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Progress of Theoretical Physics

An Effective Hamiltonian in Nonequilibrium Thermo Field Dynamics with a Reservoir

H. Matsumoto

Published Mar 1, 1988• 4 citations
Publisher

Abstract

General properties of an effective Hamiltonian in thermo field dynamics, which includes reservoir effects through coarse graining, are investigated. The necessary and sufficient conditions for the effective Hamiltonian to describe the reservoir effect are obtained. The final canonical distribution is discussed with relation to each given interaction. The result gives a quick and easy guide to construct an effective Hamiltonian and to identify the final equilibrium state.

DOI
10.1143/ptp.79.627

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General properties of an effective Hamiltonian in thermo field dynamics, which includes reservoir effects through coarse graining, are investigated. The necessary and sufficient conditions for the effective Hamiltonian to describe the reservoir effect are obtained. The final canonical distribution is discussed with relation to each given interaction. The result gives a quick and easy guide to construct an effective Hamiltonian and to identify the final equilibrium state.
A fundamental framework for the construction of a quantum field theory for open systems, which will be called non-equilibrium thermo field dynamics (NETFD), is built upon two concepts. One is the thermal state in the thermal-Liouville space, by which the quasi-particle superoperator is introduced generally from the thermal state condition. The other is the coarse graining which is realized by projecting out some partial space from the complete thermal-Liouville space, and by which the dissipation is introduced. Most properties of the usual quantum field theory (e.g., the operator formalism, the time-ordered formulation of the Green’s functions, the Feynman diagram method in real time, etc.) are preserved in NETFD. The entropy for the nonequilibrium state can be also introduced in NETFD.
We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.