Generalized Master Equation Research Articles

Effect of the conformational transitions on electron transfer in biological systems

The influence of the conformational changes in the system on the electron transfer (ET) is considered. In the present work it was assumed that the electron interacts with two baths. The influence of one bath is through the stochastic function c( t) that characterizes the conformational changes of the system, and the other is a set of harmonic oscillators. The rate of ET was derived in the master equation approach. The solution of the generalized master equation for the gating processes was obtained. It was shown that the contribution to the temperature dependence of the rate constant that comes from electronic coupling could be dominant. The presented theory was used to provide an alternative elucidation of the cytochrome c oxidation in Chromatium vinosum photosynthetic bacteria.

Is the dynamics of open quantum systems always linear?

We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e. the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.

Equation of state and phase transitions in surface monolayer

A novel thermodynamic approach based on excluded area is suggested for deriving an equation of state for a surface monolayer. The article presents a general master equation accounting for the concentration dependence of the excluded area, whose particular results are the classical equations by van Laar and Frumkin as the zero approximation, the 2D van der Waals equation as the first approximation, and a new more precise state equation as the second approximation. The equation of state obtained also accounts for the effect of molecular orientation in a surface monolayer and allows the explanation of the role of molecular orientation in 2D first-order and second-order phase transitions.

Nonequilibrium stabilization of charge states in double quantum dots

We analyze the decoherence of charge states in double quantum dots due to cotunneling. The system is treated using the Bloch-Redfield generalized master equation for the Schrieffer-Wolff transformed Hamiltonian. We show that the decoherence, characterized through a relaxation ${\ensuremath{\tau}}_{r}$ and a dephasing time ${\ensuremath{\tau}}_{\ensuremath{\varphi}},$ can be controlled through the external voltage and that the optimum point, where these times are maximum, is not necessarily in equilibrium. We outline the mechanism of this nonequilibrium-induced enhancement of lifetime and coherence. We discuss the relevance of our results for recent charge qubit experiments.

Self-organization in a simple model of adaptive agents playing 2 x 2 games with arbitrary payoff matrices.

We analyze, both analytically and numerically, the self-organization of a system of "selfish" adaptive agents playing an arbitrary iterated pairwise game (defined by a 2 x 2 payoff matrix). Examples of possible games to play are the prisoner's dilemma (PD) game, the chicken game, the hero game, etc. The agents have no memory, use strategies not based on direct reciprocity nor "tags" and are chosen at random, i.e., geographical vicinity is neglected. They can play two possible strategies: cooperate (C) or defect (D). The players measure their success by comparing their utilities with an estimate for the expected benefits and update their strategy following a simple rule. Two versions of the model are studied: (1) the deterministic version (the agents are either in definite states C or D) and (2) the stochastic version (the agents have a probability c of playing C). Using a general master equation we compute the equilibrium states into which the system self-organizes, characterized by their average probability of cooperation c(eq). Depending on the payoff matrix, we show that c(eq) can take five different values. We also consider the mixing of agents using two different payoff matrices and show that any value of c(eq) can be reached by tuning the proportions of agents using each payoff matrix. In particular, this can be used as a way to simulate the effect of a fraction d of "antisocial" individuals--incapable of realizing any value to cooperation--on the cooperative regime hold by a population of neutral or "normal" agents.

Anomalous hydrodynamical dispersion and the transport with multiple families of paths in porous media.

We investigate a family of probability distributions that shows anomalous hydrodynamic dispersion, by solving a particular class of coupled generalized master equations. The Fourier-Laplace solution is obtained analytically in terms of the matrix Green function method; then the Coats-Smith concentration profile is revisited in a particular case. Two models of disorder are worked out explicitly, and the mean current is asymptotically calculated. We present an approximation method to calculate the first passage time distribution for this stochastic transport process, and as an example an exact Markovian result is worked out; scaling results are also shown. We discuss the comparison with other different methods to work out complex diffusion phenomena in the presence of disordered multiple transport paths. Extensions when the models are nondiffusive can also be solved in the Fourier-Laplace representation.

Equation of state and phase transitions in surface monolayer

A novel thermodynamic approach based on excluded area is suggested for deriving an equation of state for a surface monolayer. The article presents a general master equation accounting for the concentration dependence of the excluded area, whose particular results are the classical equations by van Laar and Frumkin as the zero approximation, the 2D van der Waals equation as the first approximation, and a new more precise state equation as the second approximation. The equation of state obtained also accounts for the effect of molecular orientation in a surface monolayer and allows the explanation of the role of molecular orientation in 2D first-order and second-order phase transitions.

Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport

We study the time behavior of solute transport in a heterogeneous medium. We consider a spatially biased continuous time random walk (CTRW) governed by ψ( s ,t) , the joint probability density for an event–displacement s with an event–time t. In this effective transport framework the concentration distribution of a solute is given by a generalized master equation (GME). We present results detailing the time dependence for the resident and flux concentrations, the center of mass velocity and the macroscopic dispersion coefficients of the solute plume. The Laplace transform of the GME is converted to a spatial differential equation resembling the advection dispersion equation (ADE), with Laplace space dependent coefficients though, and can then be solved explicitly in Laplace space. The transport behavior of the solute is then determined by accurate numerical inverse Laplace transforms. The confirmation of the accuracy of our methods is demonstrated by the excellent agreement with efficient random walk simulations based on the same ψ( s ,t) . The ψ( s ,t) is given by the product of a Gaussian distribution for s and a truncated power-law distribution for t. This particular choice allows for a systematic study of the time regimes of anomalous and normal transport behavior and the transition from normal to anomalous behavior. The presented results show new aspects for the modeling of solute transport in heterogeneous media, in particular the effect of the system “memory” on plume patterns at asymptotically long times. In a specific application we solve for the contaminant flux entering a stream from a point injection of tracer in a catchment. The results are discussed as an independent test of a model of fractal stream chemistry in catchments.

Decoherence and the Appearance of a Classical World in Quantum Theory

In the last decade decoherence has become a very popular topic mainly due to the progress in experimental techniques which allow monitoring of the process of decoherence for single microscopic or mesoscopic systems. The other motivation is the rapid development of quantum information and quantum computation theory where decoherence is the main obstacle in the implementation of bold theoretical ideas. All that makes the second improved and extended edition of this book very timely. Despite the enormous efforts of many authors decoherence with its consequences still remains a rather controversial subject. It touches on, namely, the notoriously confusing issues of quantum measurement theory and interpretation of quantum mechanics. The existence of different points of view is reflected by the structure and content of the book. The first three authors (Joos, Zeh and Kiefer) accept the standard formalism of quantum mechanics but seem to reject orthodox Copenhagen interpretation, Giulini and Kupsch stick to both while Stamatescu discusses models which go beyond the standard quantum theory. Fortunately, most of the presented results are independent of the interpretation and the mathematical formalism is common for the (meta)physically different approaches. After a short introduction by Joos followed by a more detailed review of the basic concepts by Zeh, chapter 3 (the longest chapter) by Joos is devoted to the environmental decoherence. Here the author considers mostly rather `down to earth' and well-motivated mechanisms of decoherence through collisions with atoms or molecules and the processes of emission, absorption and scattering of photons. The issues of decoherence induced superselection rules and localization of objects including the possible explanation of the molecular structure are discussed in details. Many other topics are also reviewed in this chapter, e.g., the so-called Zeno effect, relationships between quantum chaos and decoherence, the role of decoherence in quantum information processing and even decoherence in the brain. The next chapter, written by Kiefer, is devoted to decoherence in quantum field theory and quantum gravity which is a much more speculative and less explored topic. Two complementary aspects are studied in this approach: decoherence of particle states by the quantum fields and decoherence of field states by the particles. Cosmological issues related to decoherence are discussed, not only within the standard Friedmann cosmology, but also using the elements of the theory of black holes, wormholes and strings. The relations between the formalism of consistent histories defined in terms of decoherence functionals and the environmental decoherence are discussed in chapter 5, also written by Kiefer. The Feynman--Vernon influence functional for the quantum open system is presented in detail as the first example of decoherence functional. Then the general theory is outlined together with possible interpretations including cosmological aspects. The next chapter by Giulini presents an overview of the superselection rules arising from physical symmetries and gauge transformations both for nonrelativistic quantum mechanics and quantum field theory. Critical discussion of kinematical superselection rules versus dynamical ones is illustrated by numerous examples like Galilei invariant quantum mechanics, quantum electrodynamics and quantum gravity. The introduction to the theory of quantum open systems and its applications to decoherence models is given in chapter 7 by Kupsch. Generalized master equations, Markovian approximation and a few Hamiltonian models relevant for decoherence are discussed. Some mathematical tools, e.g., complete positivity and entropy inequalities are also presented. The last chapter by Stamatescu is devoted to stochastic collapse models which can be interpreted either as certain representations of the dynamics of open quantum systems or as fundamental modifications of the Schr\odinger equation. The final part of the book consists of remarks by Zeh on related concepts and methods and seven appendices. The broad spectrum, mathematically-friendly presentation, inclusion of the very recent developments and the extensive bibliography (about 550 references) make this book a valuable reference for all researchers, graduate and PhD students interested in the foundations of quantum mechanics, quantum open systems and quantum information. The relative independence of the chapters and numerous redundancies allow for selective reading, which is very helpful for newcomers to this field.

Theory of Excitation Energy Transfer in the Intermediate Coupling Case of Clusters

Previously we presented a unified theory of excitation energy transfer (EET) in dimers, which is applicable to the intermediate coupling case by self-consistently solving the stochastic Liouville equation (Kimura, A.; Kakitani, T.; Yamato, T. J. Phys. Chem. B 2000 104, 9276). However, up to the present time, proper EET theory, which is applicable to the intermediate coupling case of clusters, was not established. In this paper, we contribute to such an intermediate coupling theory in cluster systems. We utilize the method of generalized master equations (GME). In the first step, we construct the memory function by the second-order perturbation theory of the excitonic interaction from a general point of view. Eventually, it is expressed by a Fourier transform of the overlap of the time-dependent fluorescence and the absorption spectra of the constituent molecules. We show that our new theory reduces to the Kenkre-Knox theory in the limit of fast vibrational relaxation and reduces to the Sumi theory of the hot transfer mechanism in the limit of fast decay of the memory function. In the next step, we introduce a renormalization function into this memory function. This renormalization includes a correction due to the contribution of powers of combination of the fourth- and the second-order correlation functions. By numerical calculations, we illustrate how the renormalization effect becomes significant as the excitonic coupling strength U becomes large. We applied this GME method to the EET between BChla molecules in B850 of the photosynthetic antenna system LH2. We found that the memory function calculated using the experimentally obtained optical spectra initially decays very rapidly (with time constant of about 5 fs) and has a plateau in the time region 10-25 fs. Because of this specific memory function, the calculated coherence length in the steady state was about 3. This GME method should be useful for the analysis of the mechanism of EET in many kinds of biological systems.

Entropy in Kerr–de Sitter Black Holes with Spin Fields

Generalized master equations due to spin fields are given. We obtain the entropy of electromagnetic, gravitational, Dirac, and scalar fields in a unified form by using the improved brick-wall method—membrane model. The results show that, as the cutoff is properly chosen, the entropy in the black hole satisfies the Bekenstein–Hawking area formula.

Generalized master equation via aging continuous-time random walks.

We discuss the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density psi(t) that is assumed to be an inverse power law with the power index micro. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where psi(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure we create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations.

Stochastic unraveling of Redfield master equations and its application to electron transfer problems

A method for stochastic unraveling of general time-local quantum master equations (QMEs) is proposed. The present kind of jump algorithm allows a numerically efficient treatment of QMEs which are not in Lindblad form, i.e., are not positive semidefinite by definition. The unraveling can be achieved by allowing for trajectories with negative weights. Such a property is necessary, e.g., to unravel the Redfield QME and to treat various related problems with high numerical efficiency. The method is successfully tested on the damped harmonic oscillator and on electron transfer models including one and two reaction coordinates. The obtained results are compared to those from a direct propagation of the reduced density matrix (RDM) as well as from the standard quantum jump method. Comparison of the numerical efficiency is performed considering both the population dynamics and the RDM in the Wigner phase space representation.

Completely positive Bloch-Boltzmann equations

The density operator of the arbitrary physical system must be positive definite. Employing the general master equation technique which preserves this property we derive equations of motion for the density operator of an active atom which interacts collisionally with the reservoir of perturber atoms. The obtained general relations applied to the two-level atom yield Bloch-Boltzmann equations (BBE). The form of the BBE obtained by us differs from that known from literature which, as we show, are not guaranteed to preserve the required positivity. We argue that our results are the correct ones and as such should be used in practical applications. Moreover, the structure and the terms which appear in our set of BBE seem to allow simpler and more straightforward physical interpretation.

Theoretical Analysis of the Energy Gap Dependence of the Reconstituted B800 → B850 Excitation Energy Transfer Rate in Bacterial LH2 Complexes

Theoretical analysis was carried out for a remarkable energy gap dependence of the rate of excitation energy transfer (EET) from reconstituted B800 to B850 in LH2 whose experimental data were published recently (Herek, J. L.; Fraser, N. J.; Pullerits, T.; Martinsson, P.; Polivka, T.; Scheer, H.; Cogdell, R. J.; Sundstrom, V. Biophys. J. 2000, 78, 2590). This energy gap dependence could not be explained by calculations using the generalized Forster theory. As an alternative method, we solved the generalized master equation for the population of the excited BChla molecules that are coupled inside the B850 ring and between B850 and B800 by adopting second-order perturbation theory for the memory function. Making use of the experimental data of optical absorption and fluorescence spectra of monomers and proper site energies in LH2, we could theoretically reproduce the above experimentally observed energy gap dependence of the EET rate very well. The result of the present analysis indicates that the rate-limit...

Energy back transfer, migration and energy transfer (Yb-to-Er and Er-to-Yb) processes in Yb,Er:YAG

A comparison and discussion of the energy back transfer, migration of energy and direct energy transfer process, from Yb-to-Er and Er-to-Yb ions, in the luminescent system Yb, Er: YAG are presented. The sample crystal was excited with a tunable optical parametric oscillator system. The Yb ions as donors were excited at 940nm and its fluorescence decay measured at 1030nm. On the other hand, the Er ions as donors were excited at 800 and 670nm, and the fluorescence decay of Yb ions as acceptors measured at 1030nm. The analysis is done through the solution to the General Energy Transfer Master Equations. The ytterbium fluorescence decays in both cases were fitted and found that the migration of the energy among Yb ions is neglected when they act as acceptors.

Towards deterministic equations for Lévy walks: the fractional material derivative.

Lévy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Lévy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Lévy walks, in which the fractional equivalent of the material derivative occurs. Our approach is expected to be useful for the dynamical formulation of Lévy walks in an external force field or in phase space, for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited.

Quantum theory of heating of a single trapped ion

The heating of trapped ions due to the interaction with a quantized environment is studied without performing the Born–Markov approximation. A generalized master equation local in time is derived and a novel theoretical approach to solve it analytically is proposed. Our master equation is in the Lindblad form with time dependent coefficients, thus allowing the simulation of the dynamics by means of the Monte Carlo Wave Function (MCWF) method.

Conditions for bounded solutions of non-Markovian quantum master equations

For the most general quantum master equations, also called Nakajima–Zwanzig or non-Markovian equations, we define suitable boundedness conditions on integral kernels and inhomogeneity terms in order to derive with mathematical rigor an upper bound on solutions, as required by the von Neumann conditions. Such equations are of importance for quantum dynamics of open systems with arbitrary couplings to environment and arbitrary entangled initial states. The derivation is based on an equivalent coherence-vector representation in finite dimension n leading to coupled Volterra integro-differential equations of second kind and convolution type in an (n 2−1)-dimensional real vector space. As examples, analytical and numerical model solutions are worked out for 2-level systems in order to test suitable trial functions for input quantities. All this is motivated by the fact that exact solutions can hardly be found but appropriate trial functions may provide a reasonable semiphenomenological description of complicated quantum dynamics.

Influence of clustering on a two-dimensional state equation

A general master equation accounting for the concentration dependence of the excluded area is presented, whose particular results are the two-dimensional van der Waals equation of state as the first approximation and a new much more precise state equation as the second approximation.