Quantum partition of energy for a free Brownian particle: Impact of dissipation

Abstract

We study the quantum counterpart of the theorem on energy equipartition for classical systems. We consider a free quantum Brownian particle modelled in terms of the Caldeira-Leggett framework: a system plus thermostat consisting of an infinite number of harmonic oscillators. By virtue of the theorem on the averaged kinetic energy E k of the quantum particle, it is expressed as E k = E k , where E k is thermal kinetic energy of the thermostat per one degree of freedom and ... denotes averaging over frequencies ω of thermostat oscillators which contribute to E k according to the probability distribution P(ω). We explore the impact of various dissipation mechanisms, via the Drude, Gaussian, algebraic and Debye spectral density functions, on the characteristic features of P(ω). The role of the system-thermostat coupling strength and the memory time on the most probable thermostat oscillator frequency as well as the kinetic energy E k of the Brownian particle is analysed. arXiv:1904.07555v1 [cond-mat.stat-mech]

Quantum Counterpart of Classical Equipartition of Energy

“…The quite natural question arises whether one can formulate a similar and universal relation for the mean kinetic energy of quantum systems at the thermodynamic equilibrium state. Recently, in a series of papers [5][6][7], the authors have proposed quantum analogue of the classical energy equipartition theorem. For a system of one degree of freedom this quantum counterpart, which is called the energy partition theorem, has the following appealing form:…”

“…The explicit form of P(ω) has been derived for two exactly solved quantum systems: a free Brownian particle [5] and a dissipative harmonic oscillator [6]. In these papers [5,6], thermostat is composed of quantum harmonic oscillators (à la Caldeira-Leggett [9][10][11]) and the above interpretation of E k (ω) as their mean kinetic energy per one degree of freedom is fully justified. Because P(ω) is a probability density, Eq.…”

On superstatistics of energy for a free quantum Brownian particle

“…Using Eqs. (5) and (8) one can calculate the commutator in Eq. (21) and the Green function (21) reads…”

“…Moreover, both E cl and ∆E cl are linearly increasing functions of temperature T . In the quantum case the mean energy has been analysed in our previous papers [8,9,17]. Here, in Fig.…”

“…The paper ends with with a summary in Sec. 8. In two Appendices we present the solution of GQLE and examples of the dissipation (memory) function appearing in GQLE which are considered in this paper.…”