Necessary and sufficient conditions for time reversal symmetry in presence of magnetic fields

TL;DR

This paper identifies the broad conditions under which time reversal symmetry can exist in classical Hamiltonian systems with magnetic fields, expanding understanding of Onsager relations in statistical mechanics.

Contribution

It derives the most general form of generalized time reversal operations and provides sufficient conditions on magnetic fields for TRI, broadening the known symmetry conditions.

Findings

TRI can hold under more general conditions than previously thought

A generalized form of time reversal operation is deduced

Common statistical mechanics examples confirm the wider applicability

Abstract

Time reversal invariance (TRI) of particles systems has many consequences, among~which the celebrated Onsager reciprocal relations, a milestone in Statistical Mechanics dating back to 1931. Because for a long time it was believed that (TRI) dos not hold in presence of a magnetic field, a modification of such relations was proposed by Casimir in 1945. Only in the last decade, the~strict traditional notion of reversibility that led to Casimir's work has been questioned. It was then found that other symmetries can be used, which allow the Onsager reciprocal relations to hold without modification. In this paper we advance this investigation for classical Hamiltonian systems, substantially increasing the number of symmetries that yield TRI in presence of a magnetic field. We~first deduce the most general form of a generalized time reversal operation on the phase space of such a system; secondly, we express sufficient conditions on the magnetic field which ensure TRI. Finally, we examine common examples from statistical mechanics and molecular dynamics. Our main result is that TRI holds in a much wider generality than previously believed, partially explaining why no experimental violation of Onsager relations has so far been reported.