On the Foundation of Statistical Mechanics under Experimentally Realistic Conditions: A Comparison between the Quantum and the Classical Case

TL;DR

This paper compares quantum and classical statistical mechanics for isolated macroscopic systems, analyzing how they evolve towards equilibrium and whether this state aligns with the microcanonical ensemble, highlighting differences in assumptions needed for equilibration and thermalization.

Contribution

It provides a comparative analysis of quantum and classical approaches to equilibration and thermalization, emphasizing realistic modeling of measurements and initial conditions.

Findings

Quantum equilibration requires weak assumptions on measurements and initial states.

Classical equilibration demands stronger assumptions, but thermalization follows more easily.

The paper clarifies conditions under which systems reach and agree with statistical ensembles.

Abstract

Focusing on isolated macroscopic systems, described either in terms of a quantum mechanical or a classical model, our two key questions are: In how far does an initial ensemble (usually far from equilibrium and largely unknown in detail) evolve towards a stationary long-time behavior ("equilibration")? In how far is this steady state in agreement with the microcanonical ensemble as predicted by Statistical Mechanics ("thermalization")? In the first part of the paper, a recently developed quantum mechanical treatment of the problem is briefly summarized, putting particular emphasis on the realistic modeling of experimental measurements and non-equilibrium initial conditions. Within this framework, equilibration can be proven under very weak assumptions about those measurements and initial conditions, while thermalization still requires quite strong additional hypotheses. In the second part, an analogous approach within the framework of classical mechanics is developed and compared with the quantum case. In particular, the assumptions to guarantee classical equilibration are now rather strong, while thermalization then follows under relatively weak additional conditions.