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arXiv:hep-ph/0104167

Nonequilibrium Temperature for Open Boson and Fermion Systems

Sang Pyo Kim

Published Apr 17, 2001• 1 citation
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Abstract

The effective theory of an open boson or fermion system is studied, which evolves out of equilibrium with time-dependent Hamiltonian $\hat{H}(t)$. A measure of nonequilibrium temperature for the open system evolving from an equilibrium is proposed as the time-averaged energy expectation value $T(t) = T_i(\bar{< \hat{H} (t) >_Ψ}/ \bar{< \hat{I} (t) >_Ψ})$, where $\hat{I} (t)$, 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.

DOI
10.48550/arxiv.hep-ph/0104167

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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.
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.
Recently, a number of authors have begun to study the evolution of quantum fields in the early Universe characterized by a time-dependent density matrix ${\ensuremath{\rho}}_{S}$(t). All of this work is predicated on the assumption that one's ``subsystem'' of interest is in some sense ``decoupled'' from the rest of the Universe, so that ${\ensuremath{\rho}}_{S}$ satisfies a Liouville--von Neumann equation which implies, e.g., an isentropic evolution. Starting from ``first principles,'' i.e., the Schr\"odinger equation for the totality of ``subsystem'' plus surroundings (``bath''), it is shown here how such a picture can be derived as the limiting case of a more complete statistical description. Quite generally, one finds that ${\ensuremath{\rho}}_{S}$ and the ``bath'' density matrix ${\ensuremath{\rho}}_{B}$ satisfy coupled nonlinear generalizations of the Liouville--von Neumann equation and evidence a nonisentropic evolution. However, in a Vlasov-type approximation, ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{B}$ satisfy instead much simpler bilinear equations which imply an isentropic evolution. And finally, in the limit that the ``back reaction'' of ${\ensuremath{\rho}}_{S}$ on ${\ensuremath{\rho}}_{B}$ can be neglected in computing the evolution of ${\ensuremath{\rho}}_{B}$, one recovers a true Liouville--von Neumann equation for the evolution of ${\ensuremath{\rho}}_{S}$ in an external field.
A definition of nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​ is proposed for dynamical Gaussian quantum open systems strongly coupled to a heat bath and the formal relation with the generating functional, the coarse-grained effective action and the influence action is indicated. For Gaussian open quantum systems exemplified by the quantum Brownian motion model studied here, a time-varying effective temperature can be introduced in a natural way, and, with it, the nonequilibrium free energy Fs{\mathcal{F}}_{\mathrm{s}}Fs​, von Neumann entropy SvN{\mathcal{S}}_{vN}SvN​ and internal energy Us{\mathcal{U}}_{\mathrm{s}}Us​ of the reduced system (SSS) can be defined accordingly. In contrast to the nonequilibrium free energy found in the literature which references the bath temperature, the nonequilibrium thermodynamic functions we find here obey the familiar relation ${\mathcal{F}}_{\mathrm{s}}(t)={\mathcal{U}}_{\mathrm{s}}(t)\ensuremath{-}{T}_{\mathrm{EFF}}(t){\mathcal{S}}_{vN}(t)$ at any and all moments of time in the system's fully nonequilibrium evolution history. After the system equilibrates they coincide, in the weak coupling limit, with their counterparts in conventional equilibrium thermodynamics. Since the effective temperature captures both the state of the system and its interaction with the bath, upon the system's equilibration, it approaches a value slightly higher than the initial bath temperature. Notably, it remains nonzero for a zero-temperature bath, signaling the existence of system-bath entanglement. Reasonably, at high bath temperatures and under ultraweak couplings, it becomes indistinguishable from the bath temperature. The nonequilibrium thermodynamic functions and relations discovered here for dynamical Gaussian quantum systems should open up useful pathways toward establishing meaningful theories of nonequilibrium quantum thermodynamics.
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.