There are two sorts of nonequilibrium thermodynamics: easy and hard. The (comparatively) easy cases are the ones in systems which are, in a sense, close to equilibrium. (I'll explain the sense of distance below.) Then there is a well-developed and experimentally validated theory, called ``linear'' or ``irreversible'' thermodynamics, which was founded by Lars Onsager in the 1920s and 1930s, and attained its definitive form in the hands of the Brussels School in the '40s and '50s. That leaves the hard stuff, which also comes in two varieties. One is to develop, not just a nonequilibrium thermodynamics, but a nonequilibrium statistical mechanics, i.e., to connect the theory of macroscopic variables to atomic-level, Hamiltonian dynamics in phase space. This is a fascinating and important but horridly difficult subject, on which remarkable progress has been made in recent years. It is also one on which the author of the book under review frankly punts, and I shall follow his wise example. This book rather is about thermodynamics --- about extensive quantities which are averages over many molecules --- in situations far from equilibrium, where the standard Onsager theory does not apply.

The first chapter opens with preliminaries about ensembles, statistical descriptions, and, rather more importantly, about stochastic processes. It is a remarkable fact that one can go all the way through a quite competent graduate-level course in statistical mechanics and never encounter a coherent definition of a stochastic process, much less learn how to solve a stochastic differential equation. Keizer warms up with the Wiener process, and then gets to the tools he will use over and over, the Fokker-Planck and Langevin equations, and the (Itô) stochastic calculus.

Chapter 2 surveys the classical approaches to nonequilibrium phenomena, i.e., things the reader is likely to have run across in an earlier course. On the one hand, there is, as mentioned, Onsager's theory of irreversible processes and fluctuations about equilibrium, which is very powerful and elegant, but only valid in the ``linear regime,'' i.e., when a linear expansion about the equilibrium point is a good approximation. More exactly, the (mean) time derivatives of the extensive variables are assumed to be linear combinations of the deviations from equilibrium of the intensive variables. When this is valid, various quite powerful results can be derived from the time-reversibility of the microscopic equations of motion. Apart from that last, the Onsager theory is purely thermodynamic, i.e., it deals with extensive quantities which are summed over the entire system of interest. The other classical approach to nonequilibrium phenomena is the Boltzmann equation. This is of course much older than Onsager's work, and is even quite nonlinear, and in principle doesn't care about the distance from equilibrium. Unfortunately, it is also entirely about the mean behavior of the ensemble (it ignores fluctuations), and valid only for dilute fluids.

The Boltzmann equation is at the lowest level of description which Keizer uses. Like the Onsager theory, it is also about the rates of change of extensive variables, but now those are the number of molecules in small volumes of the single-molecule phase space. (Dealing with densities in the whole-system phase space would put us in statistical mechanics proper.) Intermediate between this and the thermodynamic level is the hydrodynamic level of description, briefly touched upon in this chapter, which uses the thermodynamic variables in many small cells of physical space, passing in the limit to the density of extensive variables.

Chapter three takes the Boltzmann equation and shows how it may be extended to include fluctuations; the equations of chemical kinetics are then similarly fuzzed up. These exercises are warm-ups for chapter four, which presents the ``mechanistic statistical theory of nonequilibrium thermodynamics,'' which Keizer also calls the ``canonical theory''. The canonical theory assumes that we care about a set of extensive variables, whose values are altered by a number of distinct ``elementary processes,'' and, possibly, externally-imposed fluxes. Each time a particular elementary process takes place, it always changes the extensive variables by the same absolute amount. There is also a reverse elementary process, which changes the variables by the opposite amounts. The probability per unit time that any elementary process takes place is a function of the current values of the intensive variables alone, and Keizer postulates a certain form for this function. (The rates of the forward and reverse directions for an elementary process need not be equal; when those rates are equal for all processes separately, we have equilibrium.) In continuous time, this leads to a deterministic differential equation for the evolution of the ensemble mean, and a Fokker-Planck equation, with state-dependent noise, for the fluctuations about the mean. In the thermodynamic limit, it gives a generalization of the fluctuation-dissipation theorem, reducing to the usual form in the vicinity of equilibrium, i.e., a stationary Gaussian Markov process.

Chapter five applies the canonical theory to various chemical and electrochemical processes. There is a detailed comparison of a model based on the formalism to actual experimental data for a calcium-regulated potassium channel in muscle cells, yielding remarkably close agreement (especially since the channel is really just a single molecule!). Chapter six continues the applications to the hydrodynamic level of description, i.e., to local densities of extensive variables. There is a nice treatment here of how molecular fluctuations can be measured by light- and neutron- scattering experiments, whose results, in turn, can only be explained by taking account of those fluctuations.

Chapters seven and eight describe nonequilibrium steady states (= fixed points), their stability and the general character of fluctuations about them. Since none of the proposed ``universal evolution criteria'' for determining the stability of steady states actually works (as shown by Rolf Landauer, Keizer, and Keizer's long-time collaborator Ronald Fox), we must fall back on linear stability analysis. But Keizer does shows that stability is related to the covariance matrix of the fluctuations: a steady state is asymptotically stable if and only if the covariance matrix approaches a (finite) limit as time goes to infinity. In this case, fluctuations about the steady state generally look like fluctuations about equilibrium, i.e., they form a stationary Gaussian Markov process. The covariance matrix acts like the matrix of second partial derivatives of the entropy in equilibrium, and from it Keizer constructs a ``generalized entropy,'' and indeed a whole slew of ``generalized'' quantities analogous to those of thermodynamics, e.g., temperatures and free energies. All of this only works away from critical points; some consideration is given to what to do in their vicinity, and there's a nice discussion of critical slowing-down.

Chapter nine discusses how to relate different levels of description, mostly
by looking at how to derive coefficients of higher-level descriptions from
lower-level models. There is, for instance, a detailed calculation of how the
viscosity and heat-transport coefficients of a fluid can be gotten from
Boltzmann-level information about molecular collisions. There's a lot on what
tricks you can use when some processes are much faster than those you care
about, but curiously little on how to handle processes *slower* than
your favorites.

The last chapter looks, very briefly, at situations other than stable
states: transients, periodic trajectories, and chaos. Most of this is actually
taken up with discussing some basics of nonlinear dynamics, and affirming, what
is almost certainly true, that the canonical formalism applies *in
principle* to these problems. The difficulty is that, since even the mean
behavior is strongly nonlinear, and fluctuations are coupled to the mean
trajectory, very little can be said, analytically, about the fluctuations, and
in general one will have to resort to simulation and numerical integration of
the relevant Fokker-Planck equations. The discussion of the relative roles of
noise due to variability in the ensemble of initial conditions and noise due to
fluctuations is particularly nice. (Some of
Keizer's later papers address these issues.)

Overall, the development is admirable. Theoretical postulates are
explicitly stated and precise; the reasoning is clear, well-motivated, and
builds progressively; approximations, fudges, and what little speculation there
is are all said to be such. As I've indicated, there are a great number of
examples, many of which involve actual experimental data which are compared to
models! (These virtues, especially the last named, are *not* shared by
a distressingly large fraction of other books on nonequilibrium statistical
physics. Offending titles will be listed upon request.) Keizer does have the
habit of interrupting a physical derivation to prove a convenient bit of math,
and it would be nice if these asides were more clearly distinguished for those
of us who are willing to take his algebra on trust. There are, I am sorry to
say, numerous mis-prints in the equations --- I noticed them particularly in
chs. 2, 3, and 4, but unfortunately didn't note them all down as I read.
Keizer writes the probability of the event *A* conditional on the event
*B* as *P*(*B*|*A*), whereas every other author
known to me writes *P*(*A*|*B*). The reason for this
perversion is never explained.

A solid grounding in equilibrium statistical mechanics is, as I said at the beginning, essential to following this book. The high price is prohibitive for individuals (though good-condition used copies seem to run from $40 to $60), but it definitely belongs in libraries. There are no problems, but filling in the details in the proofs and examples seems to do admirably. This book is of great interest to statistical physicists, physical chemists, biophysicists and materials scientists, and anyone else interested in matter far from equilibrium.

Keizer was, until his premature death in May, 1999, an active and talented scientist who played a significant role not merely in the development of the formal structure of far from equilibrium thermodynamics, but also in its application to experiment, especially in biology. Unlike a number of others who have attempted such cross-overs, he made it work, and a volume entitled Joel Keizer's Computational Cell Biology is supposed to be published before the end of 2000; I look forward to it eagerly.

xiv + 506 pp., 59 figures, selected references at the end of each chapter, index of names and subjects (partially analytic)

Condensed Matter / Hydrodynamics / Physics

Currently in print as a hardback, ISBN 0-387-96501-7 (or ISBN 3-540-96501-7), US$140. LoC QC318.I7 K 42

5 July 2000 (typo fixed 7 July)