Statistical Mechanics: The Basics

This chapter introduces the reader to the basics of statistical mechanics. Gibbsian and neo-Boltzmannian approaches are outlined. It includes a statistical-mechanical analogue of the second law of thermodynamics, and a proof of the Poincaré recurrence theorem. It is argued that the differences between Gibbsian and neo-Boltzmannian approaches have been exaggerated.

Strong Poincaré recurrence theorem in MV-algebras

The classical Poincaré strong recurrence theorem states that for any probability space (Ω, ℒ, P), any P-measure preserving transformation T, and any A ∈ ℒ, almost all points of A return to A infinitely many times. In the present paper the Poincaré theorem is proved when the σ-algebra ℒ is substituted by an MV-algebra of a special type. Another approach is used in [RIEČAN, B.: Poincaré recurrence theorem in MV-algebras. In: Proc. IFSA-EUSFLAT 2009 (To appear)], where the weak variant of the theorem is proved, of course, for arbitrary MV-algebras. Such generalizations were already done in the literature, e.g. for quantum logic, see [DVUREČENSKIJ, A.: On some properties of transformations of a logic, Math. Slovaca 26 (1976), 131–137.