Regular Series


Vol. 27 (1996), No. 3, pp. 617 – 810


Schrödinger’s Interpolation Problem through Feynman–Kac Kernels

abstract

We discuss the so-called Schrödinger problem of deducing the microscopic (basically stochastic) evolution that is consistent with given positive boundary probability densities for a process covering a finite fixed time interval. The sought for dynamics may preserve the probability measure or induce its evolution, and is known to be uniquely reproducible, if the Markov property is required. Feynman–Kac type kernels are the principal ingredients of the solution and determine the transition probability density of the corresponding stochastic process. The result applies to a large variety of nonequilibrium statistical physics and quantum situations.


Relaxation in the Random Map Model

abstract

Properties of two sets of finite deterministic cellular automata are compared: the set of local homogeneous one-dimensional automata and the set of all possible automata (random map model). We investigate the following properties: relaxation time, number and length of limit cycles and the distribution of the size of bassins of attraction.


On Quantum \(L_p\)-Space Technique

abstract

We show that using recently introduced quantum \(L_p\)-spaces one can give an explicit constructions of spin-flip and diffusive type quantum dynamics.


Transport in Ratchet-Type Systems

abstract

Diffusion motion of overdamped Brownian particles in a spatially periodic asymmetrical potential and driven by a zero-mean time-periodic external force and zero-mean Gaussian white noise is considered. The influence of asymmetry of the spatial potential, amplitude of the driving force and intensity of the noise on an average translocation velocity of particles is analyzed. The adiabatic approximation for calculation of a stationary particle current is used and verified by comparison with results obtained from computer simulations of the process considered.


Cellular Automata with Voting Rule

abstract

The chosen local interaction the voting (majority) rule applied to the square lattice is known to cause the non ergodic cellular automata behaviour. Presented computer simulation results verify two cases of non ergodicity. The first one is implicated by the noise introduced to the local interactions and the second one follows properties of the initial lattice configuration selected at random. For the simplified voting rule non symmetric voting, the critical behaviour has been explained rigorously.


Simple Mathematical Tool for Statistical Description of Dynamical Systems under Random Actions. I

abstract

In the lectures, the differentiation formulae (DF) for statistical averages is introduced in a compact form. The first part of these lectures is devoted to a general description of the DF method. The procedure of obtaining exact and closed equations for mean values and probability distributions of linear and nonlinear macroscopic systems driven by coloured noise is illustrated. As models of coloured random perturbations, both random jump processes (Kubo–Anderson and kongaroo processes) and diffusion processes (Ornstein–Uhlenbeck, Rayleigh and Pearson processes) are considered.


Growth Models with Internal Competition

abstract

Combined statistical physics and computation modelling give new instruments for the study of non-equilibrium systems. We briefly review generalized Eden and Diffusion-Limited Aggregation models as applied to spreading phenomena. We indicate the occurrence of non-universal behaviors.


Nonequilibrium Spatial Correlations of Reagents in Model Reaction Diffusion System

abstract

The mesoscopic description of systems with chemical reactions predicts that if the detailed balance condition is not satisfied then the nonequilibrium spatial correlations between concentrations of reactants may appear. The present work is concerned with molecular dynamics simulations of these correlations. The correlations appearing in a stationary state of a multicomponent chemical system and in a time dependent state of an “enzymatic” reaction are studied. Nonequilibrium correlations between reactants observed in simulations are compared with results of theory based on the master equation for a spatially distributed system.


Noise-Induced Transitions in a Bistable Process Driven by non-Markovian Noise: Stationary States

abstract

Stationary states of the Verhulst process driven by non-Markovian dichotomic noise containing both Markovian and exponentially damped explicitly non-Markovian components are investigated. Noise-induced stationary states are compared with such states induced by purely Markovian dichotomic noise. It is found that the non-Markovianity of the driving noise may result in the appearance of new noise-induced stationary states, but in some cases, especially for higher values of the deterministic bifurcation parameter, non-Markovianity may result in the damping of Markovian noise-induced states. Besides, non-Markovianity generally diminishes the dispersion of noise-broadened states.


Kinetics of Microdomain Formation in Two Dimensional Assemblies

abstract

A novel phenomenological approach to the microdomain structure formation or phase transformation in two-dimensional cooperative systems is proposed. The theory offered states that a new structure consists of pieces of islands, microdomains, germs, etc. and deals with modeling of the pattern formation process with increase of area of a new structure or phase. The kinetics of the process is studied. Probabilistic characteristics are obtained and first three moments of the process are analyzed.


Nonequilibrium Velocity Distribution of Reactive Foreign Gas: Simulation and Phenomenology

abstract

The velocity distribution function of a homogeneous foreign gas reacting with a carrier gas is studied by means of the Monte Carlo simulation and it is calculated using an approximate phenomenological approach assuming the Maxwellian form of the distribution with nonequilibrium temperature. The transition of the system to the hydrodynamic regime is demonstrated. The effect of the chemical reaction on the distribution function is presented in terms of decrease in the second and fourth moments of the velocity distribution for a wide range of activation energies and ratios of molecular masses. Good agreement between the results of Monte Carlo simulations and the approximate calculations is observed.


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