Yet another surprise in the problem of classical diamagnetism

TL;DR

This paper investigates the apparent contradiction between classical statistical mechanics and fluctuation theorems regarding diamagnetism, showing conditions under which they agree or differ through analytical examples.

Contribution

It demonstrates how recent fluctuation theorems can predict non-zero diamagnetism in classical systems, challenging traditional equilibrium results.

Findings

Fluctuation theorems predict magnetic field dependence of free energy.

In some cases, fluctuation theorems align with equilibrium results.

Contradictions arise when Langevin approach is inconsistent with equilibrium.

Abstract

The well known Bohr-van Leeuwen Theorem states that the orbital diamagnetism of classical charged particles is identically zero in equilibrium. However, results based on real space-time approach using the classical Langevin equation predicts non-zero diamagnetism for classical unbounded (finite or infinite) systems. Here we show that the recently discovered Fluctuation Theorems, namely, the Jarzynski Equality or the Crooks Fluctuation Theorem surprisingly predict a free energy that depends on magnetic field as well as on the friction coefficient, in outright contradiction to the canonical equilibrium results. However, in the cases where the Langevin approach is consistent with the equilibrium results, the Fluctuation Theorems lead to results in conformity with equilibrium statistical mechanics. The latter is demonstrated analytically through a simple example that has been discussed recently.