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STATISTICAL MECHANICS: RIGOROUS RESULTS
This classic book marks the beginning of an era of vigorous mathematical progress in equilibrium statistical mechanics. Its treatment of the infinite system limit has not been superseded, and the discussion of thermodynamic functions and states remains basic for more recent work. The conceptual foundation provided by the Rigorous Results remains invaluable for the study of the spectacular developments of statistical mechanics in the second half of the 20th century.
- GenresPhysics
236 pages, Hardcover
First published January 1, 1969
17 people want to read
About the author
David Ruelle
28 books15 followersDavid Pierre Ruelle is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term strange attractor, and developed a new theory of turbulence.
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September 19, 2023
David Ruelle’s Statistical Mechanics: Rigorous Results (Imperial College Press, originally published in 1969) will be ideal for anyone who wants a tour of the current state of the art, uncompromising in its rigor. Beginning graduate students weaned on an old standby like Pathria may feel a certain unease when encountering thermodynamics in typical physicist’s fashion, where functions are frequently invoked whose very existence rests upon a statement to the effect that physical intuition suggests that there ought to be some sufficient condition or other guaranteeing that it is well defined (but we aren’t going to investigate these at all). If so, know that in fact there is probably no field of physics for which a more secure foundation exists than statistical mechanics (classical or quantum)! We have previously reviewed Walter Thirring’s excellent sketch of the elements of the subject from a rigorous point of view in the fourth volume of his Course in Mathematical Physics (see our review here), but indicated there that one might want to turn to Ruelle for more depth and breadth.
Does Ruelle deliver on his promise? One notices right away that he proves many of the same results as Thirring but in greater detail. Ruelle has a more careful accounting of the thermodynamic limit (existence of intensive state functions, classical case as well as quantum). For instance, he specifies precise notions for how the region Λ tends to infinity (roughly its volume tends to infinity while the ratio of perimeter to volume tends to zero). Other topics not to be found in Thirring at all include a nice representation of the Kirkwood-Salsburg equation in a Banach space to yield an operatorial solution and the infinite-volume limit; lattice gases; Mayer series and virial expansion to show absence of phase transitions at low density resp. activity; the cluster property and convergence of correlation functions; Lee-Yang’s circle theorem showing absence of zeros of the partition function; Peierls approach to the Ising model; group invariance of states.
Chapter seven does contain examples (classical lattice, classical continuous systems, quantum lattice systems, quantum continuous system, whether bosons or fermions). Aside from deriving the canonical commutation relations, in §7.2 he builds entropy of a classical or quantum lattice directly (compare with Peter Waters’ Introduction to Ergoic Theory; here a lot of rigamarole in defining the topological pressure can be avoided since one has an obvious measure available). Then a few of its properties are proved (convexity, upper semicontinuity, maximality in equilibrium, Gibbs phase rule). Chapter seven ends with the Kubo-Martin-Schwinger condition though, unlike Thirring, Ruelle doesn’t explain it physically or give any models.
The present text does suffer from one major omission: it does not cover the van der Waals fluid at all, the sole example to date of a phase transition in a system with continuous degrees of freedom [J. Lebowitz and O. Penrose, Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition, J. Math. Phys. 7, 98-113 (1966)]. Readers interested in a textbook presentation may turn to Radu Balescu’s Equilibrium and non-equilibrium statistical mechanics (Krieger, 1991).
By chapter two, one will be convinced that Ruelle’s proofs can be elegant, if demanding to follow. Yet the whole thing will seem like a lot of pushing of complicated formalism – it isn’t clear that it says anything deep for no realistic models are constructed and investigated and there is no confrontation with experiment, either. The subject of statistical mechanics as conveyed here then seems dissatisfying in that there remains a disproportion between the amount of expenditure of technical effort and the scope of the results obtained: apart from the Ising and related lattice models, there is hardly any adequate theory of phase transitions for realistic matter. Also mastering these techniques, while a challenging test of one’s skill at real analysis, seems unlikely to teach one much that would be applicable to other domains so one really has to be devoted to put in the time. Thus, Ruelle comes across as ornate and overrefined; probably a better use of one’s scientific energies would be to study irreversible processes which are of course what one finds everywhere in nature anyway.
Comparison of Ruelle with Thirring: Thirring, though terse is also clearer in what he does say while Ruelle leaves the formulae to speak for themselves. Thirring always defines and discusses his terms (such as relative boundedness of Kato or Friedrichs extension) while Ruelle just invokes them and expects one to look them up elsewhere. On the other hand, Ruelle is also more technically precise, as when he outlines a class of potentials including pair interactions, triple interactions up to many-body terms of any order and gives a temperedness condition which he proves to be sufficient for thermodynamic stability. Thus, one should read Thirring for a presentation of thermodynamics as a systematic science in order to learn the subject, but go to Ruelle to fill in the technical details (albeit with little insight into the physics).
A last note on Ruelle’s style versus Ludwig Boltzmann’s: the former consists in dry and exacting mathematics concerned to dot every i and cross every t in a fairly standard repertoire of already established models; the latter more free-wheeling, consisting in heuristic calculations seeking to delineate fundamental concepts by solving specific illustrative problems but never forgetting its starting point. One could say Boltzmann’s genius is more suited to the original exploration of the field of kinetic theory, Ruelle’s to its terminal decline!
Does Ruelle deliver on his promise? One notices right away that he proves many of the same results as Thirring but in greater detail. Ruelle has a more careful accounting of the thermodynamic limit (existence of intensive state functions, classical case as well as quantum). For instance, he specifies precise notions for how the region Λ tends to infinity (roughly its volume tends to infinity while the ratio of perimeter to volume tends to zero). Other topics not to be found in Thirring at all include a nice representation of the Kirkwood-Salsburg equation in a Banach space to yield an operatorial solution and the infinite-volume limit; lattice gases; Mayer series and virial expansion to show absence of phase transitions at low density resp. activity; the cluster property and convergence of correlation functions; Lee-Yang’s circle theorem showing absence of zeros of the partition function; Peierls approach to the Ising model; group invariance of states.
Chapter seven does contain examples (classical lattice, classical continuous systems, quantum lattice systems, quantum continuous system, whether bosons or fermions). Aside from deriving the canonical commutation relations, in §7.2 he builds entropy of a classical or quantum lattice directly (compare with Peter Waters’ Introduction to Ergoic Theory; here a lot of rigamarole in defining the topological pressure can be avoided since one has an obvious measure available). Then a few of its properties are proved (convexity, upper semicontinuity, maximality in equilibrium, Gibbs phase rule). Chapter seven ends with the Kubo-Martin-Schwinger condition though, unlike Thirring, Ruelle doesn’t explain it physically or give any models.
The present text does suffer from one major omission: it does not cover the van der Waals fluid at all, the sole example to date of a phase transition in a system with continuous degrees of freedom [J. Lebowitz and O. Penrose, Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition, J. Math. Phys. 7, 98-113 (1966)]. Readers interested in a textbook presentation may turn to Radu Balescu’s Equilibrium and non-equilibrium statistical mechanics (Krieger, 1991).
By chapter two, one will be convinced that Ruelle’s proofs can be elegant, if demanding to follow. Yet the whole thing will seem like a lot of pushing of complicated formalism – it isn’t clear that it says anything deep for no realistic models are constructed and investigated and there is no confrontation with experiment, either. The subject of statistical mechanics as conveyed here then seems dissatisfying in that there remains a disproportion between the amount of expenditure of technical effort and the scope of the results obtained: apart from the Ising and related lattice models, there is hardly any adequate theory of phase transitions for realistic matter. Also mastering these techniques, while a challenging test of one’s skill at real analysis, seems unlikely to teach one much that would be applicable to other domains so one really has to be devoted to put in the time. Thus, Ruelle comes across as ornate and overrefined; probably a better use of one’s scientific energies would be to study irreversible processes which are of course what one finds everywhere in nature anyway.
Comparison of Ruelle with Thirring: Thirring, though terse is also clearer in what he does say while Ruelle leaves the formulae to speak for themselves. Thirring always defines and discusses his terms (such as relative boundedness of Kato or Friedrichs extension) while Ruelle just invokes them and expects one to look them up elsewhere. On the other hand, Ruelle is also more technically precise, as when he outlines a class of potentials including pair interactions, triple interactions up to many-body terms of any order and gives a temperedness condition which he proves to be sufficient for thermodynamic stability. Thus, one should read Thirring for a presentation of thermodynamics as a systematic science in order to learn the subject, but go to Ruelle to fill in the technical details (albeit with little insight into the physics).
A last note on Ruelle’s style versus Ludwig Boltzmann’s: the former consists in dry and exacting mathematics concerned to dot every i and cross every t in a fairly standard repertoire of already established models; the latter more free-wheeling, consisting in heuristic calculations seeking to delineate fundamental concepts by solving specific illustrative problems but never forgetting its starting point. One could say Boltzmann’s genius is more suited to the original exploration of the field of kinetic theory, Ruelle’s to its terminal decline!
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