Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1904.07921 · Journal of Statistical Mechanics Theory and Experiment

On superstatistics of energy for a free quantum Brownian particle

Jakub Spiechowicz, J Łuczka

Published Jun 1, 2019• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/ab1c4e

Related papers

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

Jakub Spiechowicz, P. Bialas, J. Łuczka

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 ${E}_{k}$ of the quantum particle, it is expressed as ${E}_{k}=\ensuremath{\langle}{\mathcal{E}}_{k}\ensuremath{\rangle}$, where ${\mathcal{E}}_{k}$ 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 ${E}_{k}$ 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 ${E}_{k}$ of the Brownian particle is analyzed.

arXiv:1904.07555•Physical Review A•Nov 7, 2018•25 citations

Quantum Counterpart of Classical Equipartition of Energy

Jerzy Łuczka

Abstract It is shown that the recently proposed quantum analogue of classical energy equipartition theorem for two paradigmatic, exactly solved models (i.e., a free Brownian particle and a dissipative harmonic oscillator) also holds true for all quantum systems which are composed of an arbitrary number of non-interacting or interacting particles, subjected to any confining potentials and coupled to thermostat with arbitrary coupling strength.

arXiv:1911.06570•Journal of Statistical Physics•May 1, 2020•20 citations

Kinetic Energy of a Free Quantum Brownian Particle

Paweł Bialas, J. Łuczka

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.

Entropy•Feb 12, 2018•17 citations

Quantum analogue of energy equipartition theorem

P Bialas, Jakub Spiechowicz, J. Łuczka

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.

arXiv:1805.04012•Journal of Physics A Mathematical and Theoretical•Feb 1, 2019•31 citations

Energy of a free Brownian particle coupled to thermal vacuum

Jakub Spiechowicz, J. Łuczka

Experimentalists have come to temperatures very close to absolute zero at which physics that was once ordinary becomes extraordinary. In such a regime quantum effects and fluctuations start to play a dominant role. In this context we study the simplest open quantum system, namely, a free quantum Brownian particle coupled to thermal vacuum, i.e. thermostat in the limiting case of absolute zero temperature. We analyze the average energy [Formula: see text] of the particle from a weak to strong interaction strength c between the particle and thermal vacuum. The impact of various dissipation mechanisms is considered. In the weak coupling regime the energy tends to zero as [Formula: see text] while in the strong coupling regime it diverges to infinity as [Formula: see text]. We demonstrate it for selected examples of the dissipation mechanisms defined by the memory kernel [Formula: see text] of the Generalized Langevin Equation. We reveal how at a fixed value of c the energy E(c) depends on the dissipation model: one has to compare values of the derivative [Formula: see text] of the dissipation function [Formula: see text] at time [Formula: see text] or at the memory time [Formula: see text] which characterizes the degree of non-Markovianity of the Brownian particle dynamics. The impact of low temperature is also presented.

arXiv:2003.13567•Scientific Reports•Feb 18, 2021•17 citations

Quantum law for equipartition of energy

P. Białas, Jakub Spiechowicz, J. Łuczka

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 $E_k=k_B T/2$, where $k_B$ is the Boltzmann constant and $T$ 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 $E_k = \langle \mathcal E_k \rangle$, where $\mathcal E_k$ is thermal kinetic energy per one degree of freedom of the thermostat consisting of harmonic oscillators and $\langle ...\rangle$ denotes averaging over frequencies $\omega$ of those thermostat oscillators which contribute to $E_k$ according to the probability distribution $\mathbb P(\omega)$. 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 $E_k = \langle \mathcal E_k \rangle$ for other quantum systems.

•May 10, 2018•1 citation
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.
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.
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.
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.
Experimentalists have come to temperatures very close to absolute zero at which physics that was once ordinary becomes extraordinary. In such a regime quantum effects and fluctuations start to play a dominant role. In this context we study the simplest open quantum system, namely, a free quantum Brownian particle coupled to thermal vacuum, i.e. thermostat in the limiting case of absolute zero temperature. We analyze the average energy [Formula: see text] of the particle from a weak to strong interaction strength c between the particle and thermal vacuum. The impact of various dissipation mechanisms is considered. In the weak coupling regime the energy tends to zero as [Formula: see text] while in the strong coupling regime it diverges to infinity as [Formula: see text]. We demonstrate it for selected examples of the dissipation mechanisms defined by the memory kernel [Formula: see text] of the Generalized Langevin Equation. We reveal how at a fixed value of c the energy E(c) depends on the dissipation model: one has to compare values of the derivative [Formula: see text] of the dissipation function [Formula: see text] at time [Formula: see text] or at the memory time [Formula: see text] which characterizes the degree of non-Markovianity of the Brownian particle dynamics. The impact of low temperature is also presented.
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.