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Temperature Out of Equilibrium

Sang Pyo Kim

Published Apr 17, 2001• 2 citations
OpenAlex

Abstract

A free boson system out of equilibrium is studied with a time-dependent Hamiltonian. The density operator is determined by an operator $\\hat{I} (t)$, satisfying the quantum Liouville-von Neumann equation. When the system evolves from an initial equilibrium, the temperature is obtained as $T(t) = T_i (\\overline{< \\hat{H} (t) >_{\\Psi}}/ \\overline{< \\hat{I} (t) >_{\\Psi}})$, where the expectation value is taken with respect to the exact quantum state of Schr\\"{o}dinger equation and the time average also is taken over the period of system. It recovers the result for the adiabatic (quasi-equilibrium) and the nonadiabatic (out of equilibrium) evolution by taking into account the factor for particle production. The temperature also is obtained for fermion system.

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A free boson system out of equilibrium is studied with a time-dependent Hamiltonian. The density operator is determined by an operator hatI(t)\\hat{I} (t)hatI(t), satisfying the quantum Liouville-von Neumann equation. When the system evolves from an initial equilibrium, the temperature is obtained as T(t)=Ti(overline<hatH(t)>Psi/overline<hatI(t)>Psi)T(t) = T_i (\\overline{< \\hat{H} (t) >_{\\Psi}}/ \\overline{< \\hat{I} (t) >_{\\Psi}})T(t)=Ti​(overline<hatH(t)>Psi​/overline<hatI(t)>Psi​), where the expectation value is taken with respect to the exact quantum state of Schr\"{o}dinger equation and the time average also is taken over the period of system. It recovers the result for the adiabatic (quasi-equilibrium) and the nonadiabatic (out of equilibrium) evolution by taking into account the factor for particle production. The temperature also is obtained for fermion system.
The effective theory of an open boson or fermion system is studied, which evolves out of equilibrium with time-dependent Hamiltonian H^(t)\hat{H}(t)H^(t). A measure of nonequilibrium temperature for the open system evolving from an equilibrium is proposed as the time-averaged energy expectation value T(t)=Ti(<H^(t)>Ψˉ/<I^(t)>Ψˉ)T(t) = T_i(\bar{< \hat{H} (t) >_Ψ}/ \bar{< \hat{I} (t) >_Ψ})T(t)=Ti​(<H^(t)>Ψ​ˉ​/<I^(t)>Ψ​ˉ​), where I^(t)\hat{I} (t)I^(t), the action operator, satisfies the quantum Liouville-von Neumann equation and determines the true density operator. It recovers the result for the adiabatic (quasi-equilibrium) and nonadiabatic (out of equilibrium) evolution from one static Hamiltonian to another and takes into account the particle production due to the intermediate processes.
We formulate the thermofield dynamics for time-dependent systems by combining the Liouville-von Neumann equation, its invariant operators, and the basic notions of thermofield dynamics. The new formulation is applied to time-dependent bosons and fermions by using the time-dependent annihilation and creation operators that satisfy the Liouville-von Neumann equation. It is shown that the thermal state is the time- and temperature-dependent vacuum state and a general formula is derived to calculate the thermal expectation value of operators.
We study the behavior of two weakly coupled quantum systems, described by a separable density operator. One of them is a single oscillator, representing a microscopic system, while the other is a set of oscillators, which perform the role of a reservoir in thermal equilibrium. From the Liouville-Von Neumann equation for the reduced density operator, we devise the master equation that governs the evolution of the microscopic system, incorporating the effects of temperature via Thermofield Dynamics formalism and suitably redefining the vacuum of the macroscopic system. As applications, we study an oscillator interacting with a set of boson oscillators at finite temperature, usually employed in quantum mechanics to describe a Brownian particle; and investigate the behavior of an oscillator in the presence of a heat bath consisting of a set of fermion oscillators.