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Progress of Theoretical Physics

Relation between Method of TCL Equation with External Terms and Relaxation Method

Mizuhiko Saeki

Published Mar 1, 1993• 6 citations
Publisher

Abstract

The relation between the method of TCL equation with external driving terms and the relaxation method is discussed analytically in the quantal bath model. It is shown that the interference term in the former method gives the part of second order in powers of the system-bath interaction in the Kubo formula, and that the former method is simpler than the latter method. It is also shown that the relaxation method neglects the low temperature effects which are given by the first part of interference term, and that the relaxation method in the narrowing limit neglects the effects of the interference term completely. The general form of admittance for the case of decoupled initial condition is given by the method of TCL equation with external driving terms, and can be equal to that for the case of thermal equilibrium initial condition. The above facts are verified by applying to a quantal oscillator. Validity of the second-order perturbation calculations is discussed.

DOI
10.1143/ptp.89.607

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The relation between the method of TCL equation with external driving terms and the relaxation method is discussed analytically in the quantal bath model. It is shown that the interference term in the former method gives the part of second order in powers of the system-bath interaction in the Kubo formula, and that the former method is simpler than the latter method. It is also shown that the relaxation method neglects the low temperature effects which are given by the first part of interference term, and that the relaxation method in the narrowing limit neglects the effects of the interference term completely. The general form of admittance for the case of decoupled initial condition is given by the method of TCL equation with external driving terms, and can be equal to that for the case of thermal equilibrium initial condition. The above facts are verified by applying to a quantal oscillator. Validity of the second-order perturbation calculations is discussed.
The relation between the method of TCL equation with external driving terms and the relaxation method is discussed analytically in the quantal bath model. It is shown that the interference term in the former method gives the part of second order in powers of the system-bath interaction in the Kubo formula, and that the former method is simpler than the latter method. It is also shown that the relaxation method neglects the low temperature effects which are given by the first part of interference term, and that the relaxation method in the narrowing limit neglects the effects of the interference term completely. The general form of admittance for the case of decoupled initial condition is given by the method of TCL equation with external driving terms, and can be equal to that for the case of thermal equilibrium initial condition. The above facts are verified by applying to a quantal oscillator. Validity of the second-order perturbation calculations is discussed.
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.
Relation among various expressions of the complex admittance for quantum systems in contact with heat reservoir is studied. Exact expressions of the complex admittance are derived in various types of formulations of equations of motion under contact with heat reservoir. Namely, the complex admittance is studied in the relaxation method and the external-field method. In the former method, the admittance is calculated using the Kubo formula for quantum systems in contact with heat reservoir in no external driving fields, while in the latter method the admittance is directly calculated from equations of motion with external driving terms. In each method, two types of equation of motions are considered, i.e., the time-convolution (TC) equation and time-convolutionless (TCL) equation. That is, the full of the four cases are studied. It is turned out that the expression of the complex admittance obtained by using the relaxation method with the TC equation exactly coincides with that obtained by using the external-field method with the TC equation, while other two methods give different forms. It is also explicitly demonstrated that all the expressions of the complex admittance coincide with each other in the lowest Born approximation for the system-reservoir interaction. The formulas necessary for the higher-order expansions in powers of the system-reservoir interaction are derived, and also the expressions of the admittance in the nth order approximation are given. By transforming inverse-temperature integrals into time integrals, the admittances are also given in the formulas with time-integrals alone. To characterize the TC and TCL methods, we study the expressions of the admittances of two exactly solvable models. Each exact form of admittance is compared with the results of the two methods in the lowest Born approximation. It is found that depending on the model, either of TC and TCL would be the better method.
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.