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arXiv:2201.07721 · Journal of Statistical Mechanics Theory and Experiment

Quantum counterpart of energy equipartition theorem for fermionic systems

Jasleen Kaur, Aritra Ghosh, Malay Bandyopadhyay

Published May 1, 2022• 8 citations
arXivHTMLPDF

Abstract

Abstract In this brief report, following the recent developments on formulating a quantum analogue of the classical energy equipartition theorem for open systems where the heat bath comprises of independent oscillators, i.e. bosonic degrees of freedom, we present an analogous result for fermionic systems. The most general case where the system is connected to multiple reservoirs is considered and the mean energy in the steady state is expressed as an integral over the reservoir frequencies. Physically this would correspond to summing over the contributions of the bath degrees of freedom to the mean energy of the system over a suitable distribution function ρ ( ω ) dependent on the system parameters. This result holds for nonequilibrium steady states, even in the nonlinear regime far from equilibrium. We also analyze the zero temperature behaviour and low temperature corrections to the mean energy of the system.

DOI
10.1088/1742-5468/ac6f03

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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.
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 2024
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.
In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered.
Abstract This review provides a brief and quick introduction to the quantum Langevin equation for an oscillator, while focusing on the steady-state thermodynamic aspects. A derivation of the quantum Langevin equation is carefully outlined based on the microscopic model of the heat bath as a collection of a large number of independent quantum oscillators, the so-called independent-oscillator model. This is followed by a discussion on the relevant ‘weak-coupling’ limit. In the steady state, we analyze the quantum counterpart of energy equipartition theorem which has generated a considerable amount of interest in recent literature. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of quantum Brownian oscillators, emphasizing upon the role of confining potentials and also upon the environment-induced classical-quantum crossover. We discuss situations where the system-bath coupling is via the momentum variables by focusing on a gauge-invariant model of momentum-momentum coupling in the presence of a vector potential; for this problem, we derive the quantum Langevin equation and discuss quantum thermodynamic functions. Finally, the topic of fluctuation theorems is discussed (albeit, briefly) in the context of classical and quantum cyclotron motion of a particle coupled to a heat bath.