Master Equation Description Research Articles

Mesoscopic dynamics of chaos in chemical Lorenz model

The mesoscopic dynamics of the chemical Lorenz system has been examined within the framework of master equation description by means of ensemble stochastic simulations. The time evolution of the bin frequency, which approximates the probability of the master equation, has been checked in dynamic regimes of fixed points, limit cycles and chaos. The variances and covariances between different species are also calculated and compared. Results presented suggest a picture of mesoscopic dynamics beyond the usual deterministic description.

Reaction-controlled cooperative desorption in a one-dimensional lattice: a dynamical approach.

The spinlike dynamics of immobile reactants in a one-dimensional lattice is analyzed for two representative systems involving cooperative desorption. An exact combinatorial approach is worked out. Its failure to reproduce the results of microscopic simulations is shown to be associated with the lack of sufficiently strong ergodic properties, as a result of which the final state depends strongly on the initial conditions. A dynamical approach to the problem based on the Master equation description is subsequently developed, leading to full agreement with the microscopic simulations.

Decoherence in ion traps due to laser intensity and phase fluctuations

We consider one source of decoherence for a single trapped ion due to intensity and phase fluctuations in the exciting laser pulses. For simplicity we assume that the stochastic processes involved are white noise processes, which enables us to give a simple master equation description of this source of decoherence. This master equation is averaged over the noise, and is sufficient to describe the results of experiments that probe the oscillations in the electronic populations as energy is exchanged between the internal and electronic motion. Our results are in good qualitative agreement with recent experiments and predict that the decoherence rate will depend on vibrational quantum number in different ways depending on which vibrational excitation sideband is used.

Theory of nonequilibrium first-order phase transitions for stochastic dynamics

A dynamic definition of a first-order phase transition is given. It is based on a master equation description of the time evolution of a system. When the operator generating that time evolution has an isolated near degeneracy there is a first-order phase transition. Conversely, when phenomena describable as first-order phase transitions occur in a system, the corresponding operator has near degeneracy. Estimates relating degree of degeneracy and degree of phase separation are given. This approach harks back to early ideas on phase transitions and degeneracy, but now enjoys greater generality because it involves an operator present in a wide variety of systems. Our definition is applicable to what have intuitively been considered phase transitions in nonequilibrium systems and to problematic near equilibrium cases, such as metastability.

Stochastic versions of the Hodgkin-Huxley equations

A Hodgkin-Huxley model algorithm for the numerical simulation of noise in neurons is contracted from a master equation description (cellular automoton) into a Langevin description. This reduction reduces the time required for a simulation by about two orders of magnitude. Earlier work is summarized, condensed, and made explicit to make the algorithm transparent and facilitate applications. Two approximate treatments are reported. An extension of this approach is presented that includes spatial dependence and the propagation of a noisy action potential along an axon.

Effects of Galaxy Mergers on Their Spatial Distribution and Luminosity Function

We study how merging affects the luminosity and the spatial distributions of galaxies. By using a master equation description and a distribution of physical merging cross sections, we find that the evolution of the galaxy luminosity distribution and the spatial counts-in-cells distribution are closely related. They place strong constraints on each other. The observed evolution of the galaxy luminosity function in the Autofib Redshift Survey (Ellis et al.) implies at least a 10% decrease in the galaxy comoving number density at z < 0.7. This significantly affects the evolution of galaxy spatial counts-in-cells distribution. The observed evolution of the galaxy luminosity function also requires a form of differential dimming for the less luminous galaxies even if galaxy merging drives the evolution. A counts-in-cells analysis for distant galaxies would help clarify the nature of the excess counts of faint-blue galaxies at high redshifts.

Coherence with incoherent light in hydrogen-like systems

We predict that coherence effects may occur in the fluorescence light of the Lyman-α transition of hydrogenlike systems though the driving field is incoherent. The premise is the existence of an additional constant electric field. Using a master equation description we find quantum beats in the intensity correlation function g(2)(τ) and we give approximate results for their amplitude and frequency. We find a connection between the familiar quantum beats (Andra [2]) and these coherence effects with incoherent light.

Pumping atoms into a Bose-Einstein condensate in the boson-accumulation regime.

We consider the behavior of an atomic Bose-Einstein condensate in the presence of an atom in an (internal) excited electronic level. We analyze the boson-accumulation regime, defined by the relation ${\mathit{N}}_{0}$\ensuremath{\gg}a,N-${\mathit{N}}_{0}$, where N is the total number of atoms in the ground electronic state, ${\mathit{N}}_{0}$ is the number of atoms in the condensate, and a is the number of levels to which the excited atom can decay. In this regime, quantum statistical effects related to the boson nature of the atoms are predominant in the process of spontaneous emission. Simple arguments suggest that the proportion of atoms in the condensate decreases after the spontaneous decay. A more elaborated model based on a master equation description shows that, in general, these arguments may lead to erroneous conclusions. Under certain conditions it predicts that the proportion of atoms in the condensate increases. We give an interpretation of this phenomenon in terms of quantum interferences between processes that include reabsorption of the emitted photons, and dipole-dipole interactions between the atoms. The physical effect considered here may help to pump atoms into a condensate and to compensate for atomic losses in the atomic trap. \textcopyright{} 1996 The American Physical Society.

Maximum information entropy approach to non-markovian random jump processes with long memory: application to surprisal analysis in molecular dynamics

It is shown that non-markovian random jump processes in continuous time and with discrete state variables can be expressed in terms of a variational principle for the information entropy provided that the constraints describe the correlations among a set of dynamic variables at any moment in the past. The approach encompasses a broad class of stochastic processes ranging from independent processes through markovian and semi-markovian processes to random processes with complete connections. Two different levels of description are introduced: (a) a microscopic one defined in terms of a set of microscopic state variables; and (b) a mesoscopic one which gives the stochastic properties of the dynamic variables in terms of which the constrains are defined. A stochastic description of both levels is given in terms of two different characteristic functionals which completely characterize the fluctuations of micro- and mesovariables. At the mesoscopic level a statistic-thermodynamic description is also possible in terms of a partition functional. The stochastic and thermodynamic descriptions of the mesoscopic level are equivalent and the comparison between these two approaches leads to a generalized fluctuation-dissipation relation. A comparison is performed between the maximum entropy and the master equation approaches to non-markovian processes. A system of generalized age-dependent master equations is derived which provides a description of stochastic processes with long memory. The general approach is applied to the problem of surprisal analysis in molecular dynamics. It is shown that the usual markovian master-equation description is compatible with the information entropy approach provided that the constraints give the evolution of the first two moments of the dynamic variables at any time in the past.

Dynamics of stochastically induced and spatially inhomogeneous impurity breakdown in semiconductors

Impact ionization of trapped carriers in high electric fields at Helium temperatures can lead to threshold switching, i.e., switching of a semiconductor sample from a nearly insulating state to a high conducting steady state due to an applied voltage that exceeds the threshold value for breakdown. This phenomenon is investigated theoretically based on a set of partial differential equations containing the relevant carrier transport mechanisms. In a first analysis the impact of fluctuations on the breakdown process within the framework of a master equation description of spatially homogeneous generation-recombination kinetics is computed. Then, taking into account spatially inhomogeneous longitudinal modes, we find that threshold switching constitutes a three-stage process, consisting of a stage of front creation and propagation, a stage dominated by a seesawlike mode, and one in which impact ionization occurs nearly homogeneously throughout the sample.

Master-equation approach to macroscopic quantum tunneling of charge in ultrasmall single-electron-tunneling double junctions.

It is known that classical Coulomb blockade effects of single-electron-tunneling (SET) devices are disturbed by higher-order tunneling effects denoted as macroscopic quantum tunneling of charge (q-MQT) or cotunneling. Starting from the von Neumann equation, a master-equation approximation of cotunneling is given for ultrasmall SET double junctions in the limit T=0. The influence of an external electromagnetic environment modeled by an additional impedance in the circuit is studied both in the low-impedance and high-impedance limits. General tunneling channels described by quasiparticle current amplitudes Im${\mathit{I}}_{\mathit{q}}$(\ensuremath{\omega}) are considered just as the Ohmic approximation which leads to some known results. The rates of cotunneling show in the high-impedance case the effect of a ``quantum'' Coulomb blockade. By regularizing the logarithmic singularities in the quantum rates using linewidth effects, the master-equation description (Markov property) is complete in the sense of the tunneling theory up to fourth order.

Stochastic equations of motion for epitaxial growth.

We report an analytic derivation of the Langevin equations of motion for the surface of a solid that evolves under typical epitaxial-growth conditions. Our treatment begins with a master-equation description of the microscopic dynamics of a solid-on-solid model and presumes that all surface processes obey Arrhenius-type rate laws. Our basic model takes account of atomic deposition from a low-density vapor, thermal desorption, and surface diffusion. Refinements to the model include the effects of hot-atom knockout processes and asymmetric energy barriers near step edges. A regularization scheme is described that permits a (nonrigorous) passage to the continuum limit when the surface is rough. The resulting stochastic differential equation for the surface-height profile generically leads to the behavior at long length and time scales first described by Kardar, Parisi, and Zhang [Phys. Rev. Lett. 56, 889 (1986)] (due to desorption). If evaporation is negligible, the asymptotic behavior is characteristic of a linear model introduced by Edwards and Wilkinson [Proc. R. Soc. London, Ser. A 381, 17 (1982)] (due to asymmetric step barriers and/or knockout events). If the latter are absent as well, the surface roughness is determined by an equation independently analyzed by Villain [J. Phys. I 1, 19 (1991)] and Lai and Das Sarma [Phys. Rev. Lett. 66, 2348 (1991)] (which includes only deposition and site-to-site hopping). The consequences of reflection-symmetry breaking in the basic microscopic processes are discussed in connection with step-barrier asymmetry and Metropolis kinetic algorithms.

Burgers's turbulence model as a stochastic dynamical system: Master equation and simulation.

By means of Burgers's equation a stochastic description of turbulent fluid flows is explained which is based on a discrete master equation. The latter governs the dynamics of a discrete multivariate stochastic process representing the random velocity field of the fluid. From the characteristic function corresponding to this stochastic process, the Hopf functional equation of turbulence is obtained. This implies that the infinite hierarchy of correlation functions can be derived from the master equation. The master-equation description naturally leads to a simple stochastic simulation algorithm which is well suited for numerical implementation. Stochastic simulations of the Burgers model of turbulence are performed and are shown to yield very accurate results.

Collective fluctuations of conserved variables in liquids

Prominent among the classes of collective excitations in liquids that one would like to study are those which are compelled to obey some sort of conservation law. The instantaneous normal modes of liquid (which must be translationally invariant or, equivalently, conserve momentum) comprise one such example. The set of relaxation pathways dictated by a master-equation description of energy transfer in a liquid—which must conserve probability—constitutes another. We show that these conservation laws do impose fairly stringent requirements on the nature of the collective behavior, but the resulting excitations can nonetheless be described by liquid-theory methods. Within linear liquid theories, the desired distribution of modes ends up being a combination of a delocalized electronic-band-like portion and a fluctuating local field contribution. We illustrate the results with an explicit calculation (at the master-equation level) of energy-transfer kinetics in a liquid.

A master equation description of fluctuating hydrodynamics

Recently a mesoscopic approach to computational fluid dynamics has been proposed which is based on a stochastic description defined by a discrete master equation. Applying van Kampen's system size expansion to one-dimensional flows, the deterministic macroscopic equations, i.e., the Navier-Stokes and continuity equation are obtained. The velocity and density fluctuations around the macroscopic dynamics are governed by a Fokker-Planck equation. From the assumption of local thermodynamic equilibrium, a relation between the velocity fluctuations and the temperature is obtained which allows to derive the linear Langevin equations of fluctuating hydrodynamics. Thus, the suggested stochastic approach makes possible the simultaneous numerical simulation of both the macroscopic balance equations and the hydrodynamic fluctuations.

Quantum-field model of the injected atomic beam in the micromaser.

A general theory of the micromaser is described. The model is based on treating the input atomic beam as a two-component quantum field so that the two-level atoms in the beam are ``quanta'' of this field. This approach makes it possible to formulate a general quantum Langevin description of the dynamics with the input atomic field as a source of quantum noise. The passage to a master-equation description is then effected by use of an adjoint-operator method, and by introducing a general class of statistical states for the atomic beam known as generalized shot noise. The result is a (non-Markovian) master equation for the field inside the cavity which is valid for a broad range of statistical properties of the input atomic beam. The approximate steady-state solutions to this master equation for the photon statistics of the cavity field for sub-Poissonian (antibunched) atomic beams found by other researchers are regained. The theory is then extended to treat super-Poissonian (bunched) atomic beams. An exact result is found in the limit of a strongly bunched beam in which the cavity-field state is shown to converge to a mixture of a thermal-field state and the state produced by a random beam of twice the intensity. A physical explanation of this result is also presented.

Dynamical scaling in anisotropic phase-separating systems in a gravitational field.

We derive a generalization of the Cahn-Hilliard equation for phase separation in the presence of a field that varies linearly in (say) the z direction, e.g., gravity. The starting point of our derivation is the master-equation description for a spin-exchange kinetic Ising model in the presence of such a field. We neglect thermal fluctuations and the effect of hydrodynamic interactions. We also mptivate the concept of generalized dynamical scaling as a means of characterizing anisotropic domain growth. We present numerical results from a two-dimensional (2D) simulation of this equation. Our results indicate that the 2D time-dependent structure factor S(k x ,k z ,t) (where k x and k z are, respectively, the x and z components of the wave vector) has the generalized dynamical scaling form S(k x ,k z ,t) =l x (t)l z (t)F(k x l x (t),k z l z (t)), where l x (t) and l z (t) are time-dependent length scales in the x direction (perpendicular to direction of gravity) and in the z direction (in the direction of gravity), respectively; and F(x,y) is a universal function of its arguments. The temporal behavior of these length scales is l z (t)~t and l x (t)~t 1/3 . The experimental observability of these numerical results is also discussed.

Hydrogen bond dynamics in isotopically substituted benzoic acid dimers

The hydrogen pair transfer in the hydrogen-bonded dimers of benzoic acid and its carboxyl-deuterated species is investigated in the solid. Measurements are reported for the temperature-dependent NMR relaxation time T1 in single crystals containing dimers with one or two carboxylic deuterons. Combined with previous data, the temperature dependence of the measurements is analyzed in terms of a master-equation description for a one- or two-dimensional quantum-mechanical model of the transfer motion. The description by a one-dimensional model is found to be inadequate as it yields unrealistic isotope effects in the model parameters. On the other hand, reasonable results are obtained for a two-dimensional model which includes, apart from the transfer motion of the hydrogens, a heavy atom mode with properties suggested by x-ray structural data. This model explains the thermal activation of the transfer process mainly as a result of promotion of tunneling by heavy atom rocking. Activation energies remain considerably smaller than the barrier height and indicate predominance of tunneling even at room temperature.

Averages of additive quantities for the master equation description of the dynamics of a system

Stationary ensemble averages such as &lt; X n &gt;, &lt; XY &gt; etc. where X and Y are stochastic variables whose changes in the course of time t are additive, are examined for a system with dynamics described by a master equation. Results are obtained for the two important cases of ( a ) thermal equilibrium and ( b ) steady-state situations brought about by the imposition of external fields. The averages are expressed as integrals with respect to time of appropriate time-correlation functions. For appropriate quantities of this type one may express corresponding linear response coefficients in terms of these averages. The results represent many-body generalizations of the elementary Einstein relations between diffusion coefficient and mean-square displacement, and between mobility and mean-square displacement. While the former type of relation holds only for normal diffusive systems the second type holds also for anomalous diffusive systems.

Transition from quantum to classical dissipation in the problem of escape from a metastable well: Application to Josephson junctions.

For a dissipative quantum system in the low-damping regime, we show that the master-equation description of the reduced density matrix reduces, in the semiclassical limit, to the energy-diffusion version of the Fokker-Planck equation. This result can be generalized to include non-Ohmic damping and an external harmonic perturbation. Results of calculations based on the master equation show good agreement with experiments on the resistanceless-to-resistive transition in Josephson junctions in the classical regime and with results of purely classical calculations.