Showing papers on "Master equation published in 2005"
TL;DR: From a numerical solution of the master equation for hopping transport in a disordered energy landscape with a Gaussian density of states, the dependence of the charge-carrier mobility on temperature, carrier density, and electric field is determined.
Abstract: From a numerical solution of the master equation for hopping transport in a disordered energy landscape with a Gaussian density of states, we determine the dependence of the charge-carrier mobility on temperature, carrier density, and electric field. Experimental current-voltage characteristics in devices based on semiconducting polymers are excellently reproduced with this unified description of the mobility. At room temperature it is mainly the dependence on carrier density that plays an important role, whereas at low temperatures and high fields the electric field dependence becomes important. Omission in the past of the carrier-density dependence has led to an underestimation of the hopping distance and the width of the density of states in these polymers.
809 citations
TL;DR: A hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the integral form of the "Next Reaction" variant of the stochastically simulation algorithm is described.
Abstract: The dynamical solution of a well-mixed, nonlinear stochastic chemical kinetic system, described by the Master equation, may be exactly computed using the stochastic simulation algorithm. However, because the computational cost scales with the number of reaction occurrences, systems with one or more “fast” reactions become costly to simulate. This paper describes a hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the integral form of the “Next Reaction” variant of the stochastic simulation algorithm. The key innovation of this method is its mechanism of efficiently monitoring the occurrences of slow, discrete events while simultaneously simulating the dynamics of a continuous, stochastic or deterministic process. In addition, by introducing an approximation in which multiple slow reactions may occur within a time ste...
367 citations
TL;DR: In this paper, a model describing passive mode locking, a set of differential equations with time delay, was derived and analyzed in a parameter range typical of semiconductor lasers and the limit of a slow saturable absorber was analyzed analytically.
Abstract: We derive and study a model describing passive mode locking---a set of differential equations with time delay. Unlike classical mode locking models based on the Haus master equation, this model does not assume small gain and loss per cavity round trip. Therefore, it is valid in a parameter range typical of semiconductor lasers. The limit of a slow saturable absorber is analyzed analytically. Bifurcations responsible for the appearance and breakup of the mode locking regime are studied numerically.
284 citations
TL;DR: An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field.
Abstract: An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.
272 citations
TL;DR: In this paper, the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction-diffusion problems is reviewed, with a focus on the generic directed percolation universality class and the most prominent exception to this class: even-offspring branching and annihilating random walks.
Abstract: We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction–diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction–diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction kA → �A (� < k ). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization and Callan–Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative � = dc − d expansions with respect to the upper critical dimension dc. With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, L´ evy flights, persistence problems and the influence of spatial boundaries. We also analyse field-theoretic approaches to non-equilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: even-offspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.
238 citations
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TL;DR: In this article, the authors review different approaches to the modeling of quantum effects in electrostatic collisionless plasmas using the Wigner equation and the Hartree formalism, which is related to the multi-stream approach of classical plasma physics.
Abstract: Traditional plasma physics has mainly focused on regimes characterized by high temperatures and low densities, for which quantum-mechanical effects have virtually no impact. However, recent technological advances (particularly on miniaturized semiconductor devices and nanoscale objects) have made it possible to envisage practical applications of plasma physics where the quantum nature of the particles plays a crucial role. Here, I shall review different approaches to the modeling of quantum effects in electrostatic collisionless plasmas. The full kinetic model is provided by the Wigner equation, which is the quantum analog of the Vlasov equation. The Wigner formalism is particularly attractive, as it recasts quantum mechanics in the familiar classical phase space, although this comes at the cost of dealing with negative distribution functions. Equivalently, the Wigner model can be expressed in terms of $N$ one-particle Schr{o}dinger equations, coupled by Poisson's equation: this is the Hartree formalism, which is related to the `multi-stream' approach of classical plasma physics. In order to reduce the complexity of the above approaches, it is possible to develop a quantum fluid model by taking velocity-space moments of the Wigner equation. Finally, certain regimes at large excitation energies can be described by semiclassical kinetic models (Vlasov-Poisson), provided that the initial ground-state equilibrium is treated quantum-mechanically. The above models are validated and compared both in the linear and nonlinear regimes.
238 citations
TL;DR: A two transition state model is applied to the study of the addition of hydroxyl radical to ethylene, which serves as a prototypical example of a radical-molecule reaction with a negative activation energy in the high-pressure limit.
Abstract: A two transition state model is applied to the study of the addition of hydroxyl radical to ethylene. This reaction serves as a prototypical example of a radical−molecule reaction with a negative activation energy in the high-pressure limit. The model incorporates variational treatments of both inner and outer transition states. The outer transition state is treated with a recently derived long-range transition state theory approach focusing on the longest-ranged term in the potential. High-level quantum chemical estimates are incorporated in a variational transition state theory treatment of the inner transition state. Anharmonic effects in the inner transition state region are explored with direct phase space integration. A two-dimensional master equation is employed in treating the pressure dependence of the addition process. An accurate treatment of the two separate transition state regions at the energy and angular momentum resolved level is essential to the prediction of the temperature dependence o...
211 citations
TL;DR: A stochastic model for a general system of first-order reactions in which each reaction may be either a conversion reaction or a catalytic reaction is derived and it is shown that the distribution of all the system components is a Poisson distribution at steady state.
186 citations
TL;DR: In this article, a stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS).
Abstract: A stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial stochastic hybrid system (pSHS). pSHSs correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. We show that for pSHSs, the dynamics of the statistical moments of its continuous states, evolves according to infinite-dimensional linear ordinary differential equations (ODEs), which can be approximated by finite-dimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this types of approximation is provided.
This procedure is used to construct approximate stochastic models for a variety of chemical reactions that have appeared in literature. These reactions include a simple bimolecular reaction, for which one can solve the Master equation; a decaying–dimerizing reaction set which exhibits two distinct time scales; a reaction for which the chemical rate equations have a continuum of equilibrium points; and the bistable Schogl reaction. The accuracy of the approximate models is investigated by comparing with Monte Carlo simulations or the solution to the Master equation, when available. Copyright © 2005 John Wiley & Sons, Ltd.
174 citations
TL;DR: A stochastic description of traffic flow, called probabilistic traffic flow theory, is developed based on spatially homogeneous systems like periodically closed circular rings without on- and off-ramp effects and the calculated flux–density relation and characteristic breakdown times coincide with empirical data measured on highways.
151 citations
TL;DR: In this article, a post-Markovian quantum master equation is derived, which includes bath memory effects via a phenomenologically introduced memory kernel kstd, using as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution.
Abstract: A post-Markovian quantum master equation is derived, which includes bath memory effects via a phenomenologically introduced memory kernel kstd. The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of kstd is presented, in addition to a prescription for the experimental determination of kstd. The formalism is illustrated with examples.
TL;DR: In this article, the authors derived a new representation for spin-spin correlation functions of the finite X X Z spin-1 2 Heisenberg chain in terms of a single multiple integral, that they call the master equation.
TL;DR: In this article, the authors studied the time evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates.
Abstract: We calculate the time-evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates. In the continuous-time limit the process turns into the totally asymmetric simple exclusion process (only pieces of size 1 break off a given cluster). We express the exact solution of the master equation for the process in terms of a determinant which can be derived using the Bethe ansatz. From this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the Gaussian unitary ensemble of random matrices. This result confirms universality of the scaling form of the current distribution in the KPZ universality class and suggests that there is a link between integrable particle systems and random matrix ensembles.
TL;DR: It is shown that accurate thermodynamic and kinetic properties, such as free energy surfaces and kinetic rate coefficients, can be computed from coarse master equations obtained through Bayesian inference.
Abstract: We use Bayesian inference to derive the rate coefficients of a coarse master equation from molecular dynamics simulations. Results from multiple short simulation trajectories are used to estimate propagators. A likelihood function constructed as a product of the propagators provides a posterior distribution of the free coefficients in the rate matrix determining the Markovian master equation. Extensions to non-Markovian dynamics are discussed, using the trajectory “paths” as observations. The Markovian approach is illustrated for the filling and emptying transitions of short carbon nanotubes dissolved in water. We show that accurate thermodynamic and kinetic properties, such as free energy surfaces and kinetic rate coefficients, can be computed from coarse master equations obtained through Bayesian inference.
TL;DR: Time-dependent density functional theory is extended to include dissipative systems evolving under a master equation, providing a Hamiltonian treatment for molecular electronics and recovering the Landauer result.
Abstract: Time-dependent density functional theory is extended to include dissipative systems evolving under a master equation, providing a Hamiltonian treatment for molecular electronics. For weak electric fields, the isothermal conductivity is shown to match the adiabatic conductivity, thereby recovering the Landauer result.
TL;DR: In this article, the authors developed a theory for the full counting statistics for a class of nanoelectromechanical systems (NEMS), describable by a Markovian generalized master equation.
Abstract: We develop a theory for the full counting statistics (FCS) for a class of nanoelectromechanical systems (NEMS), describable by a Markovian generalized master equation The theory is applied to two specific examples of current interest: vibrating C60-molecules and quantum shuttles We report a numerical evaluation of the first three cumulants for the C60 setup; for the quantum shuttle we use the third cumulant to substantiate that the giant enhancement in noise observed at the shuttling transition is due to a slow switching between two competing conduction channels Especially the last example illustrates the power of the FCS
TL;DR: In this paper, the stability of spherically symmetric thin shells to linearized perturbations around static solutions is analyzed and a master equation which dictates the stable equilibrium configurations is derived.
Abstract: We analyse the stability of generic spherically symmetric thin shells to linearized perturbations around static solutions. We include the momentum flux term in the conservation identity, deduced from the ‘ADM’ constraint and the Lanczos equations. Following the Ishak–Lake analysis, we deduce a master equation which dictates the stable equilibrium configurations. Considering the transparency condition, we study the stability of thin shells around black holes, showing that our analysis is in agreement with previous results. Applying the analysis to traversable wormhole geometries, by considering specific choices for the form function, we deduce stability regions and find that the latter may be significantly increased by considering appropriate choices for the redshift function.
TL;DR: In this article, the decoherence of quantum states of continuous variable systems under the action of a quantum optical master equation resulting from the interaction with general Gaussian uncorrelated environments is quantified by relating it to the decay rates of various complementary measures of the quantum nature of a state, such as the purity, some non-classicality indicators in phase space, and, for two-mode states, entanglement measures and total correlations between the modes.
Abstract: We present a detailed report on the decoherence of quantum states of continuous variable systems under the action of a quantum optical master equation resulting from the interaction with general Gaussian uncorrelated environments. The rate of decoherence is quantified by relating it to the decay rates of various, complementary measures of the quantum nature of a state, such as the purity, some non-classicality indicators in phase space, and, for two-mode states, entanglement measures and total correlations between the modes. Different sets of physically relevant initial configurations are considered, including one- and two-mode Gaussian states, number states, and coherent superpositions. Our analysis shows that, generally, the use of initially squeezed configurations does not help to preserve the coherence of Gaussian states, whereas it can be effective in protecting coherent superpositions of both number states and Gaussian wavepackets.
TL;DR: In this article, Liouville-von Neumann equations with nonlinear decay of mixing (NLDM) and population-driven decoherence decay (PDDM) were considered.
Abstract: Electronic energy flow in an isolated molecular system involves coupling between the electronic and nuclear subsystems, and the coupled system evolves to a statistical mixture of pure states. In semiclassical theories, nuclear motion is treated using classical mechanics, and electronic motion is treated as an open quantal system coupled to a "bath" of nuclear coordinates. We have previously shown how this can be simulated by a time-dependent Schrodinger equation with coherent switching and decay of mixing, where the decay of mixing terms model the dissipative effect of the environment on the electronic subdynamics (i.e., on the reduced dynamics of the electronic subsystem). In the present paper we reformulate the problem as a Liouville-von Neumann equation of motion (i.e., we propagate the reduced density matrix of the electronic subsystem), and we introduce the assumption of first-order linear decay. We specifically examine the cases of equal relaxation times for both longitudinal (i.e., population) decay and transverse decay (i.e., dephasing) and of longitudinal relaxation only, yielding the linear decay of mixing (LDM) and the population-driven decay of mixing (PDDM) schemes, respectively. Because we do not generally know the basis in which coherence decays, that is, the pointer basis, we judge the semiclassical methods in part by their ability to give good results in both the adiabatic and diabatic bases. The accuracy in the prediction of physical observables is shown to be robust not only with respect to basis but also with respect to the way in which demixing is incorporated into the master equation for the density matrix. The success of the PDDM scheme is particularly interesting because it incorporates the least amount of decoherence (i.e., the PDDM scheme is the most similar of the methods discussed to the fully coherent semiclassical Ehrenfest method). For both the new and previous decay of mixing schemes, four kinds of decoherent state switching algorithms are analyzed and compared to one another: natural switching (NS), self-consistent switching (SCS), coherent switching (CS), and globally coherent switching (GCS). The CS formulations are examples of a non-Markovian method, in which the system retains some memory of its history, whereas the GCS, SCS, and NS schemes are Markovian (time local). These methods are tested against accurate quantum mechanical results using 17 multidimensional atom-diatom test cases. The test cases include avoided crossings, conical interactions, and systems with noncrossing diabatic potential energy surfaces. The CS switching algorithm, in which the state populations are controlled by a coherent stochastic algorithm for each complete passage through a strong interaction region, but successive strong-interaction regions are not mutually coherent, is shown to be the most accurate of the switching algorithms tested for the LDM and PDDM methods as well as for the previous decay of mixing methods, which are reformulated here as Liouville-von Neumann equations with nonlinear decay of mixing (NLDM). We also demonstrate that one variant of the PDDM method with CS performs almost equally well in the adiabatic and diabatic representations, which is a difficult objective for semiclassical methods. Thus decay of mixing methods provides powerful mixed quantum-classical methods for modeling non-Born-Oppenheimer polyatomic dynamics including photochemistry, charge-transfer, and other electronically nonadiabatic processes.
TL;DR: An ideal-gas-like model of a trading market with quenched random saving factors for its agents is analyzed and it is shown that the steady state income (m) distribution P(m) in the model has a power law tail with Pareto index nu exactly equal to unity.
Abstract: We analyze an ideal-gas-like model of a trading market with quenched random saving factors for its agents and show that the steady state income (m) distribution P(m) in the model has a power law tail with Pareto index nu exactly equal to unity, confirming the earlier numerical studies on this model. The analysis starts with the development of a master equation for the time development of P(m) . Precise solutions are then obtained in some special cases.
TL;DR: In this paper, the quantum-to-classical transition of the cosmological fluctuations produced during inflation can be described by means of the influence functional and the master equation, and the decoherence times for the system-field modes and compare them with the other time scales of the model.
Abstract: We show how the quantum-to-classical transition of the cosmological fluctuations produced during inflation can be described by means of the influence functional and the master equation. We split the inflaton field into the system field (long-wavelength modes), and the environment, represented by its own short-wavelength modes. We compute the decoherence times for the system-field modes and compare them with the other time scales of the model. We present the renormalized stochastic Langevin equation for an homogeneous system field and then we analyze the influence of the environment on the power spectrum for some modes in the system.
TL;DR: In this article, the stability of spherically symmetric thin shells to linearized perturbations around static solutions is analyzed and a master equation which dictates the stable equilibrium configurations is derived.
Abstract: We analyze the stability of generic spherically symmetric thin shells to linearized perturbations around static solutions. We include the momentum flux term in the conservation identity, deduced from the ''ADM'' constraint and the Lanczos equations. Following the Ishak-Lake analysis, we deduce a master equation which dictates the stable equilibrium configurations. Considering the transparency condition, we study the stability of thin shells around black holes, showing that our analysis is in agreement with previous results. Applying the analysis to traversable wormhole geometries, by considering specific choices for the form function, we deduce stability regions, and find that the latter may be significantly increased by considering appropriate choices for the redshift function.
TL;DR: This work analyzes a synthetic biochemical circuit, the toggle switch, and compares the results to those obtained from a numerical solution of the master equation.
Abstract: We use large deviation methods to calculate rates of noise-induced transitions between states in multistable genetic networks. We analyze a synthetic biochemical circuit, the toggle switch, and compare the results to those obtained from a numerical solution of the master equation.
TL;DR: In this article, the authors derived a master equation for the dynamical spin-spin correlation functions of the X X Z spin-1 2 Heisenberg finite chain in an external magnetic field.
TL;DR: In this article, a collision-like model was proposed to describe the evolution of open quantum systems that are completely positive and simultaneously have a semigroup property, and the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment.
Abstract: Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.
TL;DR: The concepts of partitioning on the basis of fast and slow reactions as opposed to fast andslow species and conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation.
Abstract: This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.
TL;DR: A probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times is presented.
Abstract: Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.
01 Jan 2005-Nuclear Instruments & Methods in Physics Research Section B-beam Interactions With Materials and Atoms
TL;DR: In this paper, a modification of the master equation approach is presented to simulate the minimum yield up to a depth z equi where the equilibrium is reached and modify the initial distribution of the ions in the master-equation approach in this way that the calculated minimum yield at that depthz equi is equal to the corresponding simulated one.
Abstract: The master equation approach of dechanneling based on the channeling concept of Lindhard underestimates the Rutherford backscattering minimum yield of perfect crystals mainly at small depths where the equilibrium in the transverse energy shell is not yet reached. This paper presents a modification of the master equation approach which overcomes this problem. The main idea is to simulate the minimum yield up to a depth z equi where the equilibrium is reached and modify the initial distribution of the ions in the master equation approach in this way that the calculated minimum yield at that depth z equi is equal to the corresponding simulated one. Because the simulated depth interval is small, the numerical calculation is still fast enough to be applied for the evaluation of Rutherford backscattering data. For some examples of perfect crystals the results of the calculated minimum yields are compared with those obtained by full simulations and in two cases also with experimental data.
TL;DR: A new approach to the kinetics of nucleation was proposed, which is based on molecular interactions and does not employ the traditional thermodynamics, thus avoiding such a controversial notion as the surface tension of tiny clusters involved in nucleation.
TL;DR: A generalized conditional master equation for quantum measurement by a mesoscopic detector is derived, then the readout characteristics of qubit measurement are studied where a number of remarkable new features are found.
Abstract: In this work we first derive a generalized conditional master equation for quantum measurement by a mesoscopic detector, then study the readout characteristics of qubit measurement where a number of remarkable new features are found. The work would, in particular, highlight the qubit spontaneous relaxation effect induced by the measurement itself rather than an external thermal bath.