Showing papers on "Master equation published in 2002"

Theory of open quantum systems

Abstract: A quantum dissipation theory is constructed with the system–bath interaction being treated rigorously at the second-order cumulant level for both reduced dynamics and initial canonical boundary condition. The theory is valid for arbitrary bath correlation functions and time-dependent external driving fields, and satisfies correlated detailed-balance relation at any temperatures. The general formulation assumes a particularly simple form in driven Brownian oscillator systems in which the correlated driving-dissipation effects can be accounted for exactly in terms of local-field correction. Remarks on a class of widely used phenomenological quantum master equations that neglects the bath dispersion-induced dissipation are also made in contact with the present theory.

From the time-dependent, multiple-well master equation to phenomenological rate coefficients

Abstract: We discuss at some length the relationship between solutions to the time-dependent, multiple-well master equation and a macroscopic description of the chemistry in terms of phenomenological rate coefficients In so doing, we derive two different methods of obtaining the rate coefficients from the eigenvalues and eigenvectors of G, the transition matrix of the master equation We apply the first of the two methods to the C2H3 + C2H2 and C3H3 + C3H3 reactions, problems we have treated previously using the “experimental” (or exponential-decay) approach, and obtain considerably more and somewhat different results than we obtained in our earlier work

The stochastic Gross-Pitaevskii equation II

Abstract: We show how to adapt the ideas of local energy and momentum conservation in order to derive modifications to the Gross-Pitaevskii equation which can be used phenomenologically to describe irreversible effects in a Bose-Einstein condensate. Our approach involves the derivation of a simplified quantum kinetic theory, in which all processes are treated locally. It is shown that this kinetic theory can then be transformed into a number of phase-space representations, of which the Wigner function description, although approximate, is shown to be the most advantageous. In this description, the quantum kinetic master equation takes the form of a Gross-Pitaevskii equation with noise and damping added according to a well defined prescription - an equation we call the stochastic Gross-Pitaevskii equation. From this, a very simplified description we call the phenomenological growth equation can be derived. We use this equation to study?(i) the nucleation and growth of vortex lattices, and?(ii) nonlinear losses in a hydrogen condensate, which it is shown can lead to a curious instability phenomenon.

p-Adic description of characteristic relaxation in complex systems

Abstract: This work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p-adic analysis. We show that three types of relaxation fitted into the Kohlrausch-Williams-Watts law, the power decay law, or the logarithmic decay law, are similar random processes. Inherently, these processes are ultrametric and are described by the p-adic master equation. The physical meaning of this equation is explained in terms of a random walk constrained by a hierarchical energy landscape. We also discuss relations between the relaxation kinetics and the energy landscapes.

The Chemical Langevin and Fokker−Planck Equations for the Reversible Isomerization Reaction†

Abstract: This paper uses the simple reversible isomerization reaction to illustrate and clarify the roles played in chemical kinetics by recently proposed forms for the chemical Langevin equation and chemical Fokker−Planck equation. It is shown that the stationary solution of the chemical Fokker−Planck equation for this model reaction provides, for most purposes, an excellent approximation to the stationary solution of the chemical master equation. It is also shown that, when allowance is made for the stipulated macroscopic nature of the time increment dt in the chemical Langevin equation, the changes in molecular population during dt predicted by that equation for this model reaction closely approximate the changes prescribed by the chemical master equation. The discussion highlights the role of the chemical Langevin equation as not only a potential computational aid but also a conceptual bridge between the stochastic chemical master equation and the traditional deterministic reaction rate equation.

Rotational Relaxation of N2 Behind a Strong Shock Wave

Abstract: Using an existing expression for the state-to-state rotational transition rate coefficients, which is derived from the experimental data taken at temperatures equal to or below 1500 K, the master equation for rotational states is integrated with time for N 2 . The postshock temperature considered is from 400 to 128,000 K. From the numerical solutions of the master equation, the effective collision numbers and characteristic relaxation times are determined

Solution of Some One- and Two-Dimensional Master Equation Models for Thermal Dissociation: The Dissociation of Methane in the Low-Pressure Limit

Abstract: Using three formulations of the master equation (ME), we have investigated theoretically the dissociation of methane in the low-pressure limit. The three forms of the ME are as follows: (1) A one-dimensional model in which E, the total energy, is the independent variable (the E model). (2) The two-dimensional strong-collision-in-J model of Smith and Gilbert (Int. J. Chem. Kinet. 1988, 20, 307−329) in which e, the energy in the active degrees of freedom, and J, the total angular momentum quantum number, are the independent variables (the e,J model). (3) A two-dimensional variant of the e,J model in which E and J are the independent variables (the E,J model). The third form of the ME is the most physically realistic, and for this model we investigate the dependence of values of the energy transfer moments (〈ΔEd〉, −〈ΔE〉, and 〈ΔE2〉1/2) deduced from experiment on assumed forms of the energy transfer function, P(E,E‘), and on temperature. All three moments increase as the temperature rises; −〈ΔE〉 increases fro...

Environment-Induced Decoherence and the Transition from Quantum to Classical

Abstract: We study dynamics of quantum open systems, paying special attention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection einselection in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equations we study einselection employing the general strategy of the predictability sieve. Assumptions that are usually made in the discussion of decoherence are critically reexamined along with the ``standard lore'' to which they lead. Restoration of quantum-classical correspondence in systems that are classically chaotic is discussed. The dynamical second law -it is shown- can be traced to the same phenomena that allow for the restoration of the correspondence principle in decohering chaotic systems (where it is otherwise lost on a very short time-scale). Quantum error correction is discussed as an example of an anti-decoherence strategy. Implications of decoherence and einselection for the interpretation of quantum theory are briefly pointed out.

Fourth-order quantum master equation and its Markovian bath limit

Abstract: Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.

Adiabatic transfer of electrons in coupled quantum dots

Abstract: We investigate the influence of dissipation on one- and two-qubit rotations in coupled semiconductor quantum dots, using a (pseudo-)spin-boson model with adiabatically varying parameters. For weak dissipation, we solve a master equation, compare with direct perturbation theory, and derive an expression for the ``fidelity loss'' during a simple operation that adiabatically moves an electron between two coupled dots. We discuss the possibility of visualizing coherent quantum oscillations in electron ``pump'' currents, combining quantum adiabaticity and Coulomb blockade. In two-qubit spin-swap operations where the role of intermediate charge states has been discussed recently, we apply our formalism to calculate the fidelity loss due to charge tunneling between two dots.

Fundamentals of equations of state

Abstract: A summary of thermodynamics equation of state for an ideal gas law of equipartition of energy and effects of vibrational and rotational motions Bose-Einstein equation of state Fermi-Dirac equation of state ionization equilibrium and the Saha equation Debye-Hnckel equation of state the Thomas-Fermi and related models Grnneisen equation of state an introduction to fluid mechanics in relation to shock waves derivation of hydrodynamics from kinetic theory studies of the equations of state from high-pressure shock waves in solids equation of state and inertial confinement fusion applications of equations of state in astrophysics equations of state in elementary particle physics.

On the master equation approach to diffusive grain-surface chemistry: The H, O, CO system

Abstract: Received; accepted Abstract. We have used the master equation approach to study a moderately complex network of diffusive reactions occurring on the surfaces of interstellar dust particles. This network is meant to apply to dense clouds in which a large portion of the gas-phase carbon has already been converted to carbon monoxide. Hydrogen atoms, oxygen atoms, and CO molecules are allowed to accrete onto dust particles and their chemistry is followed. The stable molecules produced are oxygen, hydrogen, water, carbon dioxide (CO2), formaldehyde (H2CO), and methanol (CH3OH). The surface abundances calculated via the master equation approach are in good agreement with those obtained via a Monte Carlo method but can differ considerably from those obtained with standard rate equations.

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

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

On Integral Equations Arising in the First-Passage Problem for Brownian Motion

Abstract: for and , where for and is the standard normal density. These equations are derived from a single ’master equation’ which may be viewed as a Chapman-Kolmogorov equation of Volterra type. The initial idea in the derivation

On brane world cosmological perturbations

Abstract: We discuss the scalar cosmological perturbations in a 3-brane world with a 5D bulk. We first show explicitly how the effective perturbed Einstein equations on the brane (involving the Weyl fluid) are encoded into Mukohyama's master equation. We give the relation between Mukohyama's master variable and the perturbations of the Weyl fluid; we also discuss the relation between the former and the perturbations of matter and induced metric. The boundary condition for the master variable on the brane is then given in the case of a perfect fluid with adiabatic perturbations on a Randall-Sundrum or Dvali-Gabadadze-Porrati brane. This provides an easy way to solve numerically for the evolution of the perturbations and also should shed light on the various approximations done in the literature to deal with the Weyl degrees of freedom.

Five Lectures on Dissipative Master Equations

Abstract: The damped harmonic oscillator is arguably the simplest open quantum system worth studying. It is also of great practical importance because it is an essential ingredient in the theoretical description of many quantum-optical experiments. One can assume rather safely that the quantum master equation of the simple harmonic oscillator would not be studied so extensively if it did not play such a central role in the quantum theory of lasers and the masers. Not surprisingly, then, all major textbook accounts of theoretical quantum optics [1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15] contain a fair amount of detail about damped harmonic oscillators. Fock state representations or phase space functions of some sort are invariably employed in these treatments.

The correlation time of mesoscopic chemical clocks

Abstract: A formula is proved for the correlation time of nonequilibrium chemical clocks in the presence of molecular noise. The correlation time is defined as the inverse of the damping rate of the autocorrelation functions of the chemical concentrations. Using the Hamilton–Jacobi method for stochastic systems as well as a Legendre transform from the Onsager–Machlup action to a reduced action depending only on the Hamilton–Jacobi pseudoenergy, the correlation time is given in the weak-noise limit in terms of the extensivity parameter, the period of oscillations, as well as the derivative of the period with respect to the pseudoenergy. Using this result, an estimation is obtained for the minimum number of molecules required for the oscillations of the chemical concentrations to remain correlated in time. This estimation puts a fundamental lower limit on the size of chemical clocks. For typical oscillators, the minimum number of molecules is estimated between ten and one hundred, which essentially corresponds to nanometric systems.

Non-Markovian stochastic Schrödinger equations: Generalization to real-valued noise using quantum-measurement theory

Abstract: Do stochastic Schr\"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schr\"odinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schr\"odinger equation introduced by Strunz, Di\'osi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction.

Stability of pulses in the master mode-locking equation

Abstract: The master mode-locking equation is a canonical model for mode locking in solid-state lasers. We consider the dynamics and stability of the localized pulse solutions that this equation admits of. We provide analytic proof of stable mode-locking behavior along with analysis that shows that mode locking can become destabilized as a result of either a radiation-mode or a saddle-node instability. This is to our knowledge the first analytic proof of the stability of the pulse solutions that takes the time-dependent gain saturation mechanism of mode-locked lasers into account.

Exact results for hydrogen recombination on dust grain surfaces.

Abstract: The recombination of hydrogen in the interstellar medium, taking place on surfaces of microscopic dust grains, is an essential process in the evolution of chemical complexity in interstellar clouds. Molecular hydrogen plays an important role in absorbing the heat that emerges during gravitational collapse, thus enabling the formation of structure in the universe. The H2 formation process has been studied theoretically, and in recent years also by laboratory experiments. The experimental results were analyzed using a rate equation model. The parameters of the surface that are relevant to H2 formation were obtained and used in order to calculate the recombination rate under interstellar conditions. However, it turned out that, due to the microscopic size of the dust grains and the low density of H atoms, the rate equations may not always apply. A master equation approach that provides a good description of the H2 formation process was proposed. It takes into account both the discrete nature of the H atoms and the fluctuations in the number of atoms on a grain. In this paper we present a comprehensive analysis of the H2 formation process, under steady state conditions, using an exact solution of the master equation. This solution provides an exact result for the hydrogen recombination rate and its dependence on the flux, the surface temperature, and the grain size. The results are compared with those obtained from the rate equations. The relevant length scales in the problem are identified and the parameter space is divided into two domains. One domain, characterized by first order kinetics, exhibits high efficiency of H2 formation. In the other domain, characterized by second order kinetics, the efficiency of H2 formation is low. In each of these domains we identify the range of parameters in which, due to the small size of the grains, the rate equations do not account correctly for the recombination rate and the master equation is needed.

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

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

Dynamical Casimir Effect in a Leaky Cavity at Finite Temperature

Abstract: The phenomenon of particle creation within an almost resonantly vibrating cavity with losses is investigated for the example of a massless scalar field at finite temperature. A leaky cavity is designed via the insertion of a dispersive mirror into a larger ideal cavity (the reservoir). In the case of parametric resonance the rotating wave approximation allows for the construction of an effective Hamiltonian. The number of produced particles is then calculated using response theory as well as a nonperturbative approach. In addition, we study the associated master equation and briefly discuss the effects of detuning. The exponential growth of the particle numbers and the strong enhancement at finite temperatures found earlier for ideal cavities turn out to be essentially preserved. The relevance of the results for experimental tests of quantum radiation via the dynamical Casimir effect is addressed. Furthermore, the generalization to the electromagnetic field is outlined.

Intraband relaxation and temperature dependence of the fluorescence decay time of one-dimensional Frenkel excitons: The Pauli master equation approach

Abstract: In molecular J-aggregates one often observes an increase of the fluorescence decay time when increasing the temperature from 0 K. This phenomenon is usually attributed to the thermal population of the dark Frenkel exciton states that lie above the superradiant bottom state of the exciton band. In this paper, we study this effect for a homogeneous one-dimensional aggregate in a host medium and we model the scattering between different exciton states as arising from their coupling to the host vibrations. A Pauli master equation is used to describe the redistribution of excitons over the band. The rates entering this equation are calculated within the framework of first-order perturbation theory, assuming a linear on-site interaction between excitons and acoustic phonons. Solving the master equation numerically for aggregates of up to 100 molecules, we calculate the temperature dependence of the fluorescence kinetics in general and the decay time scale in particular. The proper definition of the fluorescence decay time is discussed in detail. We demonstrate that, even at a quantum yield of unity, the possibility to directly interpret fluorescence experiments in terms of a simple radiative time scale depends crucially on the initial excitation conditions in combination with the competition between spontaneous emission and intraband phonon-assisted relaxation.

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

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

Towards a unified framework for anomalous transport in heterogeneous media

Abstract: We develop a unified framework to model the anomalous transport of tracers in highly heterogeneous media. While the framework is general, our working media in this study are geological formations. The basis of our approach takes into account the different levels of uncertainty, often associated with spatial scale, in characterizing these formations. The effects on the transport of smaller spatial scale heterogeneities are treated probabilistically with a model based on a continuous time random walk (CTRW), while the larger scale variations are included deterministically. The CTRW formulation derives from the ensemble average of a disordered system, in which the transport in each realization is described by a Master Equation. A generic example of such a system – a 3D discrete fracture network (DFN) – is treated in detail with the CTRW formalism. The key step in our approach is the derivation of a physically based ψ( s ,t) , the joint probability density for a displacement s with an event-time t. We relate the ψ( s ,t) to the velocity spectrum Φ(ξ) (|ξ|=1/v, ξ = v ) of the steady flow-field in a fluid-saturated DFN. Heterogeneous porous media are often characterized by a log-normal permeability distribution; the Φ(ξ) we use in this case is an analytic form approximating the velocity spectrum derived from this distribution. The common approximation of ψ( s ,t)∼p( s )t −1−β with a constant β, is evaluated in these cases. For the former case it is necessary to include s −t coupling while the latter case points to the presence of an effective t-dependent β. The full range of these features can be included in the CTRW solution but, as is shown, not in the fractional-time derivative equation (FDE) formulation of CTRW. Finally, the methods used for the unified framework are critically examined.

Approach to quantum Kramers' equation and barrier crossing dynamics.

Abstract: We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.

Constant bond breakup probability model for reversible aggregation processes.

Abstract: Reversible aggregation processes were simulated for systems of freely diffusing sticky particles. Reversibility was introduced by allowing that all bonds in the system may break with a given probability per time interval. In order to describe the kinetics of such aggregation-fragmentation processes, a fragmentation kernel was developed and then used together with the Brownian aggregation kernel for solving the corresponding kinetic master equation. The deduced fragmentation kernel considers a single characteristic lifetime for all bonds and accounts for the cluster morphology by averaging over all possible configurations for clusters of a given size. It became evident that the simulated cluster-size distributions could be described only when an additional fragmentation effectiveness was considered. Doing so, the stochastic solutions were in good agreement with the simulated data.

Kinetic Modeling of Hyperbranched Polymerization Involving an AB2 Monomer Reacting with Substitution Effect

Abstract: A model is developed in the form of one or two partial differential equations (master Smoluchowski-like equations) that describe evolution of the size distribution of polymer species formed in a step-growth polymerization of an AB2 monomer. Groups B react with a substitution effect; i.e., they are initially equally reactive, but the reactivity of the second B group changes as the first has reacted. One master equation is sufficient to model formation of branched molecules only. Two are needed to take into account intramolecular cyclization. Monte Carlo simulations of the same process are used to verify the results of applying the kinetic model. The model can be applied to calculate various molecular parameters in polymerizing systems, including various average degrees of polymerization, size distribution of acyclic and cycle-containing polymer molecules, degree of branching, etc. Explicit formulas describing the dependence of some of these quantities on time or conversion degree are derived for the random...