Carlo Cercignani's book is a detailed and well-written account of the life and science of Ludwig Boltzmann (1844-1906), emphasising his achievements in kinetic theory.

As is well known, Boltzmann was influenced by Darwin's theory of biological evolution: he hoped to become the Darwin of physics; to overthrow the deterministic view of Newton and to introduce an evolutionary description of nature. His scientific influence was enormous. For instance, the derivation of the basic law of black-body radiation by Planck was strongly influenced by Boltzmann's kinetic approach.

The author seems to accept the popular view that the arrow of time originated in the big bang, and this is also the view expressed by Roger Penrose in his foreword. If true, this would be very strange. In nature we find both reversible and irreversible processes, but they apply to different classes of dynamical systems, whose existence were discovered by Henri Poincare at the end of the 19th century. The problem is to find the relation between the class of dynamical system and the arrow of time.

Cercignani quotes "the remarkable recent book by the philosopher H. Price who defends the viewpoint that one must take the basic time symmetry much more seriously." Indeed, Price considers irreversibility as an artefact introduced by humans - which is the opposite of Boltzmann's view.

Does it mean that Cercignani himself is not convinced by Boltzmann's arguments? He adds a strange sentence: "Boltzmann saw ... correctly but we need modern research to understand correctly. And this modern research is not the still-scanty mathematical proofs of Boltzmann's physical arguments, but rather confusing arguments about chaotic phenomena."

Modern research is by no means limited to chaotic phenomena. On the contrary, it deals with many other aspects of kinetic theory. Cercignani's book ignores practically all relevant modern research, especially in kinetic theory, such as that presented by P. Resibois and M. de Leener in * Classical Kinetic Theory of Fluids * (1977) or J. R. Dorfman's, "Advances and challenges in the kinetic theory of gases" (1981).

One should never forget that Boltzmann's approach applies only to highly dilute systems. The great surprise is that the formulation of kinetic equations becomes quite different when the density is increased. As was shown in the early 1960s, the kinetic equation becomes then "non-Markovian", that is it contains memory effects.

Benno Alder's numerical simulations for hard spheres proved the existence of long tails, which is precisely a memory effect. In fact there are three time-scales for the evolution of the velocity distribution: a short time ("Zeno's time", corresponding roughly to the transition from dynamics to dissipation), the kinetic region corresponding essentially to an exponential behaviour in time, and the long-tail period which is described by polynomials.

In Cercignani's book, kinetic theory is presented as it was at the beginning of this century. There has been progress in this field over the past few decades, but the fundamental contributions of Kubo, Green or Van Hove are still simply ignored.

While Boltzmann's theory is indeed a monument to human creativity, one cannot say that he solved the problem of the relations between dynamics and probability. His famous formula describing the relationship of entropy to probability, S=k log P, (which was later to be engraved on his tombstone), is only valid for dilute gases; for other situations the meanings of S and P are hotly disputed.

The complication arises, again, from the "non-Markovian" character of the kinetic equations.

But the main point is, how can the transition from trajectories to probabilities be justified? In 1931, B. O. Koopman showed that the descriptions in terms of trajectories (which are point transformations) or in terms of probabilities are equivalent as long as we remain in the Hilbert space of nice functions, that is functions which behave like ordinary vectors with a "length" and for which one can define a scalar product. New features, including irreversibility, appear only when one extends the description to generalised functions or fractals such as the well-known delta function.

An example is deterministic chaos. For deterministic chaotic maps there exist two formulations of classical dynamics, one based on Newton's type of equations, which is deterministic and time-reversible, and the other in which irreversibility plays a fundamental role and is applicable if the initial conditions correspond to a finite measure in phase space. (This is the realistic case, since our measurements can never produce the infinite precision which would correspond to a point.) Why, therefore, does Cercignani speak of "rather confusing arguments about chaotic phenomena"? In my view, the theory of deterministic chaotic map is rigorous (see D. Driebe's "Time symmetry breaking and deterministic chaotic maps", 1998). However, this description involves fractals and we have to go outside the Hilbert space.

This book will be of interest to people who want to know more about the history of Boltzmann's pioneering work. However, the author expresses the hope that his book could be used as the basis for a course on statistical mechanics. This would be inappropriate as the book ignores all recent developments in this fascinating and difficult field.

Ilya Prigogine, Nobel laureate, is director, Centre for Statistical Mechanics and Complex Systems, University of Texas, Austin, United States.

## Ludwig Boltzmann: The Man Who Trusted Atoms

Author - Carlo Cercignani

ISBN - 0 19 850154 4

Publisher - Oxford University Press

Price - £29.50

Pages - 329

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