Finite thermal reservoirs and the canonical distribution

TL;DR

This paper demonstrates that finite quantum systems in thermal contact naturally evolve towards a canonical distribution, resolving issues with the microcanonical ensemble and extending previous numerical results to much larger systems.

Contribution

It provides a simplified method to show that finite quantum systems approach the canonical distribution, enabling analysis of systems with over a million particles.

Findings

Finite quantum systems tend to a canonical distribution when in thermal contact.

The approach to the canonical distribution is confirmed for systems with over a million particles.

The method extends previous numerical results to much larger systems.

Abstract

The microcanonical ensemble has long been a starting point for the development of thermodynamics from statistical mechanics. However, this approach presents two problems. First, it predicts that the entropy is only defined on a discrete set of energies for finite, quantum systems, while thermodynamics requires the entropy to be a continuous function of the energy. Second, it fails to satisfy the stability condition ($\Delta S / \Delta U < 0$) for first-order transitions with both classical and quantum systems. Swendsen has recently shown that the source of these problems lies in the microcanonical ensemble itself, which contains only energy eigenstates and excludes their linear combinations. To the contrary, if the system of interest has ever been in thermal contact with another system, it will be described by a probability distribution over many eigenstates that is equivalent to the canonical ensemble for sufficiently large systems. Novotny et al. have recently supported this picture by dynamical numerical calculations for a quantum mechanical model, in which they showed the approach to a canonical distribution for up to 40 quantum spins. By simplifying the problem to calculate only the equilibrium properties, we are able to extend the demonstration to more than a million particles.