The purpose of this chapter is to convey that certain aspects of the macroscopic theory called thermodynamics do admit simple statistical models that point towards a microscopic, mechanical theory; one of the first successful examples of the latter was [Boltzmann, 1896–1898] kinetic theory of gases, which we review and assess in Sect. 3.3. Before we do that, however, we present — in Sect. 3.1 — a simple stochastic model for diffusion, and — in Sect. 3.2 — the derivation of Maxwell’s equilibrium velocity distribution in a gas. Finally, we try and capture in Sect. 3.4 the light thrown by the Ehrenfest urn model — affectionately known as the “dog-flea” model — on the controversies raised by Boltzmann’s work. The common language of routine probability praxis is used throughout this chapter, as is the underlying concept of randomness. An approach, more careful with conceptual matters, is presented in the following chapters; the present chapter is therefore to be considered as a motivational primer. It also shows a procession of models — from simple and frankly unrealistic to somewhat more complex and realistic, and back — which gives illustrations to the process of model building in the search for a microscopic theory of macroscopic phenomena.
KeywordsBoltzmann Equation Diffusion Equation Kinetic Theory Maxwell Distribution Recurrence Time
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