Generalized Master Equation Research Articles

Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition.

We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

Electron transfer through a molecular wire: Consideration of electron-vibrational coupling within the Liouville space pathway technique

To fully account for electron-vibrational coupling and vibrational relaxation in the course of electron motion through a molecular wire a density operator approach is utilized. If combined with a particular projection operator technique a generalized master equation can be derived which governs the populations of the electronic wire states. The respective memory kernels are determined beyond any perturbation theory with respect to the electron-vibrational coupling and can be classified via so-called Liouville space pathways. An ordering of the different contributions to the current-voltage characteristics becomes possible by introducing an electron transmission coefficient which describes ballistic as well as inelastic electron transport through the wire. The general derivations are illustrated by numerical calculations which demonstrate the drastic influence of the electron-vibrational coupling on the wire transmission coefficient as well as on the current-voltage characteristics.

The role of different reorganization energies within the Zusman theory of electron transfer

We consider the kinetics of electron transfer reactions in condensed media with different reorganization energies for the forward and backward processes. The starting point of our analysis is an extension of the well-known Zusman equations to the case of parabolic diabatic curves with different curvatures. A generalized master equation for the populations as well as formal expressions for their long-time limit is derived. We discuss the conditions under which the time evolution of the populations of reactants and products can be described at all times by a single exponential law. In the limit of very small tunnel splitting, a novel rate formula for the nonadiabatic transitions is obtained. It generalizes previous results derived within the contact approximation. For larger values of the tunnel splitting, we make use of the consecutive step approximation leading to a rate formula that bridges between the nonadiabatic and solvent-controlled adiabatic regimes. Finally, the analytical predictions for the long-time populations and for the rate constant are tested against precise numerical solutions of the starting set of partial differential equations.

Stochastic unraveling of time-local quantum master equations beyond the Lindblad class.

A method for stochastic unraveling of general time-local quantum master equations (QME) which involve the reduced density operator at time t only is proposed. The present kind of jump algorithm enables a numerically efficient treatment of QMEs that are not of Lindblad form. So it opens large fields of application for stochastic methods. The unraveling can be achieved by allowing for trajectories with negative weight. We present results for the quantum Brownian motion and the Redfield QMEs as test examples. The algorithm can also unravel non-Markovian QMEs when they are in a time-local form like in the time-convolutionless formalism.

Non-Markovian random processes and traveling fronts in a reaction-transport system with memory and long-range interactions.

The problem of finding the propagation rate for traveling waves in reaction-transport systems with memory and long-range interactions has been considered. Our approach makes use of the generalized master equation with logistic growth, hyperbolic scaling, and Hamilton-Jacobi theory. We consider the case when the waiting-time distribution for the underlying microscopic random walk is modeled by the family of gamma distributions, which in turn leads to non-Markovian random processes and corresponding memory effects on mesoscopic scales. We derive formulas that enable us to determine the front propagation rate and understand how the memory and long-range interactions influence the propagation rate for traveling fronts. Several examples involving the Gaussian and discrete distributions for jump densities are presented.

Quantum retrodiction in open systems

Quantum retrodiction involves finding the probabilities for various preparation events given a measurement event. This theory has been studied for some time but mainly as an interesting concept associated with time asymmetry in quantum mechanics. Recent interest in quantum communications and cryptography, however, has provided retrodiction with a potential practical application. For this purpose quantum retrodiction in open systems should be more relevant than in closed systems isolated from the environment. In this paper we study retrodiction in open systems and develop a general master equation for the backward time evolution of the measured state, which can be used for calculating preparation probabilities. We solve the master equation, by way of example, for the driven two-level atom coupled to the electromagnetic field.

On homogeneous generalized master equations

It is argued that the method recently proposed by Los (Los V F 2001 J. Phys. A: Math. Gen. 34 6389) turning inhomogeneous time-convolution generalized master equations (GMEs) into homogeneous ones is problematic in the quantum case. It is also shown here that the proposed method is inapplicable to time-convolutionless GMEs.

Influence of phonons on exciton transfer dynamics: comparison of the Redfield, Förster, and modified Redfield equations

The effect of the reorganization of phonons on exciton transfer dynamics is examined. Starting from a general quantum master equation, we obtain population transfer rates for the Redfield, the Förster, and the modified Redfield equations and compare the predictions of these three models over various ranges of parameters. It is shown that the validity of the traditional Redfield equation is justified by a broad spectrum of phonons rather than a small magnitude of electron–phonon coupling strength as is usually implied. The modified Redfield equation suggested by Zhang et al. [J. Chem. Phys. 108 (1998) 7763] is shown to have a wide range of applicability and to reduce numerically to the traditional Redfield equation and the Förster equation in the respective limiting cases.

Decoherence, irreversibility, and selection by decoherence of exclusive quantum states with definite probabilities

The problem investigated in this paper is einselection, i.e., the selection of mutually exclusive quantum states with definite probabilities through decoherence. Its study is based on a theory of decoherence resulting from the projection method in the quantum theory of irreversible processes, which is general enough for giving reliable predictions. This approach leads to a definition (or redefinition) of the coupling with the environment involving only fluctuations. The range of application of perturbation calculus is then wide, resulting in a rather general master equation. Two distinct cases of decoherence are then found: (i) a ``degenerate'' case (already encountered with solvable models) where decoherence amounts essentially to approximate diagonalization; (ii) a general case where the einselected states are essentially classical. They are mixed states. Their density operators are proportional to microlocal projection operators (or ``quasiprojectors'') that were previously introduced in the quantum expression of classical properties. It is found at various places that the main limitation in our understanding of decoherence is the lack of a systematic method for constructing collective observables.

Empirical Formula of Exciton Coherent Domain in Oligomers and Application to LH2

We investigated the mechanism by which the length of exciton coherent domain in the linear array of pigments is determined. Correctly solving a generalized master equation with memory function obtained by the second-order perturbation of the intermolecular interaction, we obtained an empirical formula for the length of exciton coherent domain, abbreviated as coherence length. In the absence of inhomogeneity, this coherence length is expressed by a linear function of the ratio between the intermolecular coupling strength U and the destructing force γ of a coherent bond. In the presence of inhomogeneity, the coherence length is expressed by a linear function of U/γ only approximately, and its deviation becomes significant when the degree of inhomogeneity is larger than about 100 cm-1. Applying this formula to the LH2 antenna complex of photosynthetic purple bacteria Rhodopseudomonas acidophila, we found that the coherence length of B850 is about 5 bacteriochlorophyll molecules in the absence of inhomogeneit...

Energy Back-Transfer and other Nonradiative Energy-Transfer Processes in Yb3 + , Er3 + : Y 3Al5 O 12

Experimental data from two yttrium aluminum garnet (YAG) crystal samples with different doping concentrations were analyzed through the calculation of the exact solution of the general nonradiative energy-transfer master equations. Besides the dipole-dipole direct energy transfer and the dipole-dipole migration processes assumed by other authors to predict the -fluorescence decay, it is shown that a quadrupole-quadrupole direct energy transfer and a dipole-dipole back-transfer process are also present. The used free parameter values predicted our experimental data as well as other experimental data reported in the literature. Further, our modeling also calculates the acceptor transients which were also compared to experimental data. Therefore, the used modeling can be applied to analyze complicated nonradiative energy-transfer processes where traditional models fail. © 2002 The Electrochemical Society. All rights reserved.

Note on the equivalence of different approximations in the relaxation theory

We briefly discuss relations between different variants of the second order generalized master equations (GME), in particular among different types of the Markov-Born approximation of time-convolution GME and Born approximation in time-convolutionless one. We prove that equivalence valid in the van Hove limit does not in general apply for other types of scaling. On the other hand, for other scalings one appropriate form of the interaction representation always exist that reproduces this equivalence known from the weak-coupling (van Hove) one.

Dissipative dynamics and electronic coherence

In the second-order approximation, the Markovian generalised master equation is applied to the spin-boson Hamiltonian to find the dynamics of a dissipative two-state subsystem (TSS). The results are compared with those obtained by the path-integral formalism and considered to evaluate the possibility of apparition of electronic coherence for the electron transfer reaction in certain photosynthetic reaction centers.

Dynamics of the spin-boson Hamiltonian by the projection operator technique: applications to electron transfer reactions.

A perturbative treatment developed by using the projection operator technique is provided to find dynamics of the spin-boson Hamiltonian in the second order approximation of the subsystem-bath interaction. In the framework of the generalized master equations and in the Markovian approximation it leads to the Redfield/Bloch-type equations. The treatment can be applied to both fast and slow bath cases; in this paper we consider the fast bath case. The relaxation times, energy splitting, and bath-induced renormalization effect of the coherence frequency are discussed and applied to a Lorentzian-Ohmic fast bath. A good agreement between our results and those obtained by the path-integral formalism is obtained. The treatment is applied to electron transfer reactions to test the possibility of apparition of the electronic coherence in certain photosynthetic reaction centers.

Influence of phonons on exciton transfer dynamics: comparison of the Redfield, Förster, and modified Redfield equations

The effect of the reorganization of phonons on exciton transfer dynamics is examined. Starting from a general quantum master equation, we obtain population transfer rates for the Redfield, the Förster, and the modified Redfield equations and compare the predictions of these three models over various ranges of parameters. It is shown that the validity of the traditional Redfield equation is justified by a broad spectrum of phonons rather than a small magnitude of electron–phonon coupling strength as is usually implied. The modified Redfield equation suggested by Zhang et al. [J. Chem. Phys. 108 (1998) 7763] is shown to have a wide range of applicability and to reduce numerically to the traditional Redfield equation and the Förster equation in the respective limiting cases.

Optimal co-doping concentrations and dynamics of energy transferprocesses for Tm3+-Tb3+ and Tm3+-Eu3+ in LiYF4crystal hosts

The dynamics of energy transfer processes in Tm+3-Tb+3 andTm3+-Eu+3 co-doped LiYF4 crystal hosts were studied fromtime-resolved Tm3+ fluorescence analysis to estimate the optimalco-doping concentrations which maximize 1.5 µm laser emission from the3H4 state of Tm3+. The analysis was carried out by findinga numerical solution to the general master equations that govern non-radiativeenergy transfer processes in crystalline materials and by using the MonteCarlo technique. Our analysis improves the description of experimentalfluorescence decay curves. The predicted optimal co-doping concentrations andlaser threshold for these luminescent systems are lower than those reportedusing traditional models for non-radiative energy transfer processes.

Homogeneous generalized master equation retaining initial correlations

Using the projection operator technique, the exact homogeneous generalized master equation (HGME) for the relevant part of a distribution function (statistical operator) is derived. The exact (mass) operator governing the evolution of the relevant part of a distribution function and comprising arbitrary initial correlations is found. Neither the Bogolyubov principle of weakening of initial correlations with time nor any other approximation such as random phase approximation has been used to obtain the HGME. These approximations are usually used to derive the approximate homogeneous equation for the relevant part of a distribution function from the conventional exact generalized master equation (GME), which has a source containing the irrelevant part (initial correlations). The HGME does not have a source and contains only the linear, relative to the relevant part of a distribution function, terms of the GME modified by the dynamics of initial correlations. The obtained equation is valid on any timescale, for any initial moment of time and any initial correlations. In particular, it describes the short-time behaviour and allows for treating the influence of initial correlations consistently. As an example, we have considered a dilute gas of classical particles. By selecting the appropriate projection operator, we have derived the homogeneous equation for a one-particle distribution function retaining initial correlations in the linear approximation on the small density parameter and for the space homogeneous case. This equation allows for considering all stages of the time evolution. It converts into the conventional Boltzmann equation on the appropriate timescale if the contribution of all initial correlations vanishes on this timescale.

A derivation of the Dirac equation in an external field based on the Poisson process

The Dirac equation has been derived from the master equation of Poisson process by analytic continuation. We extend it to the case where a particle moves in an external field. Furthermore, we show that the generalized master equation is intimately connected with three-dimensional Dirac equation in an external field.

Subsystem dynamics in mixed quantum–classical systems

Starting with the mixed quantum–classical Liouville equation, projection operator methods are used to derive an equation of motion for a quantum subsystem dissipatively interacting with a classical bath. The resulting generalized master equation is reduced to the Lindblad equation after making a Markovian approximation in the weak coupling limit. The bath subsystem dynamics is studied from a similar perspective. For situations where the classical subsystem consists of few degrees of freedom, or one is interested in the nature of the modifications of its dynamics as a result of coupling to a large, rapidly relaxing quantum bath, a classical analog of the Lindblad equation is derived and discussed.

Study of decoherence with a generalized master equation

The behavior of macroscopic system is investigated with a generalized master equation. Making use of a coarse-grained position operator based on the SO(3,1), we can derive that the macroscopic nature of this operator induces decoherence leading to classical behavior.