Classical typicality of the canonical distribution

¹  Instituto Carlos I de Física Teórica y Computacional, University of Granada - Granada, Spain, EU
²  National University La Plata, CREG-UNLP-CONICET - C.C. 727, 1900 La Plata, Argentina
³  Physics Department, University of Pretoria - Pretoria 0002, South Africa
⁴  Research Institute MOVE, Faculty of Human Movement Sciences, VU University Amsterdam van der Boechorststraat 9, 1081 BT, Amsterdam, The Netherlands, EU

Corresponding authors: arplastino@maple.up.ac.za marlow@fbw.vu.nl

Received: 9 July 2008
Accepted: 24 September 2008

Abstract

We consider the typicality of the canonical ensemble's probability distribution from a classical perspective, resuming recent discussions on quantum-mechanical aspects of canonical typicality. In the conventional derivation of the classical canonical distribution for a system S that is weakly coupled to a heat bath B, it is assumed that the composite S + B is represented by the microcanonical ensemble i.e., by a uniform probability distribution on an energy shell of the composite S + B. Here we show that for a very large heat bath almost all probability distributions defined on this energy shell behave according to the microcanonical ensemble, yielding a marginal probability distribution for S of the canonical form. Consequently, the classical canonical distribution can be regarded as much more “typical" than suggested by the standard derivation.