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**An introduction to chaos in nonequilibrium statistical mechanics.**
*(English)*
Zbl 0973.82001

Cambridge Lecture Notes in Physics. 14. Cambridge: Cambridge University Press. xiv, 287 p. (1999).

There have been attempts to make a reformulation of nonequilibrium statistical mechanics on the basis of deterministic chaos dynamics in the last years (e.g., Andrey, Cohen, Gallavotti, Ruelle). But it seems the problem is very complex and only little steps have been made so far. The reviewed book started its existence as a set of lecture notes based on a series of lectures given to fourth year students at the University of Utrecht during the spring semester of 1994. That’s important as a new reformulation of nonequilibrium statistical mechanics began in 1995. But the author tried to fill up the gap in covering the subject matter till 1997 (the book was published in 1999). In fact the book mirrors the author’s fighting with the hard subject.

The book is divided into 19 chapters of very heterogeneous character. Alas, there are some troubles with a classification of basic problems, directions and even notions. The Boltzmann equation (Ch. 2, 7, 17, 18) which is commonly classified as a basis of kinetic theory is mixed with the Liouville’s equation (Ch. 3) which is standardly used as a basis for equilibrium statistical mechanics in the case of Hamiltonian systems. The connection between mixing and chaos (Ch. 5) is not clear. The problem is that the notion of dynamic (deterministic) chaos itself is not rigorously introduced in the book. Then a jump is made to the linear response theory (Ch. 6, 14) with van Kampen’s objections. The measure-preserving baker’s transformation (map) is used as “an example of great illustrative value for the applications of chaos theory to statistical mechanics” (Ch. 7, 8) which is very questionable as such. There are some unprecise (mathematically) statements in chapters on Kolmogorov-Sinai entropy (Ch. 9) and the Frobenius-Perron equation (Ch. 10). In Ch. 11, titled open systems and escape rate, paradoxically the Smale horshoe is presented. According to the author “Chapters 7 through 18 of the book discuss current research literature on transport theory in general, and nonequilibrium steady states, in particular”. But one can strongly object to this statement as it seems that only Ch. 11 and 12 are devoted to this matter.

The book contains also chapters on Boltzmann’s ergodic hypothesis (Ch. 4), Sinai-Ruelle-Bowen (SRB) and Gibbs measures (Ch. 13), and unstable periodic orbits (Ch. 15).

The author is aware of weak points in the book. He tells us that “he has not made any attempts at mathematical rigor, uses terms in a way that differ from the usual usage in the mathematical literature”. Besides he admits “the idea of associating transport with chaotic motion is not complete, and even there are cases, where this picture appears to break down”.

Also the use of baker’s transformation in only two dimensions cannot be generalized as a universal model for more general, higher dimensional systems. Even a connection of such simplified baker’s map with the Boltzmann’s equation is very questionable. And the author critically admits “the book says nothing about quantum versions of the classically chaotic dynamic systems”.

The last statement of the book concerning the quantum analoga of Lyapunov exponents, KS entropies, etc. is also not true. There have been published papers dealing with these problems not long ago.

There is a proverb: Sometimes it happens that a little is more than a lot. It seems it is appropriate to apply it to the content of the book. Even though each chapter has at the end a section “Further reading” the treatment seems to be on the surface of the problems. The bibliography is not complete and is not ordered alphabetically on the second letter, e.g., Artuse is before Alligood, and so on.

The book is divided into 19 chapters of very heterogeneous character. Alas, there are some troubles with a classification of basic problems, directions and even notions. The Boltzmann equation (Ch. 2, 7, 17, 18) which is commonly classified as a basis of kinetic theory is mixed with the Liouville’s equation (Ch. 3) which is standardly used as a basis for equilibrium statistical mechanics in the case of Hamiltonian systems. The connection between mixing and chaos (Ch. 5) is not clear. The problem is that the notion of dynamic (deterministic) chaos itself is not rigorously introduced in the book. Then a jump is made to the linear response theory (Ch. 6, 14) with van Kampen’s objections. The measure-preserving baker’s transformation (map) is used as “an example of great illustrative value for the applications of chaos theory to statistical mechanics” (Ch. 7, 8) which is very questionable as such. There are some unprecise (mathematically) statements in chapters on Kolmogorov-Sinai entropy (Ch. 9) and the Frobenius-Perron equation (Ch. 10). In Ch. 11, titled open systems and escape rate, paradoxically the Smale horshoe is presented. According to the author “Chapters 7 through 18 of the book discuss current research literature on transport theory in general, and nonequilibrium steady states, in particular”. But one can strongly object to this statement as it seems that only Ch. 11 and 12 are devoted to this matter.

The book contains also chapters on Boltzmann’s ergodic hypothesis (Ch. 4), Sinai-Ruelle-Bowen (SRB) and Gibbs measures (Ch. 13), and unstable periodic orbits (Ch. 15).

The author is aware of weak points in the book. He tells us that “he has not made any attempts at mathematical rigor, uses terms in a way that differ from the usual usage in the mathematical literature”. Besides he admits “the idea of associating transport with chaotic motion is not complete, and even there are cases, where this picture appears to break down”.

Also the use of baker’s transformation in only two dimensions cannot be generalized as a universal model for more general, higher dimensional systems. Even a connection of such simplified baker’s map with the Boltzmann’s equation is very questionable. And the author critically admits “the book says nothing about quantum versions of the classically chaotic dynamic systems”.

The last statement of the book concerning the quantum analoga of Lyapunov exponents, KS entropies, etc. is also not true. There have been published papers dealing with these problems not long ago.

There is a proverb: Sometimes it happens that a little is more than a lot. It seems it is appropriate to apply it to the content of the book. Even though each chapter has at the end a section “Further reading” the treatment seems to be on the surface of the problems. The bibliography is not complete and is not ordered alphabetically on the second letter, e.g., Artuse is before Alligood, and so on.

Reviewer: Ladislav Andrey (Praha)

### MSC:

82-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics |

82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37A60 | Dynamical aspects of statistical mechanics |

37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |