##
**Chaos, scattering and statistical mechanics.**
*(English)*
Zbl 0915.00011

Cambridge Nonlinear Science Series. 9. Cambridge: Cambridge University Press. xx, 475 p. (1998).

To write a concise, comprehensive and in a sense intelligent book on such complicated matters as dynamical systems theory, chaotic scattering and even nonequilibrium statistical mechanics in connection with a dynamical chaos background seems to be a very hard task, requiring the very experienced author. I remember very well a statement made by the late Ryogo Kubo at a conference held in Kyoto: “The problem of ergodic hypotheses is so single and difficult I have never put it as the thesis problem for my graduate students.” The problem with the reviewed book “Chaos, Scattering and Statistical Mechanics” falls into the same category. To read the book smoothly is almost impossible. But the reason for this is not a complicated mathematical formalism but a rather semirigorous style of presentation, misunderstanding of basic notions and often misleadings in making false consequences. So on the second to the last page of the book in Chapter 10, Conclusions and Perspectives, one reads: “Many results of the present work rest on the assumption of a microscopic chaos. The experimental test of this chaotic hypothesis by the observation of the dynamical randomness of Brownian motion as we explain in Chapter 9 is of particular importance.”

Alas, the reader learns only at the end of book what was the cardinal assumption, even worse, without a definition of microscopic chaos. To be precise let us mention even the “chaos” used as the first word in the title of the book is not defined in the whole book. So it is funny and very misleading that the author exploits “topological chaos” (the title of Chapter 2), “probabilistic chaos” (the title of Chapter 4), “noises as microscopic chaos” (the title of Chapter 9) in an, unfortunately, inappropriate way. The point is one can have an ad hoc random process, e.g., a coloured noise with some dynamic properties formally equivalent to those of deterministic chaos, as Lyapunov exponents, Kolmogorov-Sinai entropy, and so on. Then a question is what is the book in fact about? For sure it is not about the deterministic chaos in dissipative dynamic systems but probably and mostly about a stochasticity of Hamiltonian conservative (?), hyperbolic dynamic systems?

Coming back to the above cited sentence of the author, it is not clear at all what is meant by a microscopic chaos in so far as the term of “molecular chaos” has been used as the basic assumption for stochastic statistical considerations. The author then makes a jump to the “chaotic hypothesis”, the notion used in the book in a very different sense when the assumption of the hyperbolicity in some dynamic systems is ad hoc made what has been proclaimed by Cohen and Gallavotti. Going on further, the author’s “reasoning” culminates in requiring the experimental test of this chaotic hypothesis “by the observation of the dynamical randomness of Brownian motion…”. This seems to be a pure speculative nonsense without any founding when the author mixes a kinetic description with the dynamic one, and vice versa.

So far we have analyzed only two sentences-assertions of the reviewed book. But the book is fat (475+xviii pages), compiled of ten very heterogeneously ordered chapters. Here one has the famous case of when a precise review of the book would require writing at least two new, even thicker books to explain misconceptions and potential misleadings. To be fair, here are the titles of the chapters of the book: Dynamical systems and their linear stability; Topological chaos; Liouville dynamics; Probabilistic chaos; Chaotic scattering; Scattering theory of transport; Hydrodynamics modes of diffusion; Systems maintained out of equilibrium; Noises as microscopic chaos and finally Conclusions and perspectives.

Here one can apply the saying that very often a little means more. So omitting chapters 1, 2, 3, 4, 7, 9, 10 and writing a book on “dynamic scattering and transport” with rigorously introduced technicalities needed would be of higher value. The reviewed book is based on many strange assumptions, hypotheses and, as the author states in the Preface, “on lectures given in the Physics Department of the Université Libre de Bruxelles during the years 1991-92 and 1992-93”. Strangely enough, the book was published in 1998. Let us add that the Cambridge UP delivered technically a perfect piece of work. Based on what has been said before, one cannot recommend the reviewed book to a common reader, nor to professionals in subject fields of research and has to warn students of possible misleadings when reading the book.

Alas, the reader learns only at the end of book what was the cardinal assumption, even worse, without a definition of microscopic chaos. To be precise let us mention even the “chaos” used as the first word in the title of the book is not defined in the whole book. So it is funny and very misleading that the author exploits “topological chaos” (the title of Chapter 2), “probabilistic chaos” (the title of Chapter 4), “noises as microscopic chaos” (the title of Chapter 9) in an, unfortunately, inappropriate way. The point is one can have an ad hoc random process, e.g., a coloured noise with some dynamic properties formally equivalent to those of deterministic chaos, as Lyapunov exponents, Kolmogorov-Sinai entropy, and so on. Then a question is what is the book in fact about? For sure it is not about the deterministic chaos in dissipative dynamic systems but probably and mostly about a stochasticity of Hamiltonian conservative (?), hyperbolic dynamic systems?

Coming back to the above cited sentence of the author, it is not clear at all what is meant by a microscopic chaos in so far as the term of “molecular chaos” has been used as the basic assumption for stochastic statistical considerations. The author then makes a jump to the “chaotic hypothesis”, the notion used in the book in a very different sense when the assumption of the hyperbolicity in some dynamic systems is ad hoc made what has been proclaimed by Cohen and Gallavotti. Going on further, the author’s “reasoning” culminates in requiring the experimental test of this chaotic hypothesis “by the observation of the dynamical randomness of Brownian motion…”. This seems to be a pure speculative nonsense without any founding when the author mixes a kinetic description with the dynamic one, and vice versa.

So far we have analyzed only two sentences-assertions of the reviewed book. But the book is fat (475+xviii pages), compiled of ten very heterogeneously ordered chapters. Here one has the famous case of when a precise review of the book would require writing at least two new, even thicker books to explain misconceptions and potential misleadings. To be fair, here are the titles of the chapters of the book: Dynamical systems and their linear stability; Topological chaos; Liouville dynamics; Probabilistic chaos; Chaotic scattering; Scattering theory of transport; Hydrodynamics modes of diffusion; Systems maintained out of equilibrium; Noises as microscopic chaos and finally Conclusions and perspectives.

Here one can apply the saying that very often a little means more. So omitting chapters 1, 2, 3, 4, 7, 9, 10 and writing a book on “dynamic scattering and transport” with rigorously introduced technicalities needed would be of higher value. The reviewed book is based on many strange assumptions, hypotheses and, as the author states in the Preface, “on lectures given in the Physics Department of the Université Libre de Bruxelles during the years 1991-92 and 1992-93”. Strangely enough, the book was published in 1998. Let us add that the Cambridge UP delivered technically a perfect piece of work. Based on what has been said before, one cannot recommend the reviewed book to a common reader, nor to professionals in subject fields of research and has to warn students of possible misleadings when reading the book.

Reviewer: L.Andrey (Praha)

### MSC:

00A79 | Physics |

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

81U99 | Quantum scattering theory |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

37D99 | Dynamical systems with hyperbolic behavior |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

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