Statistical Mechanics

Aspects of Ergodic, Qualitative and Statistical Theory of Motion

“…Among these we mention those of Boltzmann, Gibbs, Maxwell; see [Bo09], [Bo02], [Bo03], [Gi60], [Ma65] Among the works of critique on the foundations see for instance Ehrenfest and Ehrenfest, [EE59], and Krylov, [Kr79]. More recent discussions can be found in [Br99], [Ga00]. The life and scientific achievements of Boltzmann are molded together in the books [Ce99], [Li01]: the first being more informative on the scientific aspects of Boltzmann's figure while the second gives a very clear picture of the aspects of his personality that made his life look to himself rather unhappy.…”

“…The methods are based on series expansions and on the use of recurrence relations to put bounds on their addends: they apply really quite generally for Gibbs states on one-dimensional lattices but they can be extended to Gibbs states on lattices of dimension higher than 1 only at the cost of severe restrictions. We devote the present chapter to a discussion of such methods and to an attempt to present various results from the unified viewpoint which is the polymer theory and the associated cluster expansion, a more modern term to indicate a method based on equations well known in Statistical Mechanics under the name of Kirkwood-Salsburg equations and Mayer-Montroll equations, see [Ga00] and appendix to the following Section §(7.2).…”

“…of the Kirkwood-Salsburg equations in statistical mechanics, [Ga00]. Note that if Ψ(∅) = 0 and Ψ(Γ) = 0 for Γ!…”

“…For instance this happens if in the examples studied in Section §(7.2) one replaces the potential Φ with βΦ with β a parameter. A paradigmatic example, not discussed here, has been a case in which the polymers γ are the vertices of closed contours, joining points on the lattice Z 2 , of length |γ| and ζ(γ) = e −β|γ| : the latter problem arises in the Ising model theory as developed in [MS67], see [Ga00] and [GMM73].…”

Probability as typicality

“…It can be seen, even with all its restrictions, as the realization, after nearly a hundred years, of Boltzmann's intuitions as made precise by Hilbert, Grad, Morrey and many others. According to Gallavotti (1999), "this is an important confirmation, mathematically rigorous, of Boltzmann's point of view according to which reversibility, and the corresponding recurrence times, is not in contradiction with the experimental observation of irreversibility" (p. 35).…”