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Annalen der Physik

Disappearance of time integrals of exact memory functions in time‐convolution generalized master equations

Vladislav Čápek

Published Jan 1, 1998• 1 citation
Publisher

Abstract

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.

DOI
10.1002/andp.19985100305

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Disappearance of time integrals of exact memory functions in time-convolution generalized master equations

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•May 10, 2021•0 citations
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.
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 application of the sudden approximation in the derivation of a quantal generalized master equation (GME) is examined. Two different types of physical systems are considered. One is a composite system comprised of a fast primary system and slow bath compared to the time the former is coupled to the latter. The other is a composite system comprised of a slow primary system and fast bath. The resulting sudden GME’s for both cases contain non-Markovian memory kernels. In the second case, the memory kernel can be further approximated by a Markovian form. The resulting Markovian-sudden GME is identical to the GME obtained by using the adiabatic elimination method for removing the (fast) stochastic bath coordinates. Using a representation of the Schrödinger propagator for the density operator analogous to the recently developed (energy) sudden representation of the Schrödinger propagator for the wave function, the exact GME is recast into a form such that when the memory kernel and the inhomogeneity term of the equation are expanded in a perturbation series, the zeroth order equation is in the sudden approximation form. Finally, a harmonic oscillator coupled linearly to a bath of harmonic oscillators is used as an illustration. The behavior of the bath correlation functions in the Markovian and the sudden limits is examined. The reduction of the exact GME to the sudden approximation form is also considered.
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-scrD0{\mathit{scrD}}_{0}scrD0​)\ensuremath{\Vert}${\mathrm{\ensuremath{\rho}}}_{0}$) (with scrD0{\mathit{scrD}}_{0}scrD0​ 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.
We study the properties of the Nakajima-Zwanzig memory kernel for a qubit immersed in a many-body localized (i.e. disordered and interacting) bath. We argue that the memory kernel decays as a power law in both the localized and ergodic regimes, and show how this can be leveraged to extract t→∞t\to\inftyt→∞ populations for the qubit from finite time (Jt≤102J t \leq 10^2Jt≤102) data in the thermalizing phase. This allows us to quantify how the long time values of the populations approach the expected thermalized state as the bath approaches the thermodynamic limit. This approach should provide a good complement to state-of-the-art numerical methods, for which the long-time dynamics with large baths are impossible to simulate in this phase. Additionally, our numerics on finite baths reveal the possibility for long-time unbounded exponential growth in the memory kernel, a phenomenon rooted in the appearance of exceptional points in the projected Liouvillian governing the reduced dynamics. This behavior, while pathological, may have interesting connections to localization and the validity of perturbative expansions of the memory kernel.