Probability, Typicality and Emergence in Statistical Mechanics
Résumé
The relevance of probability theory is obvious in a subject called “Statistical Mechanics” (SM). On the other hand, SM arose as a microscopic description of single objects made of many (invisible) parts, thus justifying from an atomistic point of view the laws of thermodynamics. As a matter of fact, experimental measurements of thermodynamic quantities are conducted on a single system of interest, hence a fundamental problem arises in connecting probabilistic computations, e.g. the averages over ensembles of identical objects, with experiments. One of the most evident aspects of macroscopic phenomena is that they are characterized by a clear trend in time, that cannot be reverted. On the other hand, our understanding of microscopic dynamics is that they are reversible in time. With the aid of analytical computations on stochastic systems, and of numerical simulations of deterministic Hamiltonian systems, we illustrate basic features of macroscopic irreversibility, thus of the microscopic foundations of the second principle of thermodynamics, along the lines of Boltzmann’s kinetic theory. It will be evidenced that in systems characterized by a very large number of degrees of freedom, irreversibility concerns single realizations of the evolution processes, in the sense of the vast majority of the far-from-equilibrium initial conditions. That the vast majority out of a collection of realizations of a given process shares certain properties is often referred to as typicality.