Quantum–Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion

Abstract

Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.

Stochastic energetics of a colloidal particle trapped in a viscoelastic bath

“…whose functional form is consistent with the so-called Carreau-Yasuda equation that is commonly used as a phenomenological model for the shear-rate dependent viscosity of non-Newtonian fluids [88]. In (36), η 0 and η ∞ correspond to the values of the fluid viscosity at vanishing and infinite frequencies, respectively, λ is the characteristic relaxation time of the fluid, whereas α ⩾ 0 and n ⩽ 1 are exponents that depend on the specific microscopic features of the fluid and describe a power-law-like decay of η(s…”

“…The values of the parameters η 0 , η ∞ , λ, α and n for each fluid along with the values of κ for the trapping harmonic potentials in each experiment are listed in table 1. Note that in the case of the polymer solution, we have restricted the outer exponent in (36) to the value (1 − n)/α = 1 by setting n = 1 − α, which corresponds to the so-called Cross rheological model characterized by a single fitting exponent α [89]. This choice is motivated by previous rheological studies on the macroscopic non-Newtonian behavior of semidilute PEO polymeric solutions reported in the literature [87].…”

“…Moreover, the heat probability distribution has been determined for overdamped Brownian particles in non-stationary states relaxing toward thermal equilibrium after a temperature quench [33,34]. In addition, the statistics of the steady heat fluxes for systems in contact with two thermostats at different temperatures have been investigated for quantum harmonic oscillators [35,36], RC electric circuits [37,38], pairs of hydrodynamically-coupled colloidal particles [39], and harmonic networks [40][41][42][43]. Other effects on the heat distribution for Brownian particles, e.g.…”

Quantum-classical correspondence in spin-boson equilibrium states at arbitrary coupling

“…Bohr's correspondence principle [1] played an essential role in the early development of quantum mechanics. Since then, a variety of interpretations and applications of the correspondence principle have been explored [2][3][4][5][6][7][8][9]. One form asks if the statistical properties of a quantum system approach those of its classical counterpart in the limit of large quantum numbers [4,5].…”

“…They proved that the system's thermodynamic partition function Z qu S associated with the Gibbs state, converges to the corresponding classical partition function Z cl S , in the limit of large spins. Such correspondence gives insight into the conditions for a quantum thermodynamic system to be well-approximated by its classical counterpart [8,9]. While Z qu S is computationally tough to evaluate for many systems, Z cl S offers tractable expressions with which thermodynamic properties, such as free energies, susceptibilities and correlation functions, can readily be computed [2,3].…”