With the aid of simple analytical computations for the Ehrenfest model, we clarify some basic features of macroscopic irreversibility. The stochastic character of the model allows us to give a nonambiguous interpretation of the general idea that irreversibility is a typical property: for the vast majority of the realizations of the stochastic process, a single trajectory of a macroscopic observable behaves irreversibly, remaining "very close" to the deterministic evolution of its ensemble average, which can be computed using probability theory. The validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit.
I. INTRODUCTIONUnderstanding the irreversibility from first principles is an old and noble problem of Physics. The technical reason of its difficulty is rather clear: on the one hand, the microscopic world is ruled by laws (Hamilton equations) which are invariant under the transformation of time reversal (t → −t , q → q , p → −p, being q and p the positions and momenta of the system); the macroscopic world, on the other hand, is described by irreversible equations, e.g. the Fick equation for the diffusion [1][2][3][4].How is it possible to conciliate the two above facts? On this topic there is an aged debate which started with the celebrated Boltzmann's H theorem and the well known criticisms by Loschmidt (about reversibility) and Zermelo (about recurrency). Of course we cannot enter in a detailed discussion about this fascinating chapter of statistical mechanics [2,5]. Already Boltzmann and Smoluchowski understood that the criticism by Zermelo is not a real serious problem as long as macroscopic systems are considered: basically, because of the Kac's lemma, in macroscopic systems the recurrence time is so large that it cannot be observed [2,6,7]. We can summarise the conclusions of Boltzmann by saying that the irreversibility describes an empirical regularity of macroscopic objects which is valid for a "vast majority" of the possible initial conditions. Often such validity for the "vast majority" of initial conditions is called typicality. According to Lebowitz [8] (as well as many others) a certain behavior is typical if the set of microscopic states for which it occurs comprises a region whose volume fraction goes to one as the number of molecules N grows. We can state that irreversibility is an emergent property [5,6,9] which appears as the number of degrees of freedom becomes (sufficiently) large; in such a limit, a single observation of the system is enough to determine its macroscopic properties.Several mathematical results, as well as detailed numerical simulations, support the coherence of the scenario proposed by Boltzmann [2]. Among the others, Lanford's work about the rarified gases is particularly important: he was able to prove, in a rigorous fashion, the validity of the Boltzmann equation for short times (of the order of the collision time) in the so-called Boltzmann-Grad limit [10].In spite of the ...