Correspondence Between the Energy Equipartition Theorem in Classical Mechanics and its Phase-Space Formulation in Quantum Mechanics

TL;DR

This paper explores the relationship between the classical energy equipartition theorem and its quantum phase-space counterpart using the Wigner representation, highlighting the classical limit at high temperatures.

Contribution

It establishes a theoretical correspondence between classical and quantum energy distributions in phase-space, bridging a gap in understanding quantum analogs of classical theorems.

Findings

Classical energy equipartition is recovered at high temperatures.

A phase-space formulation of the quantum energy distribution is proposed.

The non-commutativity of observables affects quantum energy distribution.

Abstract

In classical physics there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what we know about the classical energy equipartition theorem and its possible counterpart in phase-space formulation in quantum mechanics based on the Wigner representation. Also, we show that in the high-temperature regime, the classical result is recovered.