Simulating quantum thermodynamics of a finite system and bath with variable temperature

Abstract

We construct a finite bath with variable temperature for quantum thermodynamic simulations in which heat flows between a system S and the bath environment E in time evolution of an initial SE pure state. The bath consists of harmonic oscillators that are not necessarily identical. Baths of various numbers of oscillators are considered; a bath with five oscillators is used in the simulations. The bath has a temperature-like level distribution. This leads to definition of a system-environment microcanonical temperature T SE (t) which varies with time. The quantum state evolves toward an equilibrium state which is thermal-like, but there is significant deviation from the ordinary energy-temperature relation that holds for an infinite quantum bath, e.g. an infinite system of identical oscillators. There are also deviations from the Einstein quantum heat capacity. The temperature of the finite bath is systematically greater for a given energy than the infinite bath temperature, and asymptotically approaches the latter as the number of oscillators increases. It is suggested that realizations of these finite-size effects may be attained in computational and experimental dynamics of small molecules.

Quantum harmonic oscillators and thermalization

“…An example of the above interaction Hamiltonian for the combined system is Ĥcoupling = mx k xk , where a set of an infinite number of coupled quantum oscillators approximates a heat bath [10]. There have been many other oscillator models of heat baths [14][15][16]. Another example of an ideal bath is a massless quantum field coupled to a harmonic oscillator [17].…”

Asymmetric temperature equilibration with heat flow from cold to hot in a quantum thermodynamic system

“…An essential element of our setup is the variable temperature baths. In a recent paper [4], we introduced a computational model for such a bath and showed that it comes to thermal equilibrium with a system, while exhibiting quantum thermodynamic effects related to the finite size of the bath. The variable temperature bath generalized earlier work [1,2,[5][6][7][8] on quantum thermodynamic simulations that used a constant temperature bath.…”

“…The body of our work in Refs. [1,2,4,5] and the current article are built around a largely self-contained exposition in the unpublished dissertation of P. C. L., available online [9]. This work is part of a broad program reexamining the foundations of statistical mechanics in the context of quantum pure states evolving in time [1,2,[4][5][6][7][8].…”

“…[1,2,4,5] and the current article are built around a largely self-contained exposition in the unpublished dissertation of P. C. L., available online [9]. This work is part of a broad program reexamining the foundations of statistical mechanics in the context of quantum pure states evolving in time [1,2,[4][5][6][7][8]. There have been other attempts at formulating the second law for quantum pure states [31][32][33], but to our knowledge none of these has yet been associated with new types of quantum thermodynamic behavior such as we have here with S Q univ in the second law.…”

“…The frequencies of the bath oscillators are taken as random numbers that are scaled to set their geometric mean (∏ η osc=1 hω osc ) 1/η = 1, in accord with Ref. [4] where this finite environment model was developed as a small variable temperature bath in simulations of a system interacting with a single environment. The frequencies we use are listed in Table I.…”