Quantum counterpart of energy equipartition theorem for a dissipative charged magneto-oscillator: Effect of dissipation, memory, and magnetic field

Abstract

In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modelled as a collection of independent quantum harmonic oscillators. We derive the closed form expressions for the mean kinetic and potential energies of the charged-dissipativemagneto-oscillator in the form E k = E k and Ep = Ep respectively, where E k and Ep denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is two-fold, the first one being over the Gibbs' canonical state for the thermostat, giving E k and Ep and the second one denoted by . being over the frequencies ω of the bath oscillators which contribute to E k and Ep according to probability distributions P k (ω) and Pp(ω) respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Ohmic, Drude and radiation bath spectral density functions, on the typical properties of P k (ω) and Pp(ω). Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.

Partition of free energy for a Brownian quantum oscillator: Effect of dissipation and magnetic field

“…Thus, in summary, the function P(ω) defined from the generalized susceptibility using eqn (12) indeed fulfills the basic requirements needed to qualify as a probability distribution function, in accordance with proposition-(1).…”

“…This feature becomes explicit when one expresses the free energy of the system at temperature T in the form F (T ) = f (ω, T ) where f (ω, T ) is the free energy of an individual bath oscillator at the same temperature. Then it raises the natural question as to whether the function P(ω) is related to the probabilities with which the system receives energy contributions from the bath degrees of freedom in accordance to the recently proposed quantum counterpart of energy equipartition [7][8][9][10][11][12]. We find that it is indeed the case, i.e.…”

“…where E(ω, T ) is the mean kinetic energy of an individual bath oscillator and P(ω) is a probability distribution function [eqn (12)] according to which the bath degrees of freedom subscribe to the mean energy of the system.…”

“…The discovery of the fact that thermodynamic principles are consistent with the quantum properties of microscopic systems has led to a flurry of research activity in the recent times. Quantum thermodynamics, as the subject is known (see for example [1,2] for a pedagogical introduction), has seen several remarkable recent advances such as the formulation of quantum thermodynamic functions [3][4][5] and the third law (see [6] and references therein), a novel proposal of a quantum analogue of energy equipartition [7][8][9][10][11][12] as well as studies on non-equilibrium steady states in nanoscale systems [13][14][15]. A particularly interesting aspect is that of the quantum counterpart of the classical energy equipartition theorem, which is being actively worked on.…”

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.…”