Free energy of a charged oscillator in a magnetic field and coupled to a heat bath through the momentum variables

TL;DR

This paper derives an exact formula for the free energy of a charged quantum oscillator in a magnetic field coupled via momentum variables to a heat bath, highlighting differences from coordinate coupling and confirming the third law of thermodynamics at low temperatures.

Contribution

It provides a novel exact expression for the free energy considering momentum-based coupling, expanding understanding of quantum thermodynamics in magnetic and dissipative systems.

Findings

Free energy expression differs from coordinate coupling cases.

Entropy vanishes at zero temperature, confirming the third law.

Evaluated free energy for specific heat-bath spectrum at low temperatures.

Abstract

We obtain an exact formula for the equilibrium free energy of a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a heat bath of independent quantum harmonic oscillators through the momentum variables. We show that the free energy has a different expression than that for the coordinate-coordinate coupling between the particle and the heat-bath oscillators. For an illustrative heat-bath spectrum, we evaluate the free energy in the low-temperature limit, thereby showing that the entropy of the charged particle vanishes at zero temperature, in agreement with the third law of thermodynamics.