Showing papers on "Master equation published in 2004"
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TL;DR: The discretised theoretical distributions matching the empirical data from the Federal Reserve System are deduced from aDiscretised seed which enjoys remarkable scaling laws and may be used to develop new methods for the computation of the value-at-risk and fixed-income derivative pricing.
Abstract: The Convolution and Master equations governing the time behavior of the term structure of Interest Rates are set up both for continuous variables and for their discretised forms. The notion of Seed is introduced. The discretised theoretical distributions matching the empirical data from the Federal Reserve System (FRS) are deduced from a discretised seed which enjoys remarkable scaling laws. In particular the tails of the distributions are very well reproduced. These results may be used to develop new methods for the computation of the value-at-risk and fixed-income derivative pricing.
3,180 citations
TL;DR: An algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems with a superoperator renormalization scheme to efficiently describe the state and the time evolving block decimation technique to efficiently update the state during a time evolution is presented.
Abstract: We present an algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems. The algorithm can be used, e.g., to construct thermal states or to simulate real time evolution given by a generic master equation. Its two main ingredients are (i) a superoperator renormalization scheme to efficiently describe the state of the system and (ii) the time evolving block decimation technique to efficiently update the state during a time evolution. The computational cost of a simulation increases significantly with the amount of correlations between subsystems, but it otherwise depends only linearly on the system size. We present simulations involving quantum spins and fermions in one spatial dimension.
513 citations
01 Dec 2004
TL;DR: Rules for when the bi-stable chemical systems lose global hysteresis by spontaneous separation into spatial domains in opposite phases are formulated in terms of geometrical constraints, diffusion rates and attractor escape times.
Abstract: Bi-stable chemical systems are the basic building blocks for intracellular memory and cell fate decision circuits. These circuits are built from molecules, which are present at low copy numbers and are slowly diffusing in complex intracellular geometries. The stochastic reaction-diffusion kinetics of a double-negative feedback system and a MAPK phosphorylation-dephosphorylation system is analysed with Monte-Carlo simulations of the reaction-diffusion master equation. The results show the geometry of intracellular reaction compartments to be important both for the duration and the locality of biochemical memory. Rules for when the systems lose global hysteresis by spontaneous separation into spatial domains in opposite phases are formulated in terms of geometrical constraints, diffusion rates and attractor escape times. The analysis is facilitated by a new efficient algorithm for exact sampling of the Markov process corresponding to the reaction-diffusion master equation.
400 citations
TL;DR: In this article, a spatially biased continuous time random walk (CTRW) governed by ψ( s,t), the joint probability density for an eventdisplacement s with an event-time t is considered.
362 citations
TL;DR: In this paper, the authors extended the formulation for perturbations of maximally symmetric black holes in higher dimensions developed by the present authors in a previous paper to a charged black hole background whose horizon is described by an Einstein manifold.
Abstract: We extend the formulation for perturbations of maximally symmetric black holes in higher dimensions developed by the present authors in a previous paper to a charged black hole background whose horizon is described by an Einstein manifold. For charged black holes, perturbations of electromagnetic fields are coupled to the vector and scalar modes of metric perturbations non-trivially. We show that by taking appropriate combinations of gauge-invariant variables for these perturbations, the perturbation equations for the Einstein-Maxwell system are reduced to two decoupled second-order wave equations describing the behaviour of the electromagnetic mode and the gravitational mode, for any value of the cosmological constant. These wave equations are transformed into Schrodinger-type ODEs through a Fourier transformation with respect to time. Using these equations, we investigate the stability of generalised black holes with charge. We also give explicit expressions for the source terms of these master equations with application to the emission problem of gravitational waves in mind.
345 citations
TL;DR: The fluctuation theorem is derived for stochastic nonequilibrium reactions ruled by the chemical master equation and verified in the Schlögl model of far-from-equilibrium bistability.
Abstract: A fluctuation theorem is derived for stochastic nonequilibrium reactions ruled by the chemical master equation. The theorem is expressed in terms of the generating and large-deviation functions characterizing the fluctuations of a quantity which measures the loss of detailed balance out of thermodynamic equilibrium. The relationship to entropy production is established and discussed. The fluctuation theorem is verified in the Schlogl model of far-from-equilibrium bistability.
238 citations
TL;DR: In this article, the authors consider the evolution of arbitrary initial states of a two-particle system under open system dynamics described by a class of master equations which produce decoherence of each particle.
Abstract: The destruction of quantum interference, decoherence, and the destruction of entanglement both appear to occur under the same circumstances. To address the connection between these two phenomena, we consider the evolution of arbitrary initial states of a two-particle system under open system dynamics described by a class of master equations which produce decoherence of each particle. We show that all initial states become separable after a finite time, and we produce the explicit form of the separated state. The result extends and amplifies an earlier result of Di\'osi. We illustrate the general result by considering the case in which the initial state is an Einstein-Podolsky-Rosen state (in which both the positions and momenta of a particle pair are perfectly correlated). This example clearly illustrates how the spreading out in phase space produced by the environment leads to certain disentanglement conditions becoming satisfied.
226 citations
TL;DR: In this paper, the reduced dynamics of a central spin coupled to a bath of spin particles arranged in a spin star configuration is investigated, and an analytical solution is obtained in the limit of an infinite number of bath spins, where the model shows complete relaxation and partial decoherence.
Abstract: The reduced dynamics of a central spin coupled to a bath of $N$ spin-$\frac{1}{2}$ particles arranged in a spin star configuration is investigated. The exact time evolution of the reduced density operator is derived, and an analytical solution is obtained in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ of an infinite number of bath spins, where the model shows complete relaxation and partial decoherence. It is demonstrated that the dynamics of the central spin cannot be treated within the Born-Markov approximation. The Nakajima-Zwanzig and the time-convolutionless projection operator technique are applied to the spin star system. The performance of the corresponding perturbation expansions of the non-Markovian equations of motion is examined through a comparison with the exact solution.
224 citations
TL;DR: A detailed study is presented for a large class of uncoupled continuous-time random walks and the master equation is solved for the Mittag-Leffler survival probability.
Abstract: A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.
206 citations
TL;DR: The Onsager and higher-order reciprocity relations are derived from a fluctuation theorem for nonequilibrium reactions ruled by the chemical master equation and the macroscopic affinities associated with the fluxes are identified by graph theory.
Abstract: The Onsager and higher-order reciprocity relations are derived from a fluctuation theorem for nonequilibrium reactions ruled by the chemical master equation. The fluctuation theorem is obtained for the generating function of the macroscopic fluxes between chemiostats maintaining the system in a nonequilibrium steady state. The macroscopic affinities associated with the fluxes are identified by graph theory. The Yamamoto–Zwanzig formulas for the reaction constants are also derived from the fluctuation theorem.
187 citations
TL;DR: In this paper, a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states was studied and the authors showed that the general large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation.
Abstract: We prove for a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.
TL;DR: In this paper, the authors developed a theory of electron transport through quantum dots that are weakly coupled to ferromagnetic leads and derived generalized rate equations for the dot's occupation and accumulated spin.
Abstract: We develop a theory of electron transport through quantum dots that are weakly coupled to ferromagnetic leads. The theory covers both the linear and nonlinear transport regimes, takes noncollinear magnetization of the leads into account, and allows for an externally applied magnetic field. We derive generalized rate equations for the dot's occupation and accumulated spin and discuss the influence of the dot's spin on the transmission. A negative differential conductance and a nontrivial dependence of the conductance on the angle between the lead magnetizations are predicted.
TL;DR: This work manipulates the Hubbard-Stratonovich transformation to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields.
Abstract: Based on the Hubbard–Stratonovich transformation, the dissipative interaction between the system of interest and the heat bath is decoupled and the separated system and bath thus evolve in common classical random fields. This manipulation allows us to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields. Within the stochastic description, the interaction between the system and the bath is reflected in the mutually induced mean fields. The relationship between the bath-induced field and the influence functional in the path integral framework is revealed. As a demonstration of this approach, we derive the exact master equations for two model systems.
TL;DR: In this article, the authors examined master equations that possess a memory kernel, leading to a replacement of white noise by colored noise, and the conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel.
Abstract: The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. In this paper we examine master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel.
TL;DR: In this article, the long-time projective behavior of the stochastic master equation describing a continuous, collective spin measurement of an atomic ensemble was characterized both analytically and numerically.
Abstract: We characterize the long-time projective behavior of the stochastic master equation describing a continuous, collective spin measurement of an atomic ensemble both analytically and numerically. By adding state-based feedback, we show that it is possible to prepare highly entangled Dicke states deterministically.
TL;DR: In this paper, the exact convolutionless master equation of the quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators is derived for non-Markovian open quantum systems.
Abstract: Stochastic Schr\"odinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schr\"odinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schr\"odinger equation.
TL;DR: It is shown that the mixed quantum-classical Liouville equation is equivalent to linearizing the forward-backward action in the influence functional in terms of either the diabatic or adiabatic basis sets.
Abstract: We show that the mixed quantum-classical Liouville equation is equivalent to linearizing the forward-backward action in the influence functional. Derivations are provided in terms of either the diabatic or adiabatic basis sets. An application of the mixed quantum-classical Liouville equation for calculating the memory kernel of the generalized quantum master equation is also presented. The accuracy and computational feasibility of such an approach is demonstrated in the case of a two-level system nonlinearly coupled to an anharmonic bath.
TL;DR: In this paper, the master equation describing the completely positive time evolution of a uniformly accelerated two-level system in weak interaction with a scalar field in the Minkowski vacuum is derived and explicitly solved.
Abstract: The master equation describing the completely positive time evolution of a uniformly accelerated two-level system in weak interaction with a scalar field in the Minkowski vacuum is derived and explicitly solved. The moving system is found to be subjected to dissipation that drives its density matrix to a purely thermal equilibrium state, exhibiting a nonvanishing probability of spontaneous excitation, this phenomenon being usually referred to as the Unruh effect. Remarkably, when the uniformly accelerating system is composed by two, independent two-level atoms, the corresponding asymptotic, equilibrium state turns out to be entangled.
TL;DR: In this article, the authors investigated the kinetics of the H+C2H2 and H+c2H4 reactions, as well as their reverse dissociations, in some detail.
Abstract: In this article we investigate the kinetics of the H + C2H2 and H + C2H4 reactions, as well as their reverse dissociations, in some detail. High level electronic structure calculations are used to characterize the potential energy surfaces, and these results are not adjusted to obtain good agreement with experiment in the subsequent kinetic analysis. An approximate two-dimensional master equation is used to determine phenomenological rate coefficients, k(T,p). The effects of angular momentum conservation, tunneling, and the use of variational transition-state theory (as opposed to conventional transition-state theory) to compute microcanonical rate coefficients are investigated in detail. For both reactions, the low-pressure limit is approached very slowly, because reaction just above threshold must occur strictly by tunneling. Assuming a single-exponential-down model for P(E,E′), we deduce from experiment values of 〈ΔEd〉, the average energy transferred in a deactivating collision, as a function of temperature for both C2H3 and C2H5 in baths of He, Ar, and N2. Our results support the idea that 〈ΔEd〉 increases roughly linearly with temperature, at least for weak colliders. The agreement between theory and experiment is remarkably good for both reactions. Values of k(T,p) for the two reactions are given in the Troe format for use in modeling.
TL;DR: Two different approaches which are based on a partial time-ordering prescription, i.e., a time-local formalism and also on a numerical decomposition of the spectral density are proposed, valid for time-independent Hamiltonians and can be given in a compact quantum master equation.
Abstract: For the description of dynamical effects in quantum mechanical systems on ultrashort time scales, memory effects play an important role. Meier and Tannor [J. Chem. Phys. 111, 3365 (1999)] developed an approach which is based on a time-nonlocal scheme employing a numerical decomposition of the spectral density. Here we propose two different approaches which are based on a partial time-ordering prescription, i.e., a time-local formalism and also on a numerical decomposition of the spectral density. In special cases such as the Debye spectral density the present scheme can be employed even without the numerical decomposition of the spectral density. One of the proposed schemes is valid for time-independent Hamiltonians and can be given in a compact quantum master equation. In the case of time-dependent Hamiltonians one has to introduce auxiliary operators which have to be propagated in time along with the density matrix. For the example of a damped harmonic oscillator these non-Markovian theories are compared among each other, to the Markovian limit neglecting memory effects and time dependencies, and to exact path integral calculations. Good agreement between the exact calculations and the non-Markovian results is obtained. Some of the non-Markovian theories mentioned above treat the time dependence in the system Hamiltonians nonperturbatively. Therefore these methods can be used for the simulation of experiments with arbitrary large laser fields.
TL;DR: In this paper, it was shown that a large class of non-Markovian quantum processes in open systems can be formulated through time-local master equations which are not in Lindblad form.
Abstract: A large class of non-Markovian quantum processes in open systems can be formulated through time-local master equations which are not in Lindblad form. It is shown that such processes can be embedded in a Markovian dynamics which involves a time-dependent Lindblad generator on an extended state space. If the state space of the open system is given by some Hilbert space $\mathcal{H}$, the extended state space is the triple Hilbert space $\mathcal{H}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ which is obtained by combining the open system with a three-state system. This embedding is used to derive an unraveling for non-Markovian time evolution by means of a stochastic process in the extended state space. The process is defined through a stochastic Schr\"odinger equation which generates genuine quantum trajectories for the state vector conditioned on a continuous monitoring of an environment. The construction leads to a continuous measurement interpretation for non-Markovian dynamics within the framework of the theory of quantum measurement.
TL;DR: In this article, the authors analyzed the dynamics of a nanomechanical resonator coupled to a single-electron transistor (SET) in the regime where the resonator behaves classically.
Abstract: We analyze the dynamics of a nanomechanical resonator coupled to a single-electron transistor (SET) in the regime where the resonator behaves classically. A master equation is derived describing the dynamics of the coupled system which is then used to obtain equations of motion for the average charge state of the SET and the average position of the resonator. We show that the action of the SET on the resonator is very similar to that of a thermal bath, as it leads to a steady-state probability distribution for the resonator which can be described by mean values of the resonator position, a renormalized frequency, an effective temperature, and an intrinsic damping constant. Including the effects of extrinsic damping and finite temperature, we find that there remain experimentally accessible regimes where the intrinsic damping of the resonator still dominates its behavior. We also obtain the average current through the SET as a function of the coupling to the resonator.
TL;DR: This work combines (non-)Markovian master equations with correlation functions in Laplace space to derive a noise formula for both weak and strong coupling to the bath, demonstrating that the dephasing and relaxation rates of the two-level systems can be extracted from noise.
Abstract: We study the current noise spectrum of qubits under transport conditions in a dissipative bosonic environment. We combine (non-)Markovian master equations with correlation functions in Laplace space to derive a noise formula for both weak and strong coupling to the bath. The coherence-induced reduction of noise is diminished by weak dissipation and/or a large level separation (bias). For weak dissipation, we demonstrate that the dephasing and relaxation rates of the two-level systems can be extracted from noise. In the strong dissipation regime, the localization-delocalization transition becomes visible in the low-frequency noise.
TL;DR: Application of the linearized semiclassical (LSC) approximation for calculating the relatively short-lived memory kernel, followed by a numerically exact solution of the GQME is found to provide an accurate description of the relaxation dynamics.
Abstract: The Nakajima–Zwanzig generalized quantum master equation (GQME) provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a, possibly anharmonic, quantum bath. In this equation, a memory kernel superoperator accounts for the influence of the bath on the dynamics of the system. In a previous paper [Q. Shi and E. Geva, J. Chem. Phys. 119, 12045 (2003)] we proposed a new approach to calculating the memory kernel, in the case of arbitrary system-bath coupling. Within this approach, the memory kernel is obtained by solving a set of two integral equations, which requires a new type of two-time system-dependent bath correlation functions as input. In the present paper, we consider the application of the linearized semiclassical (LSC) approximation for calculating those correlation functions, and subsequently the memory kernel. The new approach is tested on a benchmark spin-boson model. Application of the LSC approximation for calculating the relatively short-lived memory kernel, followed by a numerically exact solution of the GQME, is found to provide an accurate description of the relaxation dynamics. The success of the proposed LSC–GQME methodology is contrasted with the failure of both the direct application of the LSC approximation and the weak coupling treatment to provide an accurate description of the dynamics, for the same model, except at very short times. The feasibility of the new methodology to anharmonic systems is also demonstrated in the case of a two level system coupled to a chain of Lennard–Jones atoms.
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TL;DR: In this paper, a generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrodinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly.
Abstract: The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrodinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber $P$ representation, the antinormal ordering $Q$ representation and the Wigner $W$ representation. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow's transitions with energy conservation, additional transitions with energy loss, are found. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment are obtained.
TL;DR: In this paper, a systematic analysis of the behavior of the quantum Markovian master equation driven by coherent control fields is proposed, and its irreversible character is formalized using control-theoretic notions and the sets of states that can be reached via coherent controls are described.
Abstract: A systematic analysis of the behavior of the quantum Markovian master equation driven by coherent control fields is proposed. Its irreversible character is formalized using control-theoretic notions and the sets of states that can be reached via coherent controls are described. The analysis suggests to what extent (and how) it is possible to counteract the effect of dissipation.
TL;DR: In this article, the interrelationship between the non-Markovian stochastic Schrodinger equations and the corresponding non-markovian master equations is investigated in the finite-temperature regimes.
Abstract: The interrelationship between the non-Markovian stochastic Schr\"odinger equations and the corresponding non-Markovian master equations is investigated in the finite-temperature regimes. We show that the general finite-temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite-temperature reservoir in the rotating-wave approximation. The exact convolutionless master equation for the damped harmonic oscillator is obtained by averaging the quantum trajectories, relying upon no assumption of coupling strength or time scale. The master equation derived in this way automatically preserves the positivity, Hermiticity, and unity.
TL;DR: In this paper, a general analytic solution to the local community model of Hubbell's neutral theory of biodiversity is provided by recasting it as an urn model, i.e. a Markovian description of states and their transitions.
TL;DR: In this article, the authors developed methods for calculating the zero-frequency noise for quantum shuttles, i.e., nanoelectromechanical devices where the mechanical motion is quantized.
Abstract: We develop methods for calculating the zero-frequency noise for quantum shuttles, i.e., nanoelectromechanical devices where the mechanical motion is quantized. As a model system we consider a three-dot array, where the internal electronic coherence both complicates and enriches the physics. Two different formulations are presented: (i) quantum regression theorem and (ii) the counting variable approach. It is demonstrated, both analytically and numerically, that the two formulations yield identical results, when the conditions of their respective applicability are fulfilled. We describe the results of extensive numerical calculations for current and current noise (Fano factor), based on a solution of a Markovian generalized master equation. The results for the current and noise are further analyzed in terms of Wigner functions, which help to distinguish different transport regimes (in particular, shuttling versus cotunneling). In the case of weak interdot coupling, the electron transport proceeds via sequential tunneling between neighboring dots. A simple rate equation with the rates calculated analytically from the $P(E)$ theory is developed and shown to agree with the full numerics.
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.