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 $E_k=⟨{\mathcal{E}}_k⟩$, where ${\mathcal{E}}_k$ is thermal kinetic energy of the thermostat per one degree of freedom and $⟨...⟩$ denotes averaging over frequencies $\omega$ of thermostat oscillators which contribute to $E_k$ according to the probability distribution $\mathbb{P}(\omega )$. We explore the impact of various dissipation mechanisms, via the Drude, Gaussian, algebraic and Debye spectral density functions, on the characteristic features of $\mathbb{P}(\omega )$. 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.

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