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Volume 4, Issue 2, February 1963

On the Development of Nonequilibrium Thermodynamics
View Description Hide DescriptionA general view of the history of nonequilibrium thermodynamics shows how two main lines of development have recently fused into a single branch of science. The field theoretical formulation of thermodynamics (leading to a balance equation for the entropy) constitutes the framework of a theory in which the Onsager reciprocal relations form the pièce de résistance, the fundamental importance of which is outlined in this paper.

Thermodynamics of Electrical Networks and the Onsager‐Casimir Reciprocal Relations
View Description Hide DescriptionElectrical networks furnish an excellent example for the application of thermodynamics of irreversible processes. In particular, they provide one of the simplest cases where, besides Onsager coefficients, one has Casimir coefficients in the phenomenological equations. Two thermodynamic formulations of the networkequations are given, which closely resemble the Lagrangian and the Hamiltonian formalism, respectively, of classical mechanics. In the first formulation, only Onsager coefficients occur, but the thermodynamic forces are of a peculiar type in that they are Lagrangian derivatives. Incidentally, it is shown that Casimir reciprocal relations can generally be replaced by Onsager reciprocal relations if the independent variables in the linear phenomenological relations are chosen in a proper way. As a generalization of the networkequations,Maxwell's equations in continuous matter with dielectric, magnetic, and Joulean heat losses are considered. Matter is assumed to be isothermal, but not necessarily uniform nor isotropic. Under the influence of impressed electric fields,current distributions are produced. The connection of these fields is expressed by a generalized admittance function. A well‐known reciprocity theorem for electromagnetic fields is seen to hold even if all types of losses, as mentioned before, are present. This is due to the Onsager‐Casimir reciprocal relations for the dielectrictensor, the permeability tensor, and the resistivity tensor. From the reciprocity theorem, a symmetry relation can be derived for the generalized admittance function. A generalized version is given in the presence of a static magnetic field.

Theory of the Soret Effect in Electrolytic Solutions
View Description Hide DescriptionThe limiting law for the square‐root concentration dependence of the heat of transport of a simple electrolyte is calculated by considering the Soret effect. The calculation is accomplished by relating the intermolecular forces arising from the temperature gradient in the nonuniform Soret stationary state to the equilibrium gradients of chemical potential required to maintain the same concentration gradients. The contribution arising from ion‐ion interactions is identical with that determined by Helfand and Kirkwood from a consideration of heat flow accompanying diffusion. Hence, the present work provides a verification of the heat‐matter reciprocal relation without explicitly invoking time‐reversal invariance.

Structure of the Three‐Particle Scattering Operator in Classical Gases
View Description Hide DescriptionIt is shown that in all cases where the finite duration of the collision may be neglected, there is a complete equivalence between the 3‐particle scattering operator obtained by Prigogine and his co‐workers and the corresponding expression derived by Choh and Uhlenbeck using Bogolubov's method. Great emphasis is put on the ``irreducible character'' of the three‐body collision: Only situations in which the three particles are simultaneously interacting play a role in both theories.

Stochastic Liouville Equations
View Description Hide DescriptionWhen a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum‐mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.

On the Kinetic Theory of Dense Gases
View Description Hide DescriptionA statistical mechanical theory of a dense gas that is not in equilibrium is presented, which is completely analogous to the well known theory of a dense gas in equilibrium. In particular, an expansion of the pair distribution function in powers of the density for a gas not in equilibrium is given, corresponding with that in equilibrium to all orders in the density, that can be represented by the same diagrams. The expansion can be reduced to that derived by Bogolubov, Uhlenbeck, and Choh from a solution of the B‐B‐G‐K‐Y hierarchy. The conditions for the validity of the expansion are, for an infinite system at not too high density, and after the lapse of some time after t = 0: (1) a statistical assumption at t = 0; (2) some conditions on the interaction potential; (3) coarse‐grained distribution functions. A simple generalization of the Boltzmann equation to general order in the density is included. Also, the connection with a Master equation for a spatially homogeneous system is discussed.

Thermal Fluctuations in Nonlinear Systems
View Description Hide DescriptionThe simple physical assumptions on which the familiar linear fluctuation theory is based do not carry over to the nonlinear case. In order to treat this case, one has to start from the master equation, and expand it in reciprocal powers of a suitable parameter (roughly speaking, the size of the system). The successive powers yield first the phenomenological law, next the linear fluctuation theory, and subsequently the influence of nonlinearity in successive approximations. The coefficients in the expansion are connected with each other by relations of the same type as the fluctuation‐dissipation theorem in the linear theory. Whether or not these relations are sufficient to uniquely determine the coefficients from phenomenological data is not known at present.

Hydrodynamics of a Superfluid Condensate
View Description Hide DescriptionThe theory of the condensate of a weakly interactingBose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self‐consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self‐consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.

A Microscopic Approach to Superfluidity and Superconductivity
View Description Hide DescriptionThis lecture describes certain properties of interacting Bose gases and superconductors which have recently been considered at Harvard. It is a brief resumé of work by the lecturer, P.C. Hohenberg, A. Fetter, R. Lange, and C. De Dominicis, which will be reported in mathematical detail elsewhere. It concerns attempts to derive from fundamental principles, several aspects of the macroscopic two‐fluid model of London, Landau, and Tisza, the Landau phonon spectrum for bosons, and the structured condensate envisaged by Onsager and Feynman. The lecture describes new results which have been obtained concerning: (1) methods for calculating properties of large, but not necessarily infinite, condensed systems at finite temperatures; (2) methods for performing calculations in boson systems consistent with current conservation and sum rules; (3) results obtained by these methods for the phonon spectrum; (4) more accurate results for the less physical single‐particle spectrum at finite temperatures; (5) results obtained for the macroscopically structured condensate in rotating systems which agree with the macroscopically inferred results of Feynman and Onsager; (6) results which treat depletion of the condensate as a result of interaction consistently and verify that nonetheless, the superfluid density, as operationally determined, is not correspondingly depleted; (7) methods in which the entropy plays a central role, for casting the macroscopic theory in a form which lays stress on the renormalized excitations of renormalized interactions; (8) the almost identical features of the mathematics for condensed Fermi and Bose systems; (9) a restatement, in this microscopic transcription, of the Feynman‐Onsager argument for flux and vorticity quantization.

On the van der Waals Theory of the Vapor‐Liquid Equilibrium. I. Discussion of a One‐Dimensional Model
View Description Hide DescriptionFor a one‐dimensional fluid model where the pair interaction potential between the molecules consists of a hard core and an exponential attraction, Kac has shown that the partition function can be determined exactly in the thermodynamic limit. In Sec. II this calculation is reviewed and further discussed. In Sec. III, we show that in the so‐called van der Waals limit when the range of the attractive force goes to infinity while its strength becomes proportionally weaker, a phase transition appears which is described exactly by the van der Waals equation plus the Maxwell equal‐area rule. In Sec. IV the approach to the van der Waals limit is discussed by an appropriate perturbation method applied to the basic integral equation. The perturbation parameter is the ratio of the size of the hard core to the range of the attractive force. It is seen that the phase transition persists in any order of the perturbation. The two‐phase equilibrium is characterized by the fact that in this range of density, the maximum eigenvalue of the integral equation is doubly degenerate and that the corresponding two eigenfunctions do not overlap. In Sec. V we comment on the relevance of our results for the three‐dimensional problem.

On the van der Waals Theory of the Vapor‐Liquid Equilibrium. II. Discussion of the Distribution Functions
View Description Hide DescriptionFor the same one‐dimensional fluid model discussed in Part I, we have derived general expressions for the two‐ and three‐particle distribution functions. It is seen that these distribution functions depend on all the eigenvalues and eigenfunctions of the basic Kac integral equation, and the dependence is so transparent that the generalization to s particles is obvious. The fluctuation and virial theorems are discussed and shown to be consequences of our general formula. In the van der Waals limit, the behavior of the two‐point distribution function is discussed, both for distances of the order of the hard core and for distances of the order of the range of the attractive force. The long‐range behavior is, in first approximation, equivalent to the one‐dimensional version of the Ornstein‐Zernike theory, but only in the one‐phase region and not too near the critical point. In the two‐phase region, all distribution functions are linear combinations of the two corresponding distribution functions of the saturated vapor and liquid, with coefficients proportional to the mole fractions of vapor and liquid. This is shown for our model; we also give arguments for our belief that these relations are general, and express the geometrical separation of the two phases. The relation to the Ornstein‐Zernike theory is discussed in more detail, especially in connection with a recent formulation of this theory by Lebowitz and Percus. We conclude with some comments on the relevance of our results for the three‐dimensional problem.

Asymptotic Behavior of the Radial Distribution Function
View Description Hide DescriptionThe pair distribution function in a uniform classical fluid is equivalent to the one‐body density when one particle is fixed. An implicit relation for this nonuniform density is found by a functional expansion of the difference of chemical potential and external potential about its value for a system of uniform density. A linearization of this expansion, followed by retention of, at most, second derivatives of the inhomogenity, reproduces the Ornstein‐Zernicke relations for the asymptotic pair correlation. Linearization alone calls for the sum of internal potential and direct correlation function to vanish asymptotically. This relation is developed for the case of weak long‐range forces, resulting in the Debye‐Huckel expression for an electron gas, and reproducing the asymptotic correlations of the Kac‐Uhlenbeck‐Hemmer one‐dimensional model. The relation is also shown to follow from the virial expansion for the direct correlation function.

Variational Statistical Mechanics in Terms of ``Observables'' for Normal and Superfluid Systems
View Description Hide DescriptionThermodynamicalfunctions are expressed by perturbation theory as stationary functionals of the 1‐, and 2‐body potentials; the stationary conditions yield the 1‐, and 2‐body correlation functions in terms of the potentials. Through the use of diagramatic methods, it is possible to explicitly invert these relations and express the potentials in terms of the correlation functions; in turn the thermo‐dynamical functions become stationary functionals of the correlation functions. This procedure is carried out for normal (quantum and classical) systems. For superfluid systems, body potentials, and correlation functions need also be considered and the above procedure becomes imperative to eliminate the nonphysical body potentials. Its first two steps are illustrated, and various features of this formulation and its usefulness are discussed.

The Master Equation with Special Transition Probabilities
View Description Hide DescriptionThe Van Hove master equation to general order is solved, making a special ansatz for the function W _{11′}(kk′). Approximate solutions are found for strong and weak coupling. For λ = 1, a comparison is given with the exact numerical solution.

Asymptotic Form of the Structure Function for Real Systems
View Description Hide DescriptionIt is shown that the asymptotic formula for the structure function obtained by Khinchin for the case of a system of n noninteracting components, also holds for a classical system of N interacting particles. It is essential for the derivation that the interactions be such that the system behaves in the limit as N tends to infinity, as a thermodynamic system.

Lattice Statistics‐A Review and an Exact Isotherm for a Plane Lattice Gas
View Description Hide DescriptionProgress in the theory of lattice statistics is reviewed with emphasis on the use of series expansions to study the critical behavior and on the exact results obtained by transformation theory. Recent work is reported which indicates that the magnetization of the three‐dimensional Ising model vanishes as (T_{c} − T)^{β} with , while the low‐temperature susceptibility diverges as (T_{c} − T)^{−γ} with in two dimensions). An Appendix is devoted to a detailed tabulation of the exact numerical values and best estimates for the critical temperatures, energies, specific heats,entropies,magnetizations, and the ferro‐ and antiferromagneticsusceptibilities of four plane lattices and the simple, body‐centered, and face‐centered cubic lattices.
A square lattice gas with infinite nearest‐neighbor repulsions and weak second‐neighbor attractions (across alternate squares) is solved exactly for one particular temperature by transformation. The gas undergoes a transition at a density , the corresponding form of the isotherm being,where , so that the isothermal compressibility becomes logarithmically infinite.

Dimer Statistics and Phase Transitions
View Description Hide DescriptionAfter the introduction of the concept of lattice graph and a brief discussion of its role in the theory of the Ising model, a related combinatorial problem is discussed, namely that of the statistics of non‐overlapping dimers, each occupying two neighboring sites of a lattice graph. It is shown that the configurational partition function of this system can be expressed in terms of a Pfaffian, and hence calculated explicitly, if the lattice graph is planar and if the dimers occupy all lattice sites. By the examples of the quadratic and the hexagonal lattice, it is found that the dimer system may show a phase transition similar to that of a two‐dimensional Ising model, or one of a different nature, or no transition at all, depending on the activities of various classes of bonds. The Ising problem is then shown to be equivalent to a generalized dimer problem, and a rederivation, of Onsager's expression for the Ising partition function of a rectangular lattice graph is sketched on the basis of this equivalence.

Time‐Dependent Statistics of the Ising Model
View Description Hide DescriptionThe individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N‐member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time‐varying magnetic field upon the model is discussed, and the frequency‐dependent magnetic susceptibility is found in the weak‐field limit. Some fluctuation‐dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time‐dependent correlation function of the magnetization at equilibrium.

Correlations and Spontaneous Magnetization of the Two‐Dimensional Ising Model
View Description Hide DescriptionIn this paper we rederive, by simpler methods, the Onsager‐Kaufman formulas for the correlations and the Onsager formula for the spontaneous magnetization of the rectangular two‐dimensional Ising lattice. The Pfaffian approach is used to derive the correlations in terms of Pfaffians, and for the correlations in a row a single Toeplitz determinant is obtained which is proved equivalent to the Onsager‐Kaufman result. The spontaneous magnetization is obtained as the limiting value of an infinite Toeplitz determinant and its evaluation is facilitated by use of the generalization of a result first published by Szego.