Nonequilibrium Statistical Thermodynamics

By Bernard H. Lavenda

This book develops in detail the statistical foundations of nonequilibrium thermodynamics, based on the mathematical theory of Brownian motion. Author Bernard H. Lavenda demonstrates that thermodynamic criteria emerge in the limit of small thermal fluctuations and in the Gaussian limit where means and modes of the distribution coincide. His treatment assumes the theory of Brownian motion to be a general and practical model of irreversible processes that are inevitably influenced by random thermal fluctuations. This unifying approach permits the extraction of widely applicable principles from the analysis of specific models.
Arranged by argument rather than theory, the text is based on the premises that random thermal fluctuations play a decisive role in governing the evolution of nonequilibrium thermodynamic processes and that they can be viewed as a dynamic superposition of many random events. Intended for nonmathematicians working in the areas of nonequilibrium thermodynamics and statistical mechanics, this book will also be of interest to chemical physicists, condensed matter physicists, and readers in the area of nonlinear optics.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 17, 2019
• ISBN: 9780486839844

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Index

Preface

In 1898, when Boltzmann published the second volume of his Lectures on Gas Theory, he made the remark that ‘I am conscious of being only an individual struggling against the stream of time’. It is rather ironical that less than a half a century later, Boltzmann’s ideas on the statistical limitations of physical laws should become so popular and widespread. Yet, this popularity has led to a polarization in science by relatively few individuals who have tried to convince the majority that their theories explain all there is to know about the evolution of nonequilibrium systems.

These remarks are not intended to justify why this book contains a theory of nonequilibrium statistical thermodynamics as I see it. There is a chasm between the worlds of mathematical physicists and physical chemists. The former attempts to extract exact results on model systems while the latter is interested in a qualitative understanding of why nonequilibrium systems tend to evolve to equilibrium or to a nonequilibrium stationary state in the course of time. Apart from the classic papers on brownian motion (Uhlenbeck and Ornstein, Chandrasekhar, Wang and Uhlenbeck, Kramers, etc.) and its relation to irreversible processes (Onsager, Onsager and Machlup, Falkofï, Hashitsume, etc.) almost all of which are over forty years old, there does not exist a book which explains, in simple terms, why the theory of brownian motion has had such an impact on the statistical theory of nonequilibrium thermodynamics. In addition, new mathematical advances in brownian motion theory have been made, notably by the Russian school of probability theory, which is beyond the reach of a physicist or a physical chemist. In this book, an attempt is made to bridge both gaps.

The book is arranged by argument rather than by chronology. It is based on the premise that random thermal fluctuations play a decisive role in governing the evolution of nonequilibrium thermodynamic processes and they can be viewed as a dynamic superposition of a large number of random events with the important proviso that the future evolution depends only on the present, independent of its entire past history. In other words, brownian motion will be our fundamental building block from which we shall develop the statistical foundations of nonequilibrium thermodynamics.

The first chapter summarizes the phenomenological approach to non-equilibrium thermodynamics which is contained in my book Thermodynamics of Irreversible Processes. After making an entire circuit, it is this approach which is validated by showing that it arises as a limiting case of a more general statistical theory where the role of random thermal fluctuations have a negligible importance. The second chapter introduces the reader to the phenomenon of brownian motion, taking the reader from the pioneering works of Einstein, Wiener, and Kolmogorov to the more recent developments of stochastic theory. I make an effort to minimize the mathematics while maximizing the physical comprehension of the end result. In the third chapter, well known theorems in nonequilibrium thermodynamics are presented from a unifying standpoint of stochastic theory. Not only is there the pleasure of recognizing old things from a new point of view but, in addition, the approach clearly delineates their range of validity which is ordinarily masked by the purely formal proofs.

In the fourth chapter, the ascent to a statistical formulation of nonequilibrium thermodynamics is undertaken by beginning with the original Onsager–Machlup formulation of nonequilibrium, Gaussian fluctuations. Had these authors been armed with the mathematical theory of stochastic processes, there would probably be no need for this book and the theory could have been fully developed over three decades ago. In any event, the approach developed in the fifth chapter can be considered as a sequel to the Onsager and Machlup theory of Gaussian fluctuations. For it is a culmination of a train of thought that was initiated by Onsager over half a century ago: the search for a kinetic analog to Boltzmann’s principle that would offer a complete statistical description of nonequilibrium statistical thermodynamic processes that can be modeled as Markov diffusion processes. The success or failure of this principle is governed by the extent to which brownian motion adequately represents a broad and interesting class of physical processes. In the sixth chapter, the summit is reached where this principle is employed in the construction of what I call a ‘stochastic’ H-theorem. Unlike its classical counterpart, it asserts that the physical mechanism responsible for the evolution to equilibrium in the course of time is the wearing off of the statistical correlations between nonequilibrium states. The additional entropy, generated by these statistical correlations, is on the average a monotonie decreasing function of the time interval of separation.

In the seventh chapter, we return to our point of departure and show how the phenomenological laws of nonequilibrium thermodynamics arise in the limit of small thermal fluctuations. The results of this chapter are based on the probability estimates of Wentzell and Freidlin who, as they put it, seem to be digging into the probability of rare events which appears to go against the grain of probability theory. Nevertheless, it is precisely those rare events which are ‘less improbable’ which govern the evolution of nonequilibrium thermodynamic processes in the long time limit. Here, not only do we see how the phenomenological laws emerge in the limit of small thermal fluctuations but, in addition, we are able to study phenomena which lie beyond the domain of nonequilibrium thermodynamics and are responsible for the statistical interpretation of the second law of thermodynamics.

No attempt is made to provide anything even resembling a complete set of references. In order to do so, a volume much greater than the present one would be needed. However, justice is done whenever possible and credit is given where credit is due. Much of science is in the rediscovery of known results and analogies drawn between different fields of research. An effort has been made to cite original contributions although a number of important contributions are only implicitly credited. The selection was also based on those references which would be most helpful to a newcomer to the field.

I would like to express my very deep gratitude to Alfonso Maria Liquori and Enzo Ferroni for their encouragement, faith and giving me the tranquility to carry on my research. I am also indebted to Franco Fraschetti, Vincenzo Gervasio, and Paolo Verrecchia who believe that nonequilibrium thermodynamics can be an important line of research in an industrial environment. Thanks are also due to my colleague and friend Pierluigi Giacomello for his advice and interest. Last, but certainly not least, I would like to thank my wife Fanny for her constant support.

The research described here has been supported by the TEMA of the ENI group, the Consiglio Nazionale delle Ricerche, and the Ministero della Pubblica Istruzione.

Bernard H. Lavenda

Camerino, June 1984

CHAPTER 1

Nonequilibrium thermodynamics

1.1 Introductory remarks

In this chapter, we review the phenomenological structure of nonequilibrium thermodynamics. After presenting the phenomenological relations in their most general form, in Section 1.2, and the thermodynamic power relations to which they give rise, in Section 1.3, we look for their justification in terms of the thermodynamic variational principle of least dissipation of energy in Section 1.4. Depending upon the variational convention which is adopted, different classes of equations for the extremal trajectories arise. And in much the same way that Boltzmann was led to the conclusion that a mechanical theory of molecules requires a statistical interpretation of the second law of thermodynamics, we will appreciate that a mechanical interpretation of the thermodynamic variational principle will solicit a probabilistic explanation for one of the classes of extremal paths which has no deterministic significance. This leads us to consider the role of fluctuations in nonequilibrium processes in Section 1.5 and, in particular, fluctuations in a system which has been left isolated for a length of time that is sufficient to secure a state of thermodynamic equilibrium.

1.2 Phenomenological relations

The macroscopic theory of irreversible thermodynamics is phenomenological in character. It attempts to describe the relaxation of nonequilibrium processes to a stationary or equilibrium state. The theory is built upon empirical relations between flows and the forces that cause them. In order to apply the principles of detailed balance and microscopic reversibility, the flows must be time derivatives of a set of N extensive thermodynamic variables {x} which we assume to define the state of the system in hand completely from the thermodynamic (phenomenological) point of view. The macroscopic thermodynamic quantities {x} are so defined that they vanish in the reference state. We consider only extensive thermodynamic quantities because there are conceptual difficulties encountered in constructing a fluctuation theory in terms of intensive quantities. Specifically, intensive thermodynamic variables are denned as derivatives of the internal energy or entropy. The stumbling-block in the creation of a fluctuation theory for intensive parameters is that the average of a derivative need not necessarily be the same as the derivative of the average.

Irreversible thermodynamic processes are described in terms of ‘phenomenological’ relations of the form:

The generalized resistance matrix JR is taken to be symmetric and positive definite and, for nonlinear processes, can become state dependent. This will depend upon the distance the system is from the time independent reference state, regardless of whether it is the equilibrium or a nonequilibrium stationary state. Here, it is important to distinguish between what is commonly referred to a ‘linear’ irreversible thermodynamics and ‘linearized’ phenomenological relations. Linear irreversible thermodynamics implies that the reference state is the equilibrium state. The entropy is a function of the set of extensive variables {x}. It assumes a maximum value S(0) at equilibrium. The tendency of the system to seek equilibrium is measured by the (thermodynamic) forces (∂/∂x)S(x). For small deviations from equilibrium, they are linear functions of the displacements and the generalized resistance matrix is state independent. The system is necessarily isolated since it relaxes to the state of equilibrium and consequently F reduces to:

where S(x) is the entropy of the nonequilibrium state. The thermodynamic forces (1.2) are a measure of the tendency of the process to evolve to the time independent state. The forces are ‘internal’ in so much as they are due to the process’ internal mechanism to restore the system to the unperturbed, time independent state. We could assume that the forces are always given by (1.2) and do not attribute initially any definite symmetry properties to the generalized resistance matrix. Then provided the forces are defined by (1.2) and the reference state is thermodynamic equilibrium, where detailed balance holds, the symmetry of the generalized resistance matrix will follow from the principle of microscopic reversibility.

By linearized phenomenological relations, we mean that Equations (1.1) are to be linearized about a given nonequilibrium state. Again the generalized resistance matrix will be state independent since the space, from our local, short-sighted view, is a flat (Euclidean) space. It is only when we go beyond the immediate neighborhood of the stationary state that the nonlinearities will become visible. There are two possibilities open to us: the traditional approach takes the internal, thermodynamic force (1.2) into consideration explicitly and accounts implicity for the action of the external force, which prevents the system from relaxing to full equilibrium, in the possible destruction of the symmetry of the Onsager transport matrix. In place of the phenomenological relations (1.1), we would write:

where L is the Onsager transport matrix. Near and far from equilibrium situations would then be distinguished by the assured or nonassured symmetry of the transport matrix, respectively. As Onsager originally showed, for infinitesimal displacements from equilibrium, its symmetry is ensured by the principle of microscopic reversibility. While for deviations from a nonequilibrium stationary state, there is nothing to guarantee that the transport matrix will remain symmetric. An alternative approach assumes that the resistance matrix is positive definite, by definition, and takes into account the effect of the external forces explicitly in the definition of the total force, F. Of the two approaches, the second is the more instructive one since it displays the effect that the non-conservative field has upon the time evolution of the nonequilibrium thermodynamic process.

, we obtain:

denotes the quadratic form associated with the symmetric matrix R. A homogeneous, quadratic form of the velocities is defined in (1.3) which is known as the Rayleigh–Onsager (RO) dissipation function. If we view the positive definite matrix R as a metric tensor in a N-dimensional manifold, then the arc length of a curve x = x(t) is given by

which is determined entirely by the RO-dissipation function, Φ. We can say that a Riemannian structure is naturally induced by the positive definite, generalized resistance matrix.

The phenomenological relations (1.1) are oft-times written as the set of kinetic equations:

and we want to obtain a decomposition for the drift vector b. In ordinary three-dimensional space, any vector field can be decomposed into a gradient of a scalar function and the curl of a vector field whose divergence is zero. In dimensions greater than three, we can appeal to a well known theorem of differential forms which will allow us to decompose the drift field into orthogonal components. Or we can more simply assume that the vector field b(x) admits the decomposition:

where R–1 is the generalized conductance matrix, such that the orthogonality condition

is satisfied. Apart from its purely macroscopic significance, the orthogonality condition will be shown to be a necessary condition for the existence of an invariant, equilibrium probability density given by Boltzmann’s principle in the limit of small thermal fluctuations (cf. Section 2.5, Sections 6.2 and 6.5 and Section 7.4).

The drift decomposition (1.5) has frequently been interpreted as a decomposition based on time reversal behavior of its components. In this decomposition, the gradient and non-gradient parts are associated with the irreversible and reversible components of the drift, respectively. This is, however, purely formal since there is no component of the drift which behaves as a molecular velocity under time reversal, i.e., an odd function of time. Rather, the v-field should be interpreted as an externally applied field such as a Coriolis or magnetic field of intensity H. And it is not fortuitous that the original derivation of the Onsager symmetry relations of the phenomenolog-ical transport coefficients, Lij, required a reversal of the field under time inversion, i.e., Lij(H) = Lji(–H). However, it is not clear how such a functional dependence is acquired and what exactly is the functional dependence. This is why we have preferred to account for the externally applied fields explicitly as an additional component of the force and left intact the symmetry of the phenomenological transport matrix.

In the case of a vanishing υ-field, the entropy becomes the potential for the drift field with respect to the Riemannian metric connected with the generalized resistance matrix R which necessarily occurs in isolated ther-modynamic systems where the entropy gradient is the driving force acting to restore the system to the state of thermodynamic equilibrium.

1.3 Thermodynamic power relations

A variety of thermodynamic power relations can be derived from the phenomenological relations (1.1). We have already derived one such relation, (1.3), which defines the RO-dissipation function. Upon introducing the orthogonal decomposition of the drift field, (1.5), we have:

is the entropy production:

) is the rate of working of the external forces B = R · υ, namely,

We should bear in mind that the entropy production (1.8) is a linear homogeneous function of the velocities and from the discussion in the last section, it is clear that the external power, Π, is not derivable from a scalar work function. Moreover, in an isolated system, the external forces B vanish and the power relation (1.7) reduces to:

Therefore, the RO-dissipation function numerically coincides with the entropy production only in isolated systems. In terms of equilibrium thermodynamic potentials, the RO-dissipation function is equal to half the rate of the ‘dissipation’ of the Helmholtz free energy per unit temperature at constant volume and temperature. In open systems such a thermodynamic characterization is no longer possible.

Now, the scalar product of the phenomenological relations and the entropy gradient is:

Inequality (1.11) clearly shows that the entropy production is positive semi-definite. Multiplying Equations (1.1) by v and using the orthogonality condition (1.6) we get:

The positive semi-definiteness of the rate of working of the external forces is a consequence of the orthogonality of the internal, thermodynamic and external force fields. Inequality (1.12), in turn, implies that in all thermodynamically stable systems (i.e., those in which the phenomenological transport matrix is positive definite):

Inequality (1.13) attests to the fact that the RO-dissipation function is greater (or at least equal to) half the rate of production of entropy; the equality sign applies solely to isolated systems.

Finally, adding (1.11) and (1.12) results in:

where Ψ is known as the ‘generating’ function, defined by;

The second equality follows from the orthogonality condition (1.6). Upon comparing the thermodynamic power relations (1.7) and (1.14), we conclude that the two dissipation functions, Φ and Ψ, are numerically equal. But whereas Ψ is a function of state, the numerically equal dissipation function, Φ, is a function of its rate of change. Although this is what irreversible thermodynamics predicts, we will find that it is, in general, not true in the presence of fluctuations.

1.4 Nonequilibrium thermodynamic variational principles

Dissipative forces have a rather awkward role in classical mechanical variational principles in that they have to be appended to the Euler–Lagrange equations. In nonequilibrium thermodynamics they have a much more respectable role and analogous to the classical mechanical principle of least action, irreversible processes are governed by the principle of least dissipation of energy. The dissipation function was first introduced by Rayleigh in 1873 and formulated as a minimum principle in hydrodynamics by him in 1913. Its generalization to irreversible processes was accomplished by Onsager in 1931 which we later showed to be a constrained variational principle. However, the form which will be most convenient for our purposes bears a strong similarity to Gauss’ principle.

Gauss’ principle is used to determine the error between a hypothetical value of a function and its observed value in perfect analogy with the method of ‘least squares’. Consider the quadratic form:

which we will refer to as a ‘thermodynamic’ Lagrangian. Expression (1.16) is a measure of the ‘error’ committed by an irreversible process in choosing a path other than that predicted by the phenomenological relations (1.4). Expanding the quadratic form (1.16) and identifying the separate terms with the generalized thermodynamic potentials (1.3), (1.8), (1.9), and (1.15) we obtain:

which is positive semi-definite by definition. Like the RO-dissipation function, L(x) is (strictly) convex since the generalized resistance matrix R is positive definite; that is, the quadratic form

for an arbitrary real vector y. Actually, condition (1.18) is a necessary but not a sufficient condition for the thermodynamic Lagrangian to be a minimum. This is to say that a necessary condition for the thermodynamic action functional

subject to fixed endpoints, to have a minimum for the curve x = x(t) is that Legendre’s inequality

must be satisfied at every point along