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Physical Review A

Consistent approximations in generalized-master-equation theories: Application to the asymptotic-time symmetry-breaking problem

Vladislav C wápek, Petr Chvosta

Published Mar 1, 1991• 7 citations

Abstract

The time-convolution as well as the time-convolutionless generalized master equations (GME) are rewritten in a simple form permitting an exact formal solution. Using this form, it is argued that possible approximations must be performed consistently on both sides of the GME in order to control deviations of the emerging approximate solution from the exact one. Arguments are given that after such consistent approximations, the initial-condition term can no longer in general be taken as proportional to (scrI-${\mathit{scrD}}_{0}$)\ensuremath{\Vert}${\mathrm{\ensuremath{\rho}}}_{0}$) (with ${\mathit{scrD}}_{0}$ being the projector used in GME). These conclusions are illustrated for the asymptotic-time symmetry-breaking problem of the spin-boson model. Although standard approximate GME approaches do indicate a possible symmetry breaking, we argue that these results are strongly influenced by the inconsistency of the approximations used. To support this statement we prove the absence of any asymptotic-time symmetry breaking in the zero-bias spin-boson model with a partial site-occupation relaxation.

DOI: 10.1103/physreva.43.2819

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The time-convolution as well as the time-convolutionless generalized master equations (GME) are rewritten in a simple form permitting an exact formal solution. Using this form, it is argued that possible approximations must be performed consistently on both sides of the GME in order to control deviations of the emerging approximate solution from the exact one. Arguments are given that after such consistent approximations, the initial-condition term can no longer in general be taken as proportional to (scrI-

{\mathit{scrD}}_{0}

)\ensuremath{\Vert}

{\mathrm{\ensuremath{\rho}}}_{0}

) (with

{\mathit{scrD}}_{0}

being the projector used in GME). These conclusions are illustrated for the asymptotic-time symmetry-breaking problem of the spin-boson model. Although standard approximate GME approaches do indicate a possible symmetry breaking, we argue that these results are strongly influenced by the inconsistency of the approximations used. To support this statement we prove the absence of any asymptotic-time symmetry breaking in the zero-bias spin-boson model with a partial site-occupation relaxation.

Memory functions in time-convolution Generalized Master Equations (GME) for probabilities of finding a general system (interacting by a general coupling with a true thermodynamic bath) in individual states are considered without resorting to any approximation. After taking the thermodynamic bath limit, time integrals from zero to infinite times of the memories are considered. It is argued that these integrals entering, e.g., the usual naive Markov approximation converting GME the Pauli master (PME) equations are exactly zero. This implies long-time tails of memories (unobtainable by perturbational expansions) and slower-than-exponential long-time asymptotics of relaxation.

Abstract Memory functions in time‐convolution Generalized Master Equations (GME) for probabilities of finding a general system (interacting by a general coupling with a true thermodynamic bath) in individual states are considered without resorting to any approximation. After taking the thermodynamic bath limit, time integrals from zero to infinite times of the memories are considered. It is argued that these integrals entering, e.g., the usual naive Markov approximation converting GME the Pauli master (PME) equations are exactly zero. This implies long‐time tails of memories (unobtainable by perturbational expansions) and slower‐than‐exponential long‐time asymptotics of relaxation.

Most traditional discussions of the Markoffian Born approximation of relaxation phenomena (e.g., Bloch equations) are based on the assumption of widely separated time scales in the non-Markoffian weak coupling generalized master equation. By exploiting the physically reasonable Gaussian factorization property of multitime correlation functions which enter through the system reservoir interaction, recent work has shed significant new light on the relationship between the Markoffian-Born approximation and the exact generalized master equation (GME) describing the evolution of the relaxing subsystem. Two limits for which the GME becomes the Born approximation relaxation equation arc discussed: (1) the Van Hove (λ2t) limit rigorously discussed for open systems by Davies, (2) the zero memory (white noise) suggested by Middleton—Schieve and recently discussed for open systems by Gorini—Kossakawsky and Palmer. We have outlined the proof of this limit. We then discuss the physical meaning of these two exact limits in the context of the general theory of kinetic equations of Resibois. The dominant pole contributions are isolated to all orders by means of an expansion in which the terms are in norm proportional to powers of the ratio of the time scales τc/τB where τc is the two point correlation memory time and τB the relaxation time in the Born approximation. Each of the above exact limits corresponds to the vanishing of this ratio. Comments are made on nonexponential decay.

The author applied the generalised master equation (GME) to the analysis of the object function, s(t), which describes the motion of the tunnelling particle in a double-well system coupled to a reservoir consisting of independent bosonic-type elementary excitation. First, using the 'unrelaxed' initial condition and a suitable projection operator we give the exact formal solution of the GME. Then, the memory operator in the GME is explicitly calculated in the weak-tunnelling regime and the function s(t) is obtained. An independent derivation is given which enables us to obtain the exact expansion of the function s(t) by means of a purely algebraic method. This expansion forms a basis on which this GME method and the well known functional-integral approach are compared: the GME-weak-tunnelling approximation is shown to be identical to the commonly used non-interacting-blip approximation ensuing from the functional-integral method. A new approximation is analysed which is weaker than the non-interacting-blip one.

In the frame of the generalized master equation (GME) approach, an analysis of the asymptotic spin relaxation is made for different j values. The fully coupled (FC) model and the rotating-wave approximation (RWA) for the system-reservoir coupling are considered and it is seen that they give rise to structurally different GMEs. In particular, we examine the evolution of the diagonal and off-diagonal matrix elements of the density operator, the mean values of the relevant observables and the equilibrium solution for both coupling models and display their significant differences. Particularly the spin j=1 case is examined.