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Entropy

Kinetic Energy of a Free Quantum Brownian Particle

Paweł Bialas, J. Łuczka

Published Feb 12, 2018• 17 citations
Publisher

Abstract

We consider a paradigmatic model of a quantum Brownian particle coupled to a thermostat consisting of harmonic oscillators. In the framework of a generalized Langevin equation, the memory (damping) kernel is assumed to be in the form of exponentially-decaying oscillations. We discuss a quantum counterpart of the equipartition energy theorem for a free Brownian particle in a thermal equilibrium state. We conclude that the average kinetic energy of the Brownian particle is equal to thermally-averaged kinetic energy per one degree of freedom of oscillators of the environment, additionally averaged over all possible oscillators' frequencies distributed according to some probability density in which details of the particle-environment interaction are present via the parameters of the damping kernel.

DOI
10.3390/e20020123

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We consider a paradigmatic model of a quantum Brownian particle coupled to a thermostat consisting of harmonic oscillators. In the framework of a generalized Langevin equation, the memory (damping) kernel is assumed to be in the form of exponentially-decaying oscillations. We discuss a quantum counterpart of the equipartition energy theorem for a free Brownian particle in a thermal equilibrium state. We conclude that the average kinetic energy of the Brownian particle is equal to thermally-averaged kinetic energy per one degree of freedom of oscillators of the environment, additionally averaged over all possible oscillators' frequencies distributed according to some probability density in which details of the particle-environment interaction are present via the parameters of the damping kernel.
We study the quantum counterpart of the theorem on energy equipartition for classical systems. We consider a free quantum Brownian particle modeled 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 Ek{E}_{k}Ek​ of the quantum particle, it is expressed as ${E}_{k}=\ensuremath{\langle}{\mathcal{E}}_{k}\ensuremath{\rangle}$, where Ek{\mathcal{E}}_{k}Ek​ is the thermal kinetic energy of the thermostat per 1 degree of freedom and $\ensuremath{\langle}\ensuremath{\cdots}\ensuremath{\rangle}$ denotes averaging over the frequencies $\ensuremath{\omega}$ of thermostat oscillators which contribute to Ek{E}_{k}Ek​ according to the probability distribution $\mathbb{P}(\ensuremath{\omega})$. We explore the impact of various dissipation mechanisms, via the Drude, Gaussian, algebraic, and Debye spectral density functions, on the characteristic features of $\mathbb{P}(\ensuremath{\omega})$. 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 Ek{E}_{k}Ek​ of the Brownian particle is analyzed.
Abstract One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the mean kinetic energy E k equals , where is the Boltzmann constant and T is the temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years, still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and consider the simplest system, i.e. the Caldeira–Leggett model for a free quantum Brownian particle in contact with a thermostat consisting of an infinite number of harmonic oscillators. We prove that the mean kinetic energy E k of the Brownian particle equals the mean kinetic energy per one degree of freedom of the thermostat oscillators, i.e. . We show that this relation can be obtained from the fluctuation–dissipation theorem derived within the linear response theory and is universal in the sense that it holds true for any linear and non-linear systems in contact with a bosonic thermostat.
One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the average kinetic energy equals Ek=kBT/2E_k=k_B T/2Ek​=kB​T/2, where kBk_BkB​ is the Boltzmann constant and TTT is temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and formulate the quantum law for equipartition of energy in the appealing form Ek=⟨Ek⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩, where Ek\mathcal E_kEk​ is thermal kinetic energy per one degree of freedom of the thermostat consisting of harmonic oscillators and ⟨...⟩\langle ...\rangle⟨...⟩ denotes averaging over frequencies ω\omegaω of those thermostat oscillators which contribute to EkE_kEk​ according to the probability distribution P(ω)\mathbb P(\omega)P(ω). We derive it for two paradigmatic and exactly solvable models of quantum open systems: a free Brownian particle and a harmonic oscillator. We formulate conditions for validity of the relation Ek=⟨Ek⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩ for other quantum systems.
Abstract We reveal a new face of the old clichéd system: a dissipative quantum harmonic oscillator. We formulate and study a quantum counterpart of the energy equipartition theorem satisfied for classical systems. Both mean kinetic energy Ek and mean potential energy Ep of the oscillator are expressed as Ek = 〈εk〉 and Ep = 〈εp〉, where 〈εk〉 and 〈εp〉 are mean kinetic and potential energies per one degree of freedom of the thermostat which consists of harmonic oscillators too. The symbol 〈...〉 denotes two-fold averaging: (i) over the Gibbs canonical state for the thermostat and (ii) over thermostat oscillators frequencies ω which contribute to Ek and Ep according to the probability distribution Pk(ω){{\mathbb{P}}}_{k}(\omega )Pk​(ω) ℙ k ( ω ) and Pp(ω){{\mathbb{P}}}_{p}(\omega )Pp​(ω) ℙ p ( ω ) , respectively. The role of the system-thermostat coupling strength and the memory time is analysed for the exponentially decaying memory function (Drude dissipation mechanism) and the algebraically decaying damping kernel.
Abstract We consider energetics of a free quantum Brownian particle coupled to thermostat of temperature T and study this problem in terms of the lately formulated quantum analogue of the energy equipartition theorem. We show how this quantum counterpart can be derived from the Callen–Welton fluctuation-dissipation relation and rephrased in terms of superstatistics. We analyse the influence of the system-thermostat coupling strength and the memory time of the dissipation kernel on statistical characteristics of the particle energy and its specific heat.