Showing papers on "Master equation published in 2007"
TL;DR: Fluctuation theorems make use of time reversal to make predictions about entropy production in many-body systems far from thermal equilibrium as mentioned in this paper, and a wide variety of distinct, but interconnected, relations have been derived and investigated theoretically and experimentally.
Abstract: Fluctuation theorems make use of time reversal to make predictions about entropy production in many-body systems far from thermal equilibrium. Here we review the wide variety of distinct, but interconnected, relations that have been derived and investigated theoretically and experimentally. Significantly, we demonstrate, in the context of Markovian stochastic dynamics, how these different fluctuation theorems arise from a simple fundamental time-reversal symmetry of a certain class of observables. Appealing to the notion of Gibbs entropy allows for a microscopic definition of entropy production in terms of these observables. We work with the master equation approach, which leads to a mathematically straightforward proof and provides direct insight into the probabilistic meaning of the quantities involved. Finally, we point to some experiments that elucidate the practical significance of fluctuation relations.
391 citations
TL;DR: The Fokker-Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative andCaputo derivative.
232 citations
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TL;DR: A practical introduction to stochastic modelling of reaction-diffusion processes is presented, starting with the classical Gillespie algorithm for the stochastically modelling of chemical reactions and progressing to more advanced methods and problems.
Abstract: A practical introduction to stochastic modelling of reaction-diffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modelling of chemical reactions. Then stochastic algorithms for modelling molecular diffusion are given. Finally, basic stochastic reaction-diffusion methods are presented. The connections between stochastic simulations and deterministic models are explained and basic mathematical tools (e.g. chemical master equation) are presented. The article concludes with an overview of more advanced methods and problems.
224 citations
TL;DR: In this paper, a general classification of non-equilibrium steady states (NESS) is proposed, in which these currents play a central distinguishing role, and the transformations of the dynamic transition rates which leave a given NESS invariant are specified.
Abstract: One of the key features of non-equilibrium steady states (NESS) is the presence of non-trivial probability currents. We propose a general classification of NESS in which these currents play a central distinguishing role. As a corollary, we specify the transformations of the dynamic transition rates which leave a given NESS invariant. The formalism is most transparent within a continuous-time master equation framework since it allows for a general graph-theoretical representation of the NESS. We discuss the consequences of these transformations for entropy production, present several simple examples, and explore some generalizations, to discrete time and continuous variables.
219 citations
TL;DR: An alternative master equation is derived that is capable of describing a stationary energy current and, at the same time, leads to a completely positive dynamical map that paves the way for efficient numerical investigations of heat transport in larger systems based on Monte Carlo wave function techniques.
Abstract: We investigate heat transport in a spin-1/2 Heisenberg chain, coupled locally to independent thermal baths of different temperature. The analysis is carried out within the framework of the theory of open systems by means of appropriate quantum master equations. The standard microscopic derivation of the weak-coupling Lindblad equation in the secular approximation is considered, and shown to be inadequate for the description of stationary nonequilibrium properties like a nonvanishing energy current. Furthermore, we derive an alternative master equation that is capable of describing a stationary energy current and, at the same time, leads to a completely positive dynamical map. This paves the way for efficient numerical investigations of heat transport in larger systems based on Monte Carlo wave function techniques.
200 citations
TL;DR: A general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks derived from the Langevin approach is introduced.
159 citations
TL;DR: In this article, the authors derived an exact master equation with time-dependent coefficients reflecting the non-Markovian influence of the environment on entanglement dynamics of continuous-variable quantum channels in terms of an entangled squeezed state of two cavity fields.
Abstract: We investigate the entanglement dynamics of continuous-variable quantum channels in terms of an entangled squeezed state of two cavity fields in a general non-Markovian environment. Using the Feynman-Vernon influence functional theory in the coherent-state representation, we derive an exact master equation with time-dependent coefficients reflecting the non-Markovian influence of the environment. The influence of environments with different spectral densities, e.g., Ohmic, sub-Ohmic, and super-Ohmic, is numerically studied. The non-Markovian process shows its remarkable influence on the entanglement dynamics due to the sensitive time dependence of the dissipation and noise functions within the typical time scale of the environment. The Ohmic environment shows a weak dissipation-noise effect on the entanglement dynamics, while the sub-Ohmic and super-Ohmic environments induce much more severe noise. In particular, the memory of the system interacting with the environment contributes a strong decoherence effect to the entanglement dynamics in the super-Ohmic case.
150 citations
TL;DR: In this article, the authors obtained general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q = 1-p to the left.
Abstract: In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q=1-p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the limit as N goes to infinity. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schuetz for the case N=2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m'th left-most particle is at x at time t. In one of these formulas we can take the limit as N goes to infinity, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.
147 citations
TL;DR: Three integral fluctuation theorems are derived for these contributions and it is shown that they lead to the following universal inequality: An arbitrary nonequilibrium transformation always produces a change in the total entropy production greater than or equal to the one produced if the transformation is done very slowly (adiabatically).
Abstract: The total entropy production generated by the dynamics of an externally driven systems exchanging energy and matter with multiple reservoirs and described by a master equation is expressed as the sum of three contributions, each corresponding to a distinct mechanism for bringing the system out of equilibrium: Nonequilibrium initial conditions, external driving, and breaking of detailed balance. We derive three integral fluctuation theorems (FTs) for these contributions and show that they lead to the following universal inequality: An arbitrary nonequilibrium transformation always produces a change in the total entropy production greater than or equal to the one produced if the transformation is done very slowly (adiabatically). Previously derived fluctuation theorems can be recovered as special cases. We show how these FTs can be experimentally tested by performing the counting statistics of the electrons crossing a single level quantum dot coupled to two reservoirs with externally varying chemical potentials. The entropy probability distributions are simulated for driving protocols ranging from the adiabatic to the sudden switching limit.
133 citations
TL;DR: In this article, the authors provide a microscopic derivation of the master equation for the Jaynes-Cummings model with cavity losses, and single out both the differences with the phenomenological master equation used in the literature and the approximations under which the phenomenologically model correctly describes the dynamics of the atom-cavity system.
Abstract: In this paper we provide a microscopic derivation of the master equation for the Jaynes-Cummings model with cavity losses. We single out both the differences with the phenomenological master equation used in the literature and the approximations under which the phenomenological model correctly describes the dynamics of the atom-cavity system. Some examples wherein the phenomenological and the microscopic master equations give rise to different predictions are discussed in detail.
130 citations
TL;DR: In this article, the authors studied the non-Markovian dynamics of a two-mode bosonic system interacting with two uncorrelated thermal bosonic reservoirs, and analyzed the effects of short-time system-reservoir correlations on the separability thresholds and showed that the relevant parameter is the reservoir spectral density.
Abstract: We study the non-Markovian dynamics of a two-mode bosonic system interacting with two uncorrelated thermal bosonic reservoirs. We present the solution to the exact microscopic Master equation in terms of the quantum characteristic function and study in detail the dynamics of entanglement for bipartite Gaussian states. In particular, we analyze the effects of short-time system-reservoir correlations on the separability thresholds and show that the relevant parameter is the reservoir spectral density. If the frequencies of the involved modes are within the reservoir spectral density, entanglement persists for a longer time than in a Markovian channel. On the other hand, when the reservoir spectrum is out of resonance, short-time correlations lead to a faster decoherence and to the appearance of entanglement oscillations.
TL;DR: In this paper, the authors studied two continuous variable systems (or two harmonic oscillators) and investigated their entanglement evolution under the influence of non-Markovian thermal environments.
Abstract: We study two continuous variable systems (or two harmonic oscillators) and investigate their entanglement evolution under the influence of non-Markovian thermal environments. The continuous variable systems could be two modes of electromagnetic fields or two nanomechanical oscillators in the quantum domain. We use the quantum open system method to derive the non-Markovian master equations of the reduced density matrix for two different but related models of the continuous variable systems. The two models both consist of two interacting harmonic oscillators. In model A, each of the two oscillators is coupled to its own independent thermal reservoir, while in model B the two oscillators are coupled to a common reservoir. To quantify the degrees of entanglement for bipartite continuous variable systems in Gaussian states, logarithmic negativity is used. We find that the dynamics of the quantum entanglement is sensitive to the initial states, the oscillator-oscillator interaction, the oscillator-environment interaction and the coupling to a common bath or to different, independent baths.
TL;DR: For any master equation which is local in time, whether Markovian, non-Markovian of Lindblad form or not, a general procedure is given for constructing the corresponding linear map from the initia...
Abstract: For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is given for constructing the corresponding linear map from the initia...
TL;DR: A real space WKB method based on the master equation is presented, and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state.
Abstract: The extinction of a single species due to demographic stochasticity is analyzed. The discrete nature of the individual agents and the Poissonian noise related to the birth-death processes result in local extinction of a metastable population, as the system hits the absorbing state. The Fokker-Planck formulation of that problem fails to capture the statistics of large deviations from the metastable state, while approximations appropriate close to the absorbing state become, in general, invalid as the population becomes large. To connect these two regimes, a real space WKB method based on the master equation is presented, and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state. The details of the underlying microscopic process, smeared out in a mean field treatment, are shown to be crucial for an exact determination of the extinction exponent. This general scheme is shown to reproduce the known results in the field, to yield new corollaries and to fit quite precisely the numerical solutions. Moreover it allows for systematic improvement via a series expansion where the small parameter is the inverse of the number of individuals in the metastable state.
TL;DR: In this paper, the conditions of pressure isotropy are reduced to a linear, second-order differential equation which can be solved in general, and exact solutions to the Einstein-Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions.
Abstract: We find new classes of exact solutions to the Einstein–Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is reduced to a linear, second order differential equation which can be solved in general. Consequently we can find exact solutions to the Einstein–Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions. It is possible to find exact solutions which can be written explicitly in terms of elementary functions, namely polynomials and product of polynomials and algebraic functions. Uncharged solutions are regainable with our choice of electric field intensity; in particular we generate the Einstein universe for particular parameter values.
TL;DR: The master equation (ME) provides a powerful technique for modeling reactions that involve at least one potential energy well and it is demonstrated that macroscopic (phenomenological) rate coefficients derived from a ME obey detailed balance if the original ME is appropriately constructed.
Abstract: The master equation (ME) provides a powerful technique for modeling reactions that involve at least one potential energy well. It can be widely applied to reactions with several connected energy wells and multiple product channels. The application of the technique is reviewed by reference to the H + SO2 reaction, where phenomenological rate constants for use, for example, in a combustion model can be extracted through an analysis of the eigenvalues and eigenvectors of the collision matrix, M, that describes formation of the adducts HSO2 and HOSO from the source H + SO2, collisional energy transfer in the adduct wells and reaction via the product channel (sink) OH + SO. The approach is extended to systems with more than one sink and it is demonstrated that macroscopic (phenomenological) rate coefficients derived from a ME obey detailed balance if the original ME is appropriately constructed. The method has been applied to the 1-, 2-pentyl radical system, that includes isomerisation and dissociation via two channels to form C3H6 + C2H5 and C2H4 + C3H7. The calculations clearly demonstrate the importance of indirect dissociation channels, in which an isomer can dissociate to form the product set to which it is not directly connected, e.g. formation of C3H6 + C2H5 from 1-pentyl, via the energized states of 2-pentyl. As in previous studies of pentyl dissociation, there is a convergence of the chemically significant eigenvalues and the internal energy relaxation eigenvalues above ∼1000 K; the consequences of this convergence are discussed.
TL;DR: A convergence analysis of explicit tau-leaping schemes for simulating chemical reactions from the viewpoint of stochastic differential equations, to handle the non-Lipschitz property of the coefficients and jumps on the integer lattice.
Abstract: This paper builds a convergence analysis of explicit tau-leaping schemes for simulating chemical reactions from the viewpoint of stochastic differential equations Mathematically, the chemical reaction process is a pure jump process on a lattice with state-dependent intensity The stochastic differential equation form of the chemical master equation can be given via Poisson random measures Based on this form, different types of tau-leaping schemes can be proposed In order to make the problem well-posed, a modified explicit tau-leaping scheme is considered It is shown that the mean square strong convergence is of order $1/2$ and the weak convergence is of order 1 for this modified scheme The novelty of the analysis is to handle the non-Lipschitz property of the coefficients and jumps on the integer lattice
TL;DR: It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard--linear Fokker-Planck equation--may be also related to a family of nonlinear Foksker- planck equations, which is equivalent to the maximum-entropy principle.
Abstract: A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The $H$ theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard---linear Fokker-Planck equation---may be also related to a family of nonlinear Fokker-Planck equations.
TL;DR: The chemical master equation is solved by a hybrid method coupling a macroscopic, deterministic description with a mesoscopic, stochastic model, applied to three chemical systems from molecular cell biology.
TL;DR: It is demonstrated that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and the consequences of the results for optimizing simulations are discussed.
Abstract: From the underlying master equations we derive one-dimensional stochastic processes that describe generalized ensemble simulations as well as tempering (simulated and parallel) simulations. The representations obtained are either in the form of a one-dimensional Fokker-Planck equation or a hopping process on a one-dimensional chain. In particular, we discuss the conditions under which these representations are valid approximate Markovian descriptions of the random walk in order parameter or control parameter space. They allow a unified discussion of the stationary distribution on, as well as of the stationary flow across, each space. We demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and discuss the consequences of our results for optimizing simulations. Finally, we point out the limitations of these representations under conditions of broken ergodicity.
TL;DR: In this article, a state-to-state approach is used to shed light on the thermodynamic and transport properties of LTE plasmas, atomic and molecular plasms for aerospace applications and RF sustained parallel plate reactors.
Abstract: State-to-state approaches are used to shed light on (a) thermodynamic and transport properties of LTE plasmas, (b) atomic and molecular plasmas for aerospace applications and (c) RF sustained parallel plate reactors. The efforts made by the group of Bari in the kinetics and dynamics of electrons and molecular species are discussed from the point of view of either the master equation approach or the molecular dynamics of elementary processes. Recent experimental results are finally rationalized with a state-to-state kinetics based on the coupling of vibrational kinetics with the Boltzmann equation for the electron energy distribution function.
TL;DR: The kinetics of the reaction CO + HO2* --> CO2 + *OH was studied using a combination of ab initio electronic structure theory, transition state theory, and master equation modeling and shows that the overall rate coefficient is independent of pressure up to 500 atm for temperature ranging from 300 to 2500 K.
Abstract: The kinetics of the reaction CO + HO2• → CO2 + •OH was studied using a combination of ab initio electronic structure theory, transition state theory, and master equation modeling. The potential energy surface was examined with the CCSD(T) and CASPT2 methods. The classical energy barriers were found to be about 18 and 19 kcal/mol for CO + HO2• addition following the trans and cis paths, respectively. For the cis path, rate constant calculations were carried out with canonical transition state theory. For the trans path, master equation modeling was also employed to examine the pressure dependence. Special attention was paid to the hindered internal rotations of the HOOC•O adduct and transition states. The theoretical analysis shows that the overall rate coefficient is independent of pressure up to 500 atm for temperature ranging from 300 to 2500 K. On the basis of this analysis, we recommend the following rate expression for reaction R1 k(cm3/mol·s) = 1.57 × 105 T 2.18e-9030/T for 300 ≤ T ≤ 2500 K with the...
14 Mar 2007
TL;DR: In this article, the Malthus-Verhulst Problem is used to solve the problem of the generation and recombination current of semiconductors. But the Langcvin approach is used instead of the Fokker-Planck Equation.
Abstract: VII. Semiconductor , . . . . . . . . 260 VIII. Higher-Order Corrections . . . . . . 263 IX. The Malthus-Verhulst Problem . . . . . . . 265 X. The Transition Region . . . . . . . , . . 268 X1. The Langcvin Approach . . . . . . 270 XII. The Generation and Recombination Currents . . . . . 213 XIII. Example of a Multivariate Master Equation . . . . . . 215 XIV. Solution of the Multivariate Fokker-Planck Equation . . . . 279 XV. Fokker-Planck Equation with Constant Coefficients . . . . 282 XVI. Chemical Reactions . . . . . , . 284 . . . . 286 XVII. Two-step Chemical Reactions . . . . . XVIII. Fluctuations about a Limit Cycle . . . . . . 289 XIX. The Random Walk Picture . . . . . 293 XX. PhaseTransitions . . . . . . . . 297 XXI. Critical Fluctuations . . . . . . . 302
TL;DR: In this paper, a spin dependent transport in a system composed of a quantum dot coupled to a normal metal lead and a ferromagnetic lead (NM-QD-FM) was studied and the spin-resolved currents in the presence of an external bias and an intra-dot Coulomb interaction.
Abstract: We report a study of spin dependent transport in a system composed of a quantum dot coupled to a normal metal lead and a ferromagnetic lead (NM-QD-FM). We use the master equation approach to calculate the spin-resolved currents in the presence of an external bias and an intra-dot Coulomb interaction. We find that for a range of positive external biases (current flow from the normal metal to the ferromagnet) the current polarization $\wp=(I_\uparrow-I_\downarrow)/(I_\uparrow+I_\downarrow)$ is suppressed to zero, while for the corresponding negative biases (current flow from the ferromagnet to the normal metal) $\wp$ attains a relative maximum value. The system thus operates as a rectifier for spin--current polarization. This effect follows from an interplay between Coulomb interaction and nonequilibrium spin accumulation in the dot. In the parameter range considered, we also show that the above results can be obtained via nonequilibrium Green functions within a Hartree-Fock type approximation.
01 Jan 2007
TL;DR: In this paper, the basic tools of non-equilibrium statistical mechanics are introduced: linear response, Brownian motion, Langevin equation, (Pauli) master equation, and the Schrodinger equation.
Abstract: In part I of these lecture notes, I introduce the basic tools of non equilibrium statistical mechanics: linear response, Brownian motion, Langevin equation, (Pauli) master equation. Part II is devoted to a derivation of quantum master equations from the Schrodinger equation, and to some of their applications to quantum optics and to Brownian motion. Finally, in part III, I examine more recent developments: the Crooks and Jarzynski equalities and the Gallavotti-Cohen fluctuation theorem.
TL;DR: It is shown that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models, and identifies that exchanging the limits of infinite system size and infinite time is problematic.
TL;DR: In this paper, the authors give a concise and self-contained introduction into a mathematically sound theory of quantum Markovian master equations, which is intended for those who are interested in practical applications and are not experts in mathematical physics.
Abstract: The aim of the first part of these lecture notes is to give a concise and self-contained introduction into a mathematically sound theory of quantum Markovian master equations The text is intended for those who are interested in practical applications and are not experts in mathematical physics Therefore the original proofs are highly simplified or replaced by heuristic ones However, the final results are always consistent with the rigorous mathematical theory Subsection 12 is devoted to the general properties of an irreversible evolution for a quantum open system In contrast to the classical theory the problem of the most general admissible dynamical transformation of quantum mixed states is not a trivial one and leads to the notion of complete positivity The main result of this discussion is the general form of the Markovian master equation satisfying the complete positivity condition Three methods of derivation of Markovian master equations from the underlying Hamiltonian dynamics of the open system coupled to the reservoir are presented in Subsect 13 In contrast to many of the derivations which can be found in the literature the attention is paid to a mathematically proper form of the obtained equations of motion The next subsection contains few examples of possible extensions of the presented formalism The open systems influenced by varying external conditions, systems with different “channels of evolution” and the simplified description of many-body open systems in terms of nonlinear single-particle evolution equations are briefly discussed In Subsect 15 a model of N 2-level atoms interacting with the electromagnetic field in thermal equilibrium is worked out in some details and serves as an illustration of the general result
TL;DR: In this paper, the effects of the interactions between a cosmological perturbation and the environment were examined and the authors concluded that interactions due to backreaction are more than sufficient to decohere the bulk of the system within 60 e-foldings of inflation.
Abstract: We study the environment-induced decoherence of cosmological perturbations in an inflationary background. Splitting our spectrum of perturbations into two distinct sets characterized by their wavelengths (super and sub-Hubble), we identify the long-wavelength modes with our system and the remainder with an environment. We examine the effects of the interactions between our system and the environment. This interaction causes the long-wavelength modes to decohere for realistic values of the coupling and we conclude that interactions due to backreaction are more than sufficient to decohere the bulk of the system within 60 e-foldings of inflation. This is shown explicitly by obtaining an analytic solution to a master equation detailing the evolution of the density matrix of the system.
TL;DR: It is found that the escape of overdamped particles out of a cusp-shaped parabolic potential well is governed asymptotically by a power-law whose exponent depends exponentially on the ratio of barrier height and temperature.
Abstract: We present an analytic study for subdiffusive escape of overdamped particles out of a cusp-shaped parabolic potential well which are driven by thermal, fractional Gaussian noise with a $1/{\ensuremath{\omega}}^{1\ensuremath{-}\ensuremath{\alpha}}$ power spectrum. This long-standing challenge becomes mathematically tractable by use of a generalized Langevin dynamics via its corresponding non-Markovian, time-convolutionless master equation: We find that the escape is governed asymptotically by a power-law whose exponent depends exponentially on the ratio of barrier height and temperature. This result is in distinct contrast to a description with a corresponding subdiffusive fractional Fokker-Planck approach, thus providing experimentalists an amenable testbed to differentiate between the two escape scenarios.
TL;DR: The perturbative master equation (Bloch-Redfield) is extensively used to study dissipative quantum mechanics, particularly for qubits, despite the 25 year old criticism that it violates positivity (generating negative probabilities).
Abstract: The perturbative master equation (Bloch-Redfield) is extensively used to study dissipative quantum mechanics - particularly for qubits - despite the 25 year old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom, and cast its perturbative master equation (derived from a perturbative treatment of Nakajima-Zwanzig or Schoeller-Schon equations) in the form of a Lindblad master equation. We find that the equation's parameters are time-dependent. This time-dependence is rarely accounted for, and invalidates Lindblad's dynamical semigroup analysis. We analyze one such Bloch-Redfield master equation (for a two-level system coupled to an environment with a short but non-vanishing memory time), which apparently violates positivity. We show analytically that, once the time-dependence of the parameters is accounted for, positivity is preserved.