Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

Stochastic Equations of Motion for the Linear Response. I. General Theory

Mizuhiko Saeki

Published Dec 1, 1983• 7 citations
Publisher

Abstract

Time-convolution and time-convolutionless expressions for the quantal equation of motion for the linear response of a system under random perturbations to an external, time-varying driving field are derived from the stochastic quantal Liouville equation for the total system. They have forms convenient for the perturbational expansions in powers of the random perturbations. One is compared with another for two initial conditions. It is shown that the time-convolutionless form of equation prevails over the time-convolution form, especially when the effects of the random perturbations are discussed except in the narrowing limit or when the effects of interference of the external driving field and the random perturbations are discussed. The higher order expansions of these equations are mentioned.

DOI
10.1143/jpsj.52.4081

Related papers

A Generalized Equation of Motion for the Linear Response. I: General Theory

Mizuhiko Saeki

A time-convolutionless expression for the quantal equation of motion for the linear response of a system in contact with a heat bath to an external, time-varying driving field is derived from the quantal Liouville equation of motion for the total system. This is done by renormalizing the memory effect by the method of Shibata, Takahashi and Hashitsume. It has a form convenient for the perturbational expansion. It is shown that this equation coincides with the time-convolution equation of Argyres and Kelley in their Markoffian approximation in the narrowing limit and in the lowest Born approximation for the system-bath interaction. This equation is expanded up to fourth order in powers of the system-bath interaction.

Progress of Theoretical Physics•May 1, 1982•33 citations

Generalized Master Equations for Driven System

Mizuhiko Saeki

Time-convolutionless forms of the quantal master equations for driven open system are derived from the Liouville equation for the total system under an arbitrary initial condition for the two cases of weak external driving fields and of arbitrary ones. They have forms convenient for the perturbational expansions, and are respectively compared with the time-convolution forms of equations in the lowest Born approximation for the interaction H SB of the system with its surroundings. It is shown that the time-convolutionless forms of equations prevail over the time-convolution ones, and that the conventional Markoffian approximation for the latter in the narrowing limit leads to the incorrect conclusion. The time-convolutionless form of master equation is expanded up to n -th order in powers of the external driving fields in the lowest Born approximation for H SB .

Journal of the Physical Society of Japan•Jun 15, 1986•22 citations

Stochastic Liouville Equation for Weakly Driven System. II: Application to a Quantal Spin

Mizuhiko Saeki

The time-convolutionless stochastic equation derived in a previous paper is applied to a quantal spin interacting weakly with its surroundings under decoupled initial condition, up to second order in powers of random perturbations. The transverse and longitudinal susceptibilities are derived from the stationary solutions. The transverse and longitudinal spectra are displayed and are compared with the ones derived from the time-convolution equation in the conventional Markoffian approximation, and with the ones by the method of second-order spin relaxation theory and Kubo's theory. It is shown that the line shape of transverse spectrum is enhanced over the Lorentzian at the resonance region, due to the effects of interference of the external driving field and the random interaction.

Progress of Theoretical Physics•Sep 1, 1988•3 citations

A Generalized Equation of Motion for the Linear Response. III: Simple Application to a Quantal Spin

Mizuhiko Saeki

The time-convolutionless equation of motion for the linear response derived in a previous paper is applied to a simple system of a quantal spin interacting weakly with its surroundings (heat bath). The operator form of equation is derived and is transcribed into the c-number one by the generalized phase-space method for spin operators. The equations of motion for the longitudinal and transverse expectation values of the spin are derived and are solved for the case of S=1/2 (S: the magnitude of spin). The longitudinal and transverse susceptibilities are derived from the stationary solutions, and are compared with the ones derived by the method of spin relaxation theory and Kubo theory for the linear response.

Progress of Theoretical Physics•Feb 1, 1987•6 citations

A Generalized Equation of Motion for the Lineqr Response. II: Simple Application to a Quantal Oscillator

Mizuhiko Saeki

The time-convolutionless equation of motion for the linear response derived in a previous paper is applied to a simple system of a quantal oscillator. This system is assumed to be interacting weakly with its surroundings (heat bath). The two cases for the initial state of the total system are considered. One case is that the oscillator and heat bath are in thermal equilibrium, and another is that these two systems are uncoupled and in each thermal equilibrium at the same temperature. The operator forms of equations are transcribed into the c-number ones by the phase-space method. The equations of motion for the expectation values of the boson operator and boson number operator are derived from the c-number equations and are solved for the case of a non-adiabatic oscillator-bath interaction. The results are compared with the ones in the conventional Markoffian approximation.

Progress of Theoretical Physics•Mar 1, 1985•8 citations

On Statistical Mechanics of Nonequilibrium Fluctuations

M. Ichiyanagi

Without assuming detailed balance, the linear response theory of systems, in which the notation of a quantum dynamical semigroup can be defined, is discussed. This theory is used to prove a fluctuation-dissipation theorem for nonstationary nonequilibrium states. We propose an extended version of the Liouville equation in which we incorporate the stochastic behavior of the subsystem of the macroscopic motions. For this we introduce a non-Hermitian Hamiltonian and set a Liouville equation for a density matrix which is not Hermitian. We show that a response to external perturbations reduces to a sum of two contributions, one for which an improved fluctuation-dissipation theorem is proved and the other due to stochastic excitations that does not satisfy a reciprocity relation of Onsager type.

Progress of Theoretical Physics•Jun 1, 1984•0 citations
Time-convolution and time-convolutionless expressions for the quantal equation of motion for the linear response of a system under random perturbations to an external, time-varying driving field are derived from the stochastic quantal Liouville equation for the total system. They have forms convenient for the perturbational expansions in powers of the random perturbations. One is compared with another for two initial conditions. It is shown that the time-convolutionless form of equation prevails over the time-convolution form, especially when the effects of the random perturbations are discussed except in the narrowing limit or when the effects of interference of the external driving field and the random perturbations are discussed. The higher order expansions of these equations are mentioned.
A time-convolutionless expression for the quantal equation of motion for the linear response of a system in contact with a heat bath to an external, time-varying driving field is derived from the quantal Liouville equation of motion for the total system. This is done by renormalizing the memory effect by the method of Shibata, Takahashi and Hashitsume. It has a form convenient for the perturbational expansion. It is shown that this equation coincides with the time-convolution equation of Argyres and Kelley in their Markoffian approximation in the narrowing limit and in the lowest Born approximation for the system-bath interaction. This equation is expanded up to fourth order in powers of the system-bath interaction.
Time-convolutionless forms of the quantal master equations for driven open system are derived from the Liouville equation for the total system under an arbitrary initial condition for the two cases of weak external driving fields and of arbitrary ones. They have forms convenient for the perturbational expansions, and are respectively compared with the time-convolution forms of equations in the lowest Born approximation for the interaction H SB of the system with its surroundings. It is shown that the time-convolutionless forms of equations prevail over the time-convolution ones, and that the conventional Markoffian approximation for the latter in the narrowing limit leads to the incorrect conclusion. The time-convolutionless form of master equation is expanded up to n -th order in powers of the external driving fields in the lowest Born approximation for H SB .
The time-convolutionless stochastic equation derived in a previous paper is applied to a quantal spin interacting weakly with its surroundings under decoupled initial condition, up to second order in powers of random perturbations. The transverse and longitudinal susceptibilities are derived from the stationary solutions. The transverse and longitudinal spectra are displayed and are compared with the ones derived from the time-convolution equation in the conventional Markoffian approximation, and with the ones by the method of second-order spin relaxation theory and Kubo's theory. It is shown that the line shape of transverse spectrum is enhanced over the Lorentzian at the resonance region, due to the effects of interference of the external driving field and the random interaction.
The time-convolutionless equation of motion for the linear response derived in a previous paper is applied to a simple system of a quantal spin interacting weakly with its surroundings (heat bath). The operator form of equation is derived and is transcribed into the c-number one by the generalized phase-space method for spin operators. The equations of motion for the longitudinal and transverse expectation values of the spin are derived and are solved for the case of S=1/2 (S: the magnitude of spin). The longitudinal and transverse susceptibilities are derived from the stationary solutions, and are compared with the ones derived by the method of spin relaxation theory and Kubo theory for the linear response.
The time-convolutionless equation of motion for the linear response derived in a previous paper is applied to a simple system of a quantal oscillator. This system is assumed to be interacting weakly with its surroundings (heat bath). The two cases for the initial state of the total system are considered. One case is that the oscillator and heat bath are in thermal equilibrium, and another is that these two systems are uncoupled and in each thermal equilibrium at the same temperature. The operator forms of equations are transcribed into the c-number ones by the phase-space method. The equations of motion for the expectation values of the boson operator and boson number operator are derived from the c-number equations and are solved for the case of a non-adiabatic oscillator-bath interaction. The results are compared with the ones in the conventional Markoffian approximation.
Without assuming detailed balance, the linear response theory of systems, in which the notation of a quantum dynamical semigroup can be defined, is discussed. This theory is used to prove a fluctuation-dissipation theorem for nonstationary nonequilibrium states. We propose an extended version of the Liouville equation in which we incorporate the stochastic behavior of the subsystem of the macroscopic motions. For this we introduce a non-Hermitian Hamiltonian and set a Liouville equation for a density matrix which is not Hermitian. We show that a response to external perturbations reduces to a sum of two contributions, one for which an improved fluctuation-dissipation theorem is proved and the other due to stochastic excitations that does not satisfy a reciprocity relation of Onsager type.