arXiv:1911.06570 · Journal of Statistical Physics

Quantum Counterpart of Classical Equipartition of Energy

Jerzy Łuczka

Published May 1, 2020• 20 citations

Abstract

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.

DOI: 10.1007/s10955-020-02557-5

Related papers

Quantum analogue of energy equipartition theorem

P Bialas, Jakub Spiechowicz, J. Łuczka

arXiv:1805.04012Journal of Physics A Mathematical and TheoreticalFeb 1, 201931 citations

Quantum law for equipartition of energy

P. Białas, Jakub Spiechowicz, J. Łuczka

May 10, 20181 citation

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

Jakub Spiechowicz, P. Bialas, J. Łuczka

arXiv:1904.07555Physical Review ANov 7, 201825 citations

On superstatistics of energy for a free quantum Brownian particle

Jakub Spiechowicz, J Łuczka

arXiv:1904.07921Journal of Statistical Mechanics Theory and ExperimentJun 1, 201911 citations

Partition of energy for a dissipative quantum oscillator

P. Bialas, Jakub Spiechowicz, J. Łuczka

arXiv:1904.07560Scientific ReportsOct 25, 201825 citations

Quantum counterpart of equipartition theorem in quadratic systems

Xin-Hai Tong

arXiv:2309.14582Physical Review ResearchMay 13, 20245 citations

arXiv:1911.06570 · Journal of Statistical Physics

Quantum Counterpart of Classical Equipartition of Energy

Jerzy Łuczka

Published May 1, 2020• 20 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/s10955-020-02557-5

Related papers

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.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:1805.04012Journal of Physics A Mathematical and TheoreticalFeb 1, 201931 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.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, 20181 citation

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.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:1904.07555Physical Review ANov 7, 201825 citations

On superstatistics of energy for a free quantum Brownian particle

Jakub Spiechowicz, J Łuczka

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.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.

arXiv:1904.07921Journal of Statistical Mechanics Theory and ExperimentJun 1, 201911 citations

Partition of energy for a dissipative quantum oscillator

P. Bialas, Jakub Spiechowicz, J. Łuczka

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 ${{\mathbb{P}}}_{k}(\omega )$ ℙ k ( ω ) and ${{\mathbb{P}}}_{p}(\omega )$ 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 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 $${{\mathbb{P}}}_{k}(\omega )$$ ℙ k ( ω ) and $${{\mathbb{P}}}_{p}(\omega )$$ 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.

arXiv:1904.07560Scientific ReportsOct 25, 201825 citations

Quantum counterpart of equipartition theorem in quadratic systems

Xin-Hai Tong

The equipartition theorem is a fundamental law of classical statistical physics, which states that every degree of freedom contributes kBT/2 to the energy, where T is the temperature and kB is the Boltzmann constant. Recent studies have revealed the existence of a quantum version of the equipartition theorem. In the present work, we focus on how to obtain the quantum counterpart of the generalized equipartition theorem for arbitrary quadratic systems in which the multimode Brownian oscillators interact with multiple reservoirs at the same temperature. An alternative method of deriving the energy of the system is also discussed and compared with the result of the quantum version of the equipartition theorem, after which we conclude that the latter is more reasonable. Our results can be viewed as an indispensable generalization of recent works on a quantum version of the equipartition theorem. Published by the American Physical Society 2024The equipartition theorem is a fundamental law of classical statistical physics, which states that every degree of freedom contributes kBT/2 to the energy, where T is the temperature and kB is the Boltzmann constant. Recent studies have revealed the existence of a quantum version of the equipartition theorem. In the present work, we focus on how to obtain the quantum counterpart of the generalized equipartition theorem for arbitrary quadratic systems in which the multimode Brownian oscillators interact with multiple reservoirs at the same temperature. An alternative method of deriving the energy of the system is also discussed and compared with the result of the quantum version of the equipartition theorem, after which we conclude that the latter is more reasonable. Our results can be viewed as an indispensable generalization of recent works on a quantum version of the equipartition theorem. Published by the American Physical Society 2024

arXiv:2309.14582Physical Review ResearchMay 13, 20245 citations

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.

{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.

{{\mathbb{P}}}_{k}(\omega )

ℙ k ( ω ) and

{{\mathbb{P}}}_{p}(\omega )

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.