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Journal of the Physical Society of Japan

Relation between TCLE Method and Relaxation Method. II

Mizuhiko Saeki

Published Dec 15, 1999• 7 citations
Publisher

Abstract

A general form of admittance is derived using the relaxation method under thermal equilibrium initial conditions from a second-order TCL equation with no external driving terms. Its admittance is shown to include the two interference terms in the TCLE method (method of TCL equations with external driving terms). Effects of the first and second interference terms in the TCLE method are numerically discussed for a quantal spin (magnitude 1/2) interacting with a spin-like bath and with a boson-like bath, respectively. It is shown that the first interference term which represents effects of the initial correlation and which is neglected in the relaxation method under decoupled initial conditions of the spin and bath, produces the effects which enhance the peaks of line shapes slightly and which abase the line shapes at the small frequency region, and becomes larger as the characteristic frequency of the heat bath becomes smaller.

DOI
10.1143/jpsj.68.3831

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A general form of admittance is derived using the relaxation method under thermal equilibrium initial conditions from a second-order TCL equation with no external driving terms. Its admittance is shown to include the two interference terms in the TCLE method (method of TCL equations with external driving terms). Effects of the first and second interference terms in the TCLE method are numerically discussed for a quantal spin (magnitude 1/2) interacting with a spin-like bath and with a boson-like bath, respectively. It is shown that the first interference term which represents effects of the initial correlation and which is neglected in the relaxation method under decoupled initial conditions of the spin and bath, produces the effects which enhance the peaks of line shapes slightly and which abase the line shapes at the small frequency region, and becomes larger as the characteristic frequency of the heat bath becomes smaller.
The transverse susceptibility of a quantal spin (magnitude 1/2) interacting weakly with its surroundings (quantal heat bath) is discussed using the method of the TCL equation with external driving terms (the TCLE method) analytically and numerically. The susceptibility obtained by the TCLE method under decoupled initial conditions is shown to coincide with that obtained under thermal equilibrium initial conditions. The results obtained with the TCLE method are compared with those obtained with the relaxation method. It is shown that the susceptibility in the relaxation method includes the second term of the interference terms in the TCLE method, but not its first term. With numerical calculations for the case of a spin-like bath including a quickly damped subsystem, it is shown that the interference terms in the TCLE method give larger effects as the temperature becomes lower, that its first term enhances the peaks of line shapes slightly, and that the peaks of line shapes are enhanced as the temperature becomes low. A model which shows that the TCLE method is simpler than the relaxation method is investigated numerically using the TCLE method.
For a quantal spin S (magnitude 1/2) interacting weakly with heat bath, the fluctuation-dissipation theorem, dispersion relation and sum rules are discussed by the method of TCL equations with external driving terms (the TCLE method). The transverse susceptibility of the quantal spin is numerically calculated using the TCLE method for the case of a boson-like bath including a quickly damped subsystem, comparing with the results obtained using the relaxation method under decoupled initial conditions of the spin and bath. It is shown that the interference terms in the TCLE method produce the effects which enhance the peaks of line shapes and abase the line shapes at the small frequency region. It is also shown that the peaks of line shapes are enhanced as the temperature becomes low, as shown for the case of a spin-like bath. For the case of a boson-like bath including a subsystem with finite damping constants, the transverse susceptibility is calculated using the TCLE method numerically.
The TCLE method (a method in which the admittance of a system is directly calculated from time-convolutionless equations with external driving terms) formulated in terms of thermo-field dynamics, is applied to a boson system interacting with its surroundings, referred to as the “heat reservoir”, and with an external driving field. For a boson system interacting with its heat reservoir, the fluctuation-dissipation theorem, a dispersion relation, and sum rules are studied through application of the TCLE method. The effects of the interference between the external driving field and the heat reservoir, which give the effects of the initial correlation and memory, as well as the effects of the boson-boson interaction, are numerically investigated for a non-degenerate boson system interacting with its heat reservoir, up to first order in powers of the boson-boson interaction.
In the context of non-equilibrium statistical mechanics, the TCLE method (a method in which the admittance of a system is directly calculated from time-convolutionless equations with external driving terms) for the linear response, is formulated in terms of thermo-field dynamics and in this way, thermo-field dynamics is generalized to the case of a quantal system interacting with its surroundings and with an external driving field. It is applied to interacting boson systems, and an admittance formula for such systems is derived using the TCLE method.
The non-equilibrium thermo-field dynamics proposed by Arimitsu and Umezawa are generalized to the case of a fourth-order unperturbed Hamiltonian which includes not only a second-order (quadratic) part but also a fourth-order part. Fujita's analysis for effects of the initial particle correlation of a quantum gas is proved generally in terms of TFD. The forms of the quasi-particle operators for a semi-free boson field are derived. It is shown that the energies and life-times of the quasi-particles depend on the adiabatic boson-reservoir interaction which leads to the fourth-order part of the unperturbed Hamiltonian. The form of the two-point Green's function for the semi-free boson field is evaluated. A form of the admittance for a boson system interacting with its heat reservoir, which includes effects of the initial correlation and memory, is derived using the TCLE method formulated in terms of the generalized non-equilibrium thermo-field dynamics. A calculation method of the higher-order parts of the admittance in powers of the boson-boson interaction is given. Furthermore, a calculation method of the perturbation expansions of the two-point Green's function for the boson system is given.