit, equation 79 becomes hor Zr) = AP (CtdAZ Is CAG + Bu) os) C200 WAL (C-AEAD Az abs Ce ? “ and equetion 72, of course, retains the sane fora. Substitution of the constants from equation 71 into equation 73 sives FI Th Aen ta [Her Ubhesa 4h ra (7%) Aas | (a ab bcateh cet) J” where n= Lk. , - |h.) » and yy 4s the angle petween the vectors ° and a 3 Ti (@) = 124 For 1,0) we ovtain: — =k, he,coal _ p * am Ue, Then phe L Yoo) = te AS Mays a ih Ay Mis (76) Age agar ht adadeal oc (Bi hho ‘ > where we define | ._ J, 4 —— bo) Meay Me MB ‘ L(o) 4s identical, except that the index kK Se ren placed everywhere by £ . Now, in order to proceed further, we gust substitute KF —~—— (> and 2 for some specific potential and integrate over R, and AR, - If we consider the case of a couloab potential without short-range petentiels and a syster sufficiently dilute that tne Debye-Huckel radisl distribution function car be used, ? = tg! Guz (A) = -Y4IT By E ma _ (7 “5 SS me af Ung (k) coal 47f 22g e ge (79) were X is the Debye-Hiickel inverse lengtn Ss 2 so .. fF k~ = 47 E73 as Ge Be : 60) / S =. rere e, igs the conrcertration of narticles of type S in units of particles per unit voluse, ind 2 is thelr 125 charge. This case corresponds to a dilute plasma which is sufficiently hot that all of the ions are stripped of all tneir electrons. Other cases can be considered, including that of systems far erourgh from equilibrius that the equilibrium distribution functions carmot be used. Substitution of equations 74, 75, 76, 738, and 79 into equation 72 and interration over 4, and A, yield our first-order correction to the FPokker-Pleanck coefficient, but this intesration will not be carried through in this thesis. be Momentum Autocorrelation Function Ve now apply the perturbation series of ecaustions 4iw46 to the calculation of a one-body aomentua auto- correlation function, < Peo) p.t)> Pirst, tne zero order terz is just the sane as the linear trajectory approximation, so < po ROS? = = Jp) pe)y (81) waich at equiliprius is mA] . we look first at the simplest case, in whien we use the equilibriun distribution function ard nerlect correlations vetween field particles. The firsteorder correction is: € P Reply ff Ue 0:0) foldaBreyU)ddedlde . (62 126 or for pair potentials, pop” = . .t€ Ww =f; f{ “Rk: ofa YY; (lie (tb VLRAR 6 fe jrz Using the Fourier transfora of “ , equation 33 bee Cones <popl)”’= @ = sp, eae COLSON: i le x ie: (34) x fale, AE ke “ Ret Me, Ret), At equilibrium, the monentun separates and we can average the functions of the separate velocities sepi- rately. Averaging the real and imaginary parts, we find by that —% LOnS imino ® (é-t) in > REL Vie vanishes, as it must, and taat < he Simin, e W. ‘oy can be shown with souwe effort to equal [ re —Sa <_ ras 2 WF fee ge L)R ap ~_/ (Lt) (85) ( yom ‘ Also, as before, <e en t bk: i = ep aon & Re 127 Using these, we find for the first-order correction ss 7) <p) Ft) == a [uz fee Ch) Reh x. (88) X (b- cdexp3~ e) a CG, The tine oneerenes ak now be done. We let le fats _ & lee x= fae Ct) 2p me ye (89) == ale’ t° LK — he tC’ 2 fees OS , where ¢ = f-T, » and we now let fs = Ch « We find — ge fit? = E eT 23a, J ar can perfil” = LS > Ub bb me) Xv Gna Fe be ay j (91) be [ops SEQ] Look at the simplest case sgain, with the Debye-iuckel radial @istribution funetion and the coulomb potential without shorterance repulsion: u, (hk) = AT Bz E™ Gi (k)=" ~47q 22:68 (92) Me aK™ - 128 We now have LAOS’ = = RZ, -e(8 2K, Le kk y WY jer This can be shown to equal, vor ve cratedity. large Nz tt patels S G4, /E- 2] (98) Pie where C; is the concentration of species Ss , and cL, (93) 48 the integral Xs = a — (95) / This integral can be evaluated as follows (29): We use the identity ‘ae © hy KIKI eer alc * ax from which . Ke = gel (Ik due e (97) - y where = hau. . complet ine the square, one finde re) 2 rp ~X- — K/xI (98) e 14s Ax y fT AKUA, | -2D 129 which gives after again completing a square? oY, = me ®% fy of GO| From equation 94, we now nave < Pop” = - AL Frets ay x LB (oe Bic rage (109) This has exactly the behaviour we would expect, vanisning at TPC , as the linear trajectory approxigation be- comes valid, becoming negative for all times E>o > and attaining an asyaptotic value depending on wy, B, a4 oo 2, and “, for S=(,%--40 : o) é) if aT 2, “ec Yes Z 2 o> <pepe Y me C2 By (102) Now we turn our attention to the second-order correct- a4 fe fon to <poplt)>. The entire argument proceeds as before, except that there are two tine integrations, two R anterrations, and attendant complications, Prow an “ and ii we have L fopleys (te folt fh ea 0) ‘CED x _ _ (102) x Pw “Eo )p.ARAR 130 This eimplifies to: oo a, oS a ce) og POP? = [Ht [alt fio f eh; 262 EE)F Veg X = (103) x E CR, + (t,-%) Ue ) AKA Fe We replace Veg FE (+ (6-t,)Us ) with (t,t) Ve. F (etG-tyt%,) 28 we aid im the case of the Fokker-Planck coefficient. We also assume that the forces can be derived frox pair potentials, and obtain <fo fu> = | Pe Reo dt f de (EG) X XS DUG)? a VEN, Puddle, ae ie ca S naz Lc) where 2: = 7 + £2) 8 Lan and TR a 44 (4-4) €L, = btot (EQ) EB An - Now we express the potentials as Fourier integrals as before, and obtain the following result (since we have broken up 3N-dimens ional vectors into sums of 3-disen- sional vectors, we include 3-dimens ional unit vectors Cu. for clarity): 131 -s . (2) 3 —~ pW) CPO Pe)> ~ Gane fede nf Bo) xX “(de foe 4. oh ET 2& dd Lithye* im 6 (oth, REALL) cA tm —5,2 &s fle hitlhde aes n=2Jj-2 MM, "4; SE (lh, bb, ah, )e pa Nim | We now wean by this notation that e Rag is a vector, not a tensor, but that its three con nponents lie in the subspece of particle c . We interchange the order of the kK integrations and enseuble averaging, and again restrict ourselves, for the purpose of this calculation, to the case of equilibriua distribution functions, and ia addition we again neglect correlations petween field particles. The enseable averages involved are the same as those encountered before in equations 58, 58a, 60, 61, and 85. Using these results and separating the teras for Naf and WF y » we obtains (2) vw U ? a LPO po» _ Fs 5 5 aed, ad, (4 4, ) x Nc n=z jen — = JinVG,. Chee) Te (106) -™, 3) ae S Vv a => 3 , —s ~ sb $j AG, 1d) (any A orb Te) where the ty are double time interrals, and sre functions of t k,, 2, B> M,, ty snd My them ss integrals over (= t- Cy » and “y= C-t2 -. if we write they ares L,@) fro mole (ht +hz,)e 8 ~ Fy B25 G, e oe au 2. 2 ; ~2G va ZO be [bam he 6-66 G af, GC os 133 where the D £ and Ff are as follows: » 2 _ PI Ban, = yo Dy oy (Ait hh), pi- 4A pe de 9) [7 J AS fy Qa, rae pe a ee & 23Mn os | 7 A ohn ? . Y H/F mn : . eowpletion of squares and change of variables gives Foy x= ¥s: , P ffe* Zt)= |-&, ( a +! \ + 4a, (be “+ + & Qn E E ( EF ee ox, [A ; 7 | F 2 A, Care + IQ (111) — at } e” =), a. ; HA $ tes i a & “A and for <=6: ) f 7p = Le = 2.) (fe 4 fe\ pnb / a —— E x, L Ft ES. 5 (822) Va me oT z x2 E, Fe x [ & 4 - LO) | ES 3 . Here “ 2 Ebr HG 4 sou — Gy G =X 2 f= 1G © f e olx (113) 2 Bat tT € a z (114) y _ nw ; 9% (deme *9 xe” ax ; Xz “Js “ “Sf end & Ee Of be, wae ce 2 ro ar Cc; 2 = ra Z LY = (ade & J KE ie 5 (115) /» (Egt Ha), 2. 6s =e ” where (2. = De i JES _ DQ. = H. and H.. = F& Uh pe 155 If N and V are large, it is worthwhile to rewrite the sums in equation 104 so that they run over the chemical species present. We now do 80, nezlect- ine the distinction between 142 AZ-/ 4 and 1-2 where Ai is the number of particles of species o< in the system. ive vet ce. = NM W/ » and writes < Fie fey = Ls S Oe Als, lh. f A, A, \V Fan, a "I an Gn? x a) ee Guth) we f Ch al ; : }+ y 4 7 a +t Lb ce J + a a > mp 4) ie al arg , x! /&, _ 4 \oe _4& Osx ae: Ee ele = as i (116) tf so fi / / A / x Ox ee +f om La Fal 4 Ng — L. an 6 EY J a £, 2 2 6 &—<¢ 136 where we have separated off the term with the delta function factor and performed the Az integration. — The primes indicate that in the primed functions a has been replaced by _h, . We will find that tne factor in curly brackets in this term is proportional to |Al~* in the limit 0 S20 , cancelling the A's — in the numerator and leaving an integral over #, proportional to f al 3h [ace dT G. CL ) Pe - when tne ae have been evaluated, equation 116 becomes our result for 2 seneral potential. As before, 4t 48 in the form of an intezral over Fourier transform variables, and we wust choose 4 particular potential in order to proceed further. We can see that the qualitative behavior of Lf ple>? 4s what we would expect. For Tsao, < Be) Aes” vanishes, and 28 t¢—>®© it approaches some function of (27, MH, Wn y the form of which is not yet clesr. we now consider the time integrals. he evaluation of Qe 4s eleuentary, at least if we are willing to accept tabulated functions, but Ni A and LY, require that we evaluate the integral Cathe © -Ga oe x = be =} # [fe “ole de ~ (117) Lor Efforts to evaluate this for finite a were unsuccesge ful, but it can be evaluated for Z —> a2 » We thus content ourselves with en expression ror fore < Pe po> we use the series representation for the error funct- fon under the integral of Jn, letting J, = Fr : Lt) = J, fe“ ar Tet “RIG Az i! 3 (118) CO 2n aa “ c 22 _ a © Ci) & a a Che car n=o m! (Qn ! (n+) - In the limit t+ cO , the integral is a ‘e function, and ] oS - ys 2nut ~| lt) == Ls == - (119) This sum cen be evaluated by = wethod of Wheelon (30). We observe that J@pr') is the Laplace trans- form of e- Gs » and write: St are Foe Oe 4G, ale 71 Oe oe 138 where @ =-XLnz +4 . This is a geometric SUL, 8O _ r _f4/e — L, as Sa [ obe a (122) © ofr bi —I .(@) ~ if 22 = 4- = —taw'2 | (123) e. AG, J Low! 2G. The series in equation 119 is divergent for Z > i However, the right-hand side of equation 122 1s an analytic function of > forall = , 80 must hold as the correct value of pl? for sll 4 vy analytic continuation. Now that we have Lao)» we can obtain all the oo) with sone algebra and a few partial integrations. The result is Oto) = (Et fe) pas 2 ~¥ Ewt+ B/E > 1 (Be-Felee) (R= E/E) _ [4 (Re eye ( Be Gey ) [2-6 Jom Ent Se /Ey ; 139 / lw — - } AC Rer ahr ké ) 5 aes) 2) (Ext Fle) | AUR. Qiat Be) * (fez . (5% FE. /E, 7 ~The E+ GN) (Ri Be are AQHR es (126) nF Now that the * A have been evaluated for the Limit a2 » equation 116 furnishes our desired general expression for <P pil@o)>” . For our present purposes, 4t is — necessary to carry out the pp intesrations for 3a particular potential, nor ie it necessary to proceed with further applications of the perturbation series, althouzh there may be soue of considerable interest. 14060 D. DISCUSSION It must be admitted that the research reported in part III of this thesis enjoyed only a qualified success. First, the original intent was to create an approximation scheme for time-correlation functions which would be not only valid, but also directly useful, for ionic solut- fons, which are very dense systeas. The perturbation scheme here developed is formally valid for all systems for which the potentials are sums of pair potentials. Also, the assunption of convergence 4s not a drastic one. However, in the present simple form of the scheme, large nuubers of terms must perhaps be taken for systems in which the potential energy is comparable to or larger than the kinetic energy, which 4s the case in a condensed systes. Still, there is currently considerable interest in rarefied systeus such as plasmas, ana the work of this part of the thesis is directly useful for these, and provides 4& considerable improvement over treatments which neglect entirely 4ntermolecular forces, such as the work of Taylor on electric field autocorrelations in a plasma (22). The work of 2on and Kalman (19a-c), based on the work of Gasiorowicz, Neuman, and Riddell (194) is also - 141 capable of civing corrections to the zero-order approxi- mation of time-dependent ensemble averages. However, their work, although very detailed, does not present a single, unified perturbation scheue. Rather, it is a sort of Taylor series expansion around an unperturbed path, followed by approximation of the expansion variable. This can be used for low orders, but becomes progressively clumeier for higher orders, and furnishes no general expressions. Te work of Prigogine's school (10) and of Andrews (11) 4s beyond the present understanding of this author, and furnishes, by its very couplexity, impetus for such relatively simple-minded treataents as the preceding scnene. The work of this part of the thesis may be considered as a veginning, and much sore work and refinement can surely be spent with profit on this technique. For example, if infinite potentials, such as hard cores, are introduced, the Fourier transform of the potential diverges, and sethods must be sought to overcome this (10d). Means of overcoming other divergences which can arise must also be sought. For example, the zero-order Pokker=Planck coefficient, which was calculated by Yelfand, diverges for a coulomb potential, and the ine finite part is presumably cancelled rigorously by highere order terms, and a seans of renoving these divergences 142 by partial summations analogous to those of part II of this thesis would seem to be a very worthwhile research objective, Helfand renoved the divergence by introduc- ing a finite upper limit ina Fourier transfors integration, but this cutoff, althougn commonly used in plasma treate ments, is arbitrary, 2nd rigorous treatment avoiding this would of course be 2 considerable achievement. The applications of the perturbation scheme to a Fokkere-Planck coefficient and to a momentum autocorrelLate fon function are illustrative of many other epplications which could be made. In particular, it would be of anterest to attempt treatment of a system having hard- core plus coulomb potentials. The formal simplicity of our perturbation expansion would make it seen very worthwhile to seek extensions and further applications. 143 REFERENCES 1. Kirkwood, John G., J. Chem. Phys. 14, 180-201 (1944). Oppenhein, Irwin, and Bloom, Myer, Can. J. Fhys. 39, Bl 5-869 (1951). 3. Lebowitz, Joel L., and Bergmann, Peter %7., Ann. Phys. 1, 1-23 (1957). 4. Chandraseknar, S., Reve Mod. Phys. 15, 1-89 (1943). 5. Ross, John, J. Chem. Phys. 24, 375-380 (1956). 6. Helfand, Eugere, Phys. Fluids 4, 691-690 (1961). 7, Brittin, Ww. Es, Phys. fev. 106, 43-347 (1957). 8, See for example a) Herris, Robert A., and Rice, Stuart 1. J. Chem. Phys. » 1955-1058 (1960). » 2144-2155 (1961). c) Tlnatt, Alan 2., and Rice, Stuart A., J. Chen. Phys. 34, 2156-2165 (1961). — a) avis, fh. Te, Bice, S. Aw, and Sengers, J. Ve, J. Chem. Pays. 35, 2210-2233 (1961). 9, Rice, S. A., Kirkwood, J. G., Ross, J., and Zwanzig, 2. We, J. Chem. Phys. 31, 575-583 (1959). 8 10. a) See for exposition and references Prigorine, I., Non-Equilibrium Statistical Mechanics, Interscience Publishers, inc. New ork, 1961. b) Weinstock, Jerome, Phys. KeVe, 126, 341-344 (1962). b) Andrews, Prank C., - nev. 125, 1469-1472 (1962). ll. 32) Andrews, Frank C., Phys. Rev., 125, 1461-1469 (1962). c) Andrews, Frank C., Phys. Fev. 125, 1473-1477 (1962). 12. Thompson, ¥. 8., Reports on Progress in Physics 2k, 363-424 (1961). 13. ostoker, Norman, and Bosenbluth, M. N., Phys. Fluids 3, 1-14 (1960). 14, 15. 17. 20. 21. 226 236 24, 256 26. 276 28. 29 30. 144 a} Simon, A., and Harris, F. %., Phys. Fluids 3, 245=254 (1960). b) Harris, F. G., and Simon 4., Phys. Fluids 3, 255-258 (19605. — c) Simon, Albert, Phys. Fluids 4, 586-599 (1961). a) Simon, Albert, Phys. Fluids 4, 691-695 (1961). e) Dupree, Thomas Y., Phys. Fluids 4, 696-702 (1961). Lenard, Andrew, Amn. Phys. 10, 390-400 (1940). Guernsey, 2alph L., Phys. Fluids 5, 322-328 (1962). Varma, 2am K., Phys. Fluids 5, 525-533 (1962). Koga, Toyoki, Phys. Fluids 5, 705-711 (1962). a) Ron, A. and Kalman, G., Ann. Phys. ll, 240=259 (1960). b) Kalman, G. and Zon, Amiram, Ann. Phys. 16, 118-150 (1961). c) Ron, Awiras, and Kalman, G., Phys. Rev. 123 1100-1105 (1961). — = , a) Gastorowiez, S., Newnann, “., and aiddell, a. J. Jr. Phys. Rev. 102, 922-934 (1955). a) Von Roos, Oldwig, J. Math. Phys. 1, 112-120 (1960). b) Von Roos, Oldwig, J. Meth. was. t, 107-111 (1960). Vineyard, Seorge H., Pnys. Fluids 3, 339-35 (1960). Taylor, J. 8., Phys. Fluids 3, 792-796 (1960). Hill, Terrell L., Statistical Mechanice - ppsoeipiee reYa) and Selected Applications, pp. 5-7, wooraw~! Co. Y. Vs 19 e , Pines, David, and Bohan, David, Phys. Rev. 85, 335-353 (1952). : Goldstein, Herbert, Classical Mechanics, pp. 19-22, Addison-Wesley Publishing Coe, inc. Heading, Mass., Lebowitz, J. L., Frisch, H. L., and Helfand, E., Phys. Fluids 3, 325-338 (1960). Mazo, Robert M., Je Che. Phys. 35, 331-835 (1961). Feynman, Richard P. Phys. Rev. 84, 108-128 (1951). Mazo, Re Mey Private communication Wheelon, Albert D., J. Applied Phys., 23» 113-118 (1954). 145 PROPOSITION I It is proposed that the results of section I of this thesis be applied to a three-level maser, contradicting published work of Barker (1) and Seovil and Schulz-DuBois (2). Equation 39 of part I of this thesis gives the entropy production for a system undergoing photochemical reactions under conditions ef local equilibrium. It is: a 7 ) 2 = = A Avy E€ <6 — mee 1 where gp is the entropy production, the $s are pro- gress variables in molecules per unit volume, Ap is the negative of the free energy change per molecule of the reaction (the affinity), Ti is the thermal temperature, | fF is the density, © is the rate per unit mass of energy absorption from the radiation, and = / is the radiation temperature. We idealize a three-level maser as N identical particles, independent of each other, but interacting with the radiation field, and each able to occupy any one of three energy levels. As does Barker, we will consider the systens in each state to be a separate “chemical species," By (il, 2, 3), 146 and transitions between levels to be "chemical reactions.” We consider six reactions Boo 8, 3 Aau-u dead e-4a.3 MD Bs as B5 A, =A; A453 As =d4,u, =-asn > (IIT) 3) RLY 1| |v KH - S — Bo +A, 5 A, = As Ps Az, =A, MN, -~of, Nz - Avi) Here Mn is Planck's constant the vs are frequencies, anc the /(‘ are chemical potentials. We assume now that the part of the radiation field important for maser operation can be adequately represented by two radiation temperatures (not necessarily different), one for a narrow band of frequencies around Y, and another for a narrow band around Vs; . In other parts of the spectrum, and perhaps in directions other than in certain small solid angles, the radiation temperature may be near “/, . Equation 1 now becomes 147 CG. paah DA, bbs — Ay, hy fe (1 ” b=) At by y _ Ai | _t. a / / ge (FT, = Auge EF) (2) In the operation of our system as 4 maser, an external source provides radiation at temperature a in the narrow band around >), , causing reaction Wy to proceed to the left. A weak signal is introduced, causing re- action VI to proceed to the right by stimulated emission, giving a coherently amplified signal. The other reactions proceed in some manner. We will now neglect spontaneous emission, as well as any emission or absorption outside of the small solid angles in which the pump and signal radiation is con- centrated. Thus ro Oo , and E, and Sy are due only to stimulated emission and absorption. A simple calculation shows that = ~, 4 =8 (3) Le f for a steady state, which is the case Barker treats. However, he assumes that the only source of entropy production is from reactions I, II, and III, which is shown by equation 3 to be incorrect. 1428 After alia for the entropy production, Barker expresses the £ in terms of transition probabilities and affinities, after assuming Ay AT ‘ A<cATl , Since he ig treating the steady state, he minimizes gp with respect to A, » at constant A, (constant pump in- tensity). Although he is able to obtain a population expression which resembles one obtained from rate equations, his result can not be correct, since he does not use the correct entropy production. Several attempts were made at obtaining a population expression by minimizing the correct entropy production expression, with the constraint A, = constant. Since the exact expression led to a complicated transcendental equation, various approximations were attempted, but no reasonable result was obtained. The probable cause for this is that the theorem requiring minimum entropy pro~ duction for a steady state requires linear dependence of fluxes on forces for its derivation, and a maser certainly does not operate in the linear range if there is one. There may not be a linear range since it is not apparent how to express the term pene) in the form LX ~ . We can obtain a useful piece of information from equation 2, using equation 3; the second law, p =O ; 149 and our assumption, -— xO . we find Osta (i =i > | (5) 23 3¢ Te Te, ) If we define the maser efficiency, ys ee “33, > ) we find from equation 5, _/ Sf — => He Ties whieh is a ratio of Carnot limiting efficiencies. This differs from a result of Scovil and Schulz-DuBois, which - / / / y < hie - cae (3) The numerical difference will generally be small, but it is apparent that their result would correspond to an infinite 7, in the output signal, which is in principle unattainable. REFERENCES 1. Barker, William A., Phys. Bev-, 12, 1%-128 (1961). Letters, 2, 262-263 (1959). 159 PROPOSITION II A derivation of a many-particle Fokker-Planck equation for the ions in an Louie solution is proposed. Kirkwood gives (1), without proof, a wany-particle equation of this form, which is slightly different from the one we shall obtain. Beginning with the Neparticle Liouville equation, we integrate over the coordinates and momenta of all solvent molecules, and apply Green's theorem. The reduced distribution functions are those of part If (see equation II 4) but are now timeedependent. je obtain me} mei . (n Te = (n) oe eo fi UE + 2 | Ve Ne f ; _ ot = (om 3 os 4 AEs ren) MA >a We have assumed pairwise additivity of forces, and call F the force on particle S due to particle AL ° The set 7 is the set of ions, and N-Uu is the solvent, and we use Y) P and Ke je for their respective coordinates and mowenta. ve shall use tn and 4 tor the set Vv : At this point, we introduce time-smoothing. This is the artifice that introduces irreversibility into the 151 reversible Liouville equation. The timeesmoothed dis- tribution function is defined: . =p @ Pept) = ef Fields ° After time-smoothing, equation 1 becomes where (w) =, . (Mp re nay = Lf {f fieot (todtdRads . (4) We shall use the method of Ross (2). His phase space transformation function is defined by: Equations 9 through 17 in Ross’ article are a N) applicable. Using our equation 5 to express iY) (£+5) and Ross’ equatfon 17, we find n NV »~1 Z. n= tf ( [EG \[I- 2 ARV] * (6) ad TY Marta Kite) {Be BA Bl eA pede dR 152 where we now use P and R fox the set N-n-Z& ° Using Ross' equations 22 through 24, we have AS W/) (N oe = — (1) \ (A ‘ k= = We shall discard all terns of higher order in Ap” 5 those linear in Ap® » etc., as we proceed. Ross discusses these approximations in his paper. fy) Now we rewrite , defining the correlation o C4 function, Ps and +‘ /M) » the conditional distribution ee - (N) n u Me wztes) = £ ep wt {ER b45) 0) = % “Gas. ‘t+5) LG, Ro, walt ea t+s) We shall require this written for the set (+ instead (3) of for Ae ‘ . () ¢ With Ross, we assume that the f 5 acted on by the (n+!) Vo. and the Fa +2 can be replaced by the equilibrium functions. Thus _ fn Ye fC, Pe.) = f£ (n. 77) (9) ate ) and equation 7 becomes im + Mra 6 . Se Mey (6 he tee. k= 153 (#) We now obtain the following expression for &L . y (4) | aC = . in Bie =) fff C1- 2 An Ve VF Sle rnd Sect d uth 2 1) bo _ (w) [1B 2AM Hee 1 F (8,8; tes dtd RdpadudPdRds (11) =f (Als, f-& SAR 0) Gq [fi 83 a5 Selo ap xX (ew), "Fe ed aD o (4+ We J9- a Only the terms for « in the set 7” will survive in the < sum. Note that in equation ll, F a +2 ‘ n+') has been approximated by the equilibrium function, Pare = ut! (a +!) Lxp 3- jf Wy at ° W is a potential of mean force, defined as in part II of the thesis. We drop terms of O(a) ,;ané from equation 3 we obtain 22.8 HUE, SL Op iF” f= Jt ye 6m; J Leh, J J nm W t ots =~ 2g 2. [4 [fie FC ER: tisldsdgdgd Par yet £=ntl —~ +t tr a Se [I f fetaroz E(Ras't)¥ 16% fa -t4s) (o Neg. tesa ddkdsds (12) TE f [-ffGoWEew'§ ) BE Carter ee tes)aad dans ? +h nr) “EI ((. Fa leyteh (hts). Pe Z (rarest i PR tsp de dPAR dsdls . vO/JO~ 154 Making the standard assumptions concerning Plateau values in the time correlation function's integrals, we find after again time-smoothing: - (nA) . ; rs - (2) *& Ww OF PFO GREG = ot ca ; sf" 3 F Pp fe, 2% a pp Here we use the same vector notation in phase space as (13) I] Kirkwood, and equation 13 is the same as his result (reference (1), equation 60) except for the definitions of some quantities in it. We define ro = <5) +S, [se Elan, trelndg + +A (14) L= ner ) —T where te is a correction term to absorb all the ~ approximations made, and is hopefully small, J (16) 155 =? N= where = =< > ; fF d la=nei ¢ We use the notation of Kirkwood. A left superscript on an ensemble average indicates particles not averaged over, and a right super zero means an equilibrium average. Note that equations 15 and 16 differ from Kirkwoods definitions. Kirkwood's &//” has no WwW” in ie, and his is has only .=, terms in it and has a for the second F . In addition, of course, we have F (Rts £ ) instead of evp (ish) ECR) » as does Ross in his oneebody equation. Up to this point, our results apply, within the approximations made, to any system. Our particular application in which 2 is the set of fons in a solution, is no different from the case of wcharged species only, since all of coordinates of the particles interacting with long range forces have been left unintegrated. Note that at this point, we can, if we wish, insert the well-known approximation for "Fy given by ah a 2.2,€7/(dln-y)) . Beginning with equation 13, we can now proceed to integrate over the coordinates and momenta of part of the ions, say all but the set 7% , where +7 << -» Either by a considerable amount of manipulation or by recognizing that the derivation of equation 19 is formally acceptable for longerange forces, we obtain an equation identical with equation 13 except that % is replaced by ~. 156 However, when we write equation 13 for the set “72e 5 we must use the proper cluster expansions for the ensemble averages in Fa™ 3 » and ee » since these quantities are sums of formally divergent terms, as discussed in part II of this thesis. "eK E> poses no problem not treated in part II, nor do the potentials of mean force in Ee However, the time correlation functions, although evaluated to a linear trajectory approximation in the Rose theory, involve all particles, and must be treated in a different way. REFERENCES lL. Kirkwood, John G., J. Chem. Phys. 14, 180-201 (1946) ed 2. Ross, John, J. Chem. Phys. 24, 375-380 (1956) An alternative perturbation schese for tine- correlation functions, based on a method of von Zoos (1), is proposed. A formal solution of Liouville’s equation nas been published (1). However, this result, which is 4 series of scattering-type operators operating on a distribution function, cannot be applied directly to calculations of time-correlstion functions*, so we derive a result that can be. f Von Roos uses a phase space transformation function similar to that of Ross (2). The only difference is that the Ce t| RUE) of von Boos is the retarded Green's function for the Liouville equation, while the Ka (RF t| , &; te) of Ross satisfies the Liouville equation. Thue, for times later than the ini- tial tine, the two functions are essentially identical. We define a tineedisplaced distribution function, 7\W’ Al) A Cu) 7 N)z q (REC te, B; t.) =K COt\ Rt) E ie #, ts ) | (2) /(M) Since K 4s the probability density that the systen a phase point be at Kf at time tae 4t was known to be at te Te at time te, g (W) 48 the joint probability * See p. 107 of this thesis. 158 that the pair of pointe be observed to be the location of the phase point at the pair of times. ve could also ) kw use G instead of » arbitrarily introducing asyumetry in time at this point, but otherwise changing nothing. Equation 1 can be integrated over the coordinates of A/-2 particles at each time, giving (20 2 ZVF4y . ie 7 16; pt Ir B, to) =K Ch pele plod t Cr, fyto) 2 (2) (2) where Kas the reduced function of Bice and Allnatt (3). We can go through an analysis of the von 4oos type for qo” » but we work directly with <b pWaHp)> as follows: & aba) = [Aawdalny) SU LVR Uy ll lVal AT, ¢3) 5 W) where we use the G of von Hoos. In the case - (3 “es S Cr-/)SCVo-Vvel) J lr-r) Siv-v)) ¢ pbedaclt)> = gow Vit} Yuet), (4) ‘ in so the analysis of 7 ar ineluded as a special case of the analysis of <3 lt). of course, any < 3(G)x (> ean be obtained with a / » since 159 aif Blts)ec é)»> -|(3% Yo) ACV) 9st ) toe b)olrdvdnv, 5) (/) Now we can use the expansion of G given by von Roos (nis #S and // ) directly, except that we notice that his Liouville equation should have F- V, = f= Ws instead of FeV, on the right-hand side. We correct this minor error in the following, using — es > — EMV ~ EM, + BV ype ¢ fy Vu, The geroth order term is: < alts) alt) be [/2Gem) adbov) (=U) S [RR (tLe) | x 5(é-t) FCA Uz ted KAVAR AU (6) = | Ba w)ocloy) Slv-u) § Len-velety) SEL) X _ ‘nfa) (7) x FO Gauate) £ Lee, RUIN tA REAVandY, - Now change integration variables to fo, Vo, KV, R-V(t-be), V. The Jacobian for this transformation is a triangular 160 determinant with unit diagonal elements, so is unity. Now: Jbl ~ - fi BCav) Guy) Sev-u)d Lr-ve vole te) ] SCL) X c% utedolrall aly, cl, Ml Siz Vat.) | (8) = otassat Vo(t-te) ) Fs “Cave taolr Ave , which is of course the rectilinear approximation. Now we “tnd at the nth tera: or E, Gai <auoS'= 6) via! felt, he latnwie (qu) X x aldol Tol dt; ----dUn X x S free VEU EV E,4)- UES) x (9) x Elrvl-t)1-0, (rv ERv et Go. YSU) - --- PLRTE2)-3F GG) Ulere}]- Ve SOW) {YR te) Here the / a Kh, and U integrations have been performed. Integration over he replaces R. by tp- VE) - £5 -G) -Vilt-t) 161 in and Ve by rve-t)- Ev. (E-b )-Ve (En be) in (3 - We replace Vor 6Y-Ve) by — Vy. S(U;- Ud , etc., and then integration over “U. gives: € bn Kaela fall, --— [clés Rol Ad, ote X to te Xatny) Flaevilet] YSN) ---- (10) x Fle ves)-2YE4I- LA At -(Yg flabewle shy eG) -nlen thi] X y lu) : sina. ff ig x eviet) LEE b) wt bIP) We rewrite the expression in the manner of von Roos, using operators that operate only on the first or second argument of (3 or of & instead of on the JJ. variables. Now V7~ 1s veplaced by Vy - (L,-te) fot Vy - Gar to) Vie where these operate only as follows: The first term operates on the second argument of iS » the second on the first argunent of (nN (3 » the third on the second argunent of a and e pW? the last on the first argument of a e 162 Further integrations over the U- variables give the final result, ¢ plelalt)s= ©) "a bo TT ; Prev Gel) Los alrve-oy) x & fo (tele (rv) x x fYfe vere] 2 (11) where be Vy ~ (b- to) Vigt Y, -(ta-te )y, -& eta > In p » the first four operators operate only as before, and each Vg in the sus operates on the argument of the mal Brg. U(t-ty )q » only if v=8 ‘ This is like the work of von Roos except for the presence of the (2 , 80 we can use diagraas Like his and re- interpret the line to the exterior vertex to represent the first four terms of our Br « Equation 11 will be represented by rn f diagrams. Tne integration over “ and “U~ aust still be done in equation 11. This is 4nconvenient in the form shown, so we seek another way of writing it, to avoid 163 the gradients operating on the distribution function. Consider one term of equation 11, say the one represented by the diagram aoe aN 4 (12) with A (4LéE n) lines to the external vertex. We leave the teras of operators operating on (SE unchanged # and look at the 2 terms composed of products of Vax 5 and Ve S operating on - Say there are b Yes and t-b Vy'S in the product of LTS ° f Now we change fron 0, U- variables to Ww, Vy where 4e'= 0-VU (E-te) . The Jacobian is unity, and since the V'S are simply a way of writing part of the { ae '‘S , they are replaced by Vie" S The term we are considering now becomes: = Cn) £ 4 at (ee Te [% lech tale av &-O)s) BLiNJX xf Peet) BIE Lavstl] 9), (13) CW SFLR Neel] Covet) Me yh) wrt 164 / where we use |! to include the vertices from which an i] internal line begins ( nt of these), IT to include Mf / the b V/s ana TT to include A-b Uo S » Now we apply Ureen’s theorem b times to the UV / integration and (-b times to the J< integration and obtain (assuming the surface integral vanishes): &é 6 eo om “TG: ia ih, jf (i) ge YX) dlr4vto), v é Crys (Leet Gt) Ba) x oe where T/ is TH e The divergences operate on the entire factor to their right, which is a cartesian tensor with 4 free indices (of the lth rank). The Carn notation is interpreted so that by (J-)* we mean “V Vt , etc. ais is now an ensemble average of a function of only one set of coordinates and momenta. 1. 36 165 REFERENCES von foos, Cldwig, J. Math. Physe, 1, 107-111 (1960). Ross, John, J. Chem. Phys., 24, 375-380 (1956). Rice, S. A. and Allnatt, A. 8, Je Chen. Phys., Ba, 2144-2155 (1961). 166 PROPOSITION IV An approximate solution for the temperature distribution in a reaction vessel immersed in a constant-temperature bath is proposed. It is known (1) that temperature gradients in re- action vessels can be appreciable for cases in which no mixing occurs. In cases for which convection is negli- gible, the temperature distribution very nearly obeys the equation (2): Xx Here “1 48 the temperature, AK 4s the coefficient of thermal conductivity, 0 48 the specific reaction rate, H 48 the heat of reaction, f is the density, and C 4s the heat capacity. Benson (1) nas published a solution for equation 1 based on the assumption that Oa, #, f » <—, ana K are constants. Wilson (2) has published a steady-state solution (setting ITA =O > for the case in which a 6 Ege) ff =k 62 = A enp CERT) Lit BE, | = 7 ame < frité o /p7.2 | , where E's the Arrhenius activation energy, At is a constant, “C= /-—/e , 7o is the wall temperature, 167 and R 4s the gas constant. We consider the case of a spherical reaction vessel of radius Ve . Equation 1 becones ot. kL Da + RH —— Benson’s work corresponds to a solution of equation 3 with CR replaced by WK ; ‘We attempt an approxinate solution usins equation 2 for a . Benson's solution is: ntmprkKe 2, \r .£ - = = eos [herr Be, Ei) denM) 2 Yo *Pe 4] ee Co ° / ne = — in We let we » and equation 3 becomes, using equation 23 Op = Pre + Rr y ME" p Dt je 5 peer f+ 8) We assume, with Benson, that the reaction is sufficiently slow that WW. does not vary appreciably with time or position in the cell. Since the ye are a complete set of functions, at least for functions having the right syanetry and obey- ing the right boundary conditions, we attempt 4s solution 168 4n the formu P= Bebe [ rn +S LO Ye | . CK We write IP _ A) 4 @AneS Ja ot em "6K nw Ot ic > vit Z where (apt), is the derivative with respect to time if b,, were a constant. Putting equation 7 into equation 5, we find } (28 ) + BAR sy bn Ye << \ ot Ab bK naw Ov 3° 4 + Kir , Ke HE” £ a Fe ae “pe poR re” But 7 ; (3% ) — K ee Y + &B fir Ot Po dr! PS ? Hn S tb, YW) - BHE™ cp bok guot — PeRhe Now we multiply equation 10 by Yn and integrate over |” from . to “so, We obtain, Ps (6) (7) (8) (9) (10) Le Ha™ oy Rte | (oe) roby (a foe ne _& Yh LE Yad Fem | axpeR Te* 169 © due to the orthogonality of the Y. e After doing the integrations and performing soue manipulations, we obtain oben = Ke HE [ep EXE phe | dt eeRT* re2PC ai The solution of equation 12 iss It) = 2x ee HET PeRE i m P gpcRT.* { mK _ @uet TO as (13) mpc PCRT~ ~ Our solution is now: not tK Ys Goel + fe 9 biota nr) oF oy r 6K | wom ? where Le is given by equation 13. We notice that, in order for the solution not to diverge in the limit $20 » we must require ke” Z witke FE ter rpc This is not surprising, since our treatment is Limited anyway to slow reactions. 170 For the following values of the parameters: er = 20 keal anole ~! TT = B00 oleg Se po = | cal cm ° deg ~ kK = |x 10 “Beal oun gear ‘cleg” REC! hry, = SS we find that "(t> must be smaller than about 2 x 107" mole liter™+ second™*. This is fairly slow, but there are reactions of interest that proceed this slowly. Now that we have the temperature distribution, we can, in principle, integrate a given teuperature- dependent rate equation, but this is likely to ve ime possible without a high-speed computer. REFERENCES 1. Benson, Sidney W., J. Chem. Phys. 22, 46-50 (1954). 2, Wilson, David J., J. Phys. Chem. 62, 653-655 (1958). 171 PROPOSITION V A modification is proposed of the model of Saunders and Rossa(1) for the reaction between polyadenylic acid and polyuridylic acid, This reaction, which forms a two-stranded helix, is found to exhibit a decreasing rate with increasing tempera- ture. The model of Saunders and Ross proposes a series of steps, each having ordinary temperature-dependence, and each adding one more cross-link to the double chain, and is able to fit experimental data reasonably well, although using more than one free parameter to do so. The rate of the initiation reaction is assumed to follow: VY, = k&AB- #2, a) where % is the rate, ke is the forward rate constant, A and B are the concentrations of the two single- strand species, Rep is the backward rate constant and C 1 is the concentration of "dimers" with only one link. The rate of the reaction which closes link number v +) in the “zipping-up" of the double helix is assumed 172 to be KC 7 ty Cr+, ‘ (2) Saunders and Goss assume that all the steps after the first have the same Ry and R, » and thet Ay 208 Ky are egucl. The first may oe a reasonable approxi- mation, especially for z. re-ccmably large, and seens an unavoidable avproxtnation. However, it would seea unlikely the ky, woull equal an since one reaction simply unzips the chains by one hydrogen-bond of aany, while the other reaction completely separates the chains. We assume hth, and proceed in the nmamer of Saunders and toss. *ultiplication of equation 2 vy (Ay [by )' vives: al L MG 4, (4+) « (Ze) Che . (3) over © from / to/-/ (the nuaber We sum equation 3 of links in the chain st completion) and add equation 1, odtsining: wv -/ : a w-} 2s ee = £, ABA. th - 4B ) Cy. oy) tle now consider the rate of “zipping-up", V , without considerstion cf how far along the chain the L723 reaction is occurring. (E & = AABAG 86 ~h(4e)" Cy 59 Tne sum is seen to be s seonmetric progression, N & (fe) = i— or, b hep) (6) Rp I Hk ? Chk” [lhe be) (7) _ Ay(& 5 Jee) Kone 1-4 Mpg >. Now we assume thet the temperature is less than the helix melting temperature, so that As Ja, ae: . (8) If this is so, / ip fe (A4,)” << | 9 (9) since A/ >> / is assumed. Equation 7 becomes V = 4,4B(\-fe ) +h.) C.(/- 44/4.) . (10) The result of Saunders and Ross is lacking the last / tern, We assume the temperature dependence for the e's, 174 z) + . ha = fa wep 6 ~ HE IRT ¢ k= Obtr. (11) and obtain a V = ABL ew CURT | 7) [1 fe wap (AMT ) £ t 1 ~WeART - UY AH/RT +¢,|he ° ~fe RN I-he ( J, (12) where AH = He = My is the enthalpy change for one step. The snalozous result cf Saunders and Joss, wnich consists of the first line only, is used to fit w some experimental data, using xo and AHX 2 kcal mole™+, In order to see if the data can be fit using all of equation 12, we transform equation 12 to use the independent variable g = axp(AH /er) since this was the independent variable on the zraph of date, against whicn V was a straight line of nezative slope. Equation 12 becomes - Ks It 7 fe Vv=AB f, I Soe Cf i hg eg a yo (13) 175 We differentiate this to obtain the slope, obtaining: ut / dv =AB(- hha” AK fly a4 - & % = + —He 7 \ f - Hn Au 4 eae. be ale jt CG (> J — He) Hu £ ¢ f ul | + fut q AN fy Me Lat arg A f, 0 AH ) aan 4 ue MAR feb Hal ote fH) (14) f, AH te AH? ; Now if Saunders and Ross assume As =O , they get a constant slope and can fit their data, since their result corresponds to the first line only of equation 14, However, we can also cet a constant slope, enabling us to fit s straight line, if we set in equation 14s 4: 4 + H, =f = Hy = © (15) enabling us to obtain 4 value for aH - Perhaps other combinations of parameters can also be found to nake dv fg constant. We heve shown thet the model of ioss and Saunders, altnougsh fitting the data, is not tne only one to do BO, and that one which is a@ priori nore appealing can also fit the data. 3EPERENCE Saunders, Martin, and loss, Philip D., Biochem. and