Quantum counterpart of energy equipartition theorem for fermionic systems

Abstract

In this brief report, following the recent developments on formulating a quantum analogue of the classical energy equipartition theorem for open systems where the heat bath comprises of independent oscillators, i.e. bosonic degrees of freedom, we present an analogous result for fermionic systems. The most general case where the system is connected to multiple reservoirs is considered and the mean energy in the steady state is expressed as an integral over the reservoir frequencies. Physically this would correspond to summing over the contributions of the bath degrees of freedom to the mean energy of the system over a suitable distribution function ρ(ω) dependent on the system parameters. This result holds for nonequilibrium steady states, even in the nonlinear regime far from equilibrium. We also analyze the zero temperature behaviour and low temperature corrections to the mean energy of the system.

Independent-oscillator model and the quantum Langevin equation for an oscillator: a review

“…Although it may come as a surprise that thermodynamics can be consistently formulated for a single particle, which is far from the thermodynamic limit that one invokes in traditional thermodynamic studies, it turns out that not only does a thermal description follow naturally from the framework of classical and quantum Brownian motion, it is also consistent with all the laws of thermodynamics, including the third law (see, for example, [26][27][28]35]). We will contrast the expressions of average energy between classical and quantum Brownian oscillators, which will lead us to the recently-proposed quantum counterpart of energy equipartition theorem [36][37][38][39][40][41][42][43][44][45][46]. As an application of the framework relevant to condensed-matter physics, we shall discuss dissipative diamagnetism [46,47] and emphasize on the role of confining potentials.…”

“…In recent times, there have been research activities aimed at the articulation of the quantum analogue of the energy equipartition theorem [36][37][38][39][40][41][42][43][44][45][46]81]. It has been observed that unlike the classical case, the average energy of an open quantum system, modeled as a system interacting with a collection of an infinite number of independent harmonic oscillators, can be understood as being the sum of contributions from individuallyequilibrated oscillators distributed over the entire frequency spectrum of the heat bath.…”

“…It may be shown that P k (ω) is both positive definite and is normalized, justifying its interpretation as a bona fide probability density (see appendix B). In recent literature, equation ( 63) has been termed as the quantum counterpart of energy equipartition theorem [36][37][38][39][40][41][42][43][44][45][46]81]. The justification behind this terminology can be understood from the high-temperature limit of equation ( 63), for which, upon using lim Let us turn our attention to the potential energy.…”

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and Its Phase-Space Formulation in Quantum Mechanics

“…Currently, the authors working in the area agree that the equipartition of energy no longer holds in the quantum regime, and the energetic distribution follows a better-called energy partition theorem supported in the construction of a distribution function [ 3 , 4 , 5 ]. Applications to a few models have been made with satisfactory results and holding the correspondence with the classical theorem at a high-temperature regime [ 4 , 6 , 7 ]. Nonetheless, these works are based on the same conceptual and mathematical framework, and none of them is formulated in phase-space formalism.…”

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

“…Currently, the authors working in the area agree that the equipartition of energy no longer holds in the quantum regime and the energetic distribution follows a better-called energy partition theorem supported in the construction of a distribution function [3,4,5].Applications to a few models have been made with satisfactory results and holding the correspondence with the classical theorem at the high-temperature regime [4,6,7]. Due to the recent exploration of this area, the works about it are very homogeneous and based on the same conceptual and mathematical framework.…”