System Size Expansion Research Articles

Two-level atom coupled to a squeezed vacuum inside a coherently driven cavity

A single two-level atom in a coherently driven cavity and damped by a broadband squeezed vacuum centered about the atomic transition frequency is studied. A second-order Fokker-Planck equation for this system is obtained without using system size expansion. Effects of detunings and cavity decays are also incorporated in the Fokker-Planck equation. This equation is used to study atomic inversion, fluorescent spectrum, and the intensity correlations of the transmitted and fluorescent photons in the bad-cavity limit. Several interesting effects in the atomic inversion, spectrum, and intensity correlations due to the squeezed vacuum are presented. These results are also compared with an atom that is damped by a thermal reservoir.

Fokker-Planck equation for lattice deposition models.

An asymptotically exact Fokker-Planck equation for the height fluctuations of lattice deposition models is derived from a Van Kampen expansion of the master equation. Using an Edwards-Wilkinson-type model as an example, the solution of the equivalent Langevin equation reproduces the surface roughness and lateral height correlations obtained with kinetic Monte Carlo (KMC) simulations. Our discrete equations of motion thereby provide an exact analytic and computational alternative to KMC simulations of these models.

THE VAN KAMPEN EXPANSION FOR LINKED DUFFING-LINEAR OSCILLATORS EXCITED BY COLORED NOISE

This paper expands on previous work by Rodriguez and van Kampen, and by Weinstein and Benaroya. The original work by Rodriguez and van Kampen outlined a method of extracting information from the Fokker-Planck equation without having to solve the equation itself. In the van Kampen expansion, the Fokker-Planck equation is expanded about the random component of the response. This expansion is used to derive a series of first order differential equations in time for the moments of the response. These can then be solved for moments as a function of time. In the original work, this method was demonstrated on the case of a Duffing oscillator excited by white noise. Weinstein and Benaroya developed this method for the case of a Duffing ocsillator excited by colored noise, performed parametric studies on the system parameters, and verified their results by comparing them to Monte Carlo experiments. In this work, the van Kampen expansion is applied to systems of linked Duffing-linear oscillators excited by colored noise. Again, parametric studies are performed on the system parameters and the results compared to those of Monte Carlo experiments. It is found that the analytical results compare closely with the numerical results as long as the initial assumptions of the expansion are not violated. The closeness of the comparisons indicates that the van Kampen expansion is a valid technique for the study of this class of problem and is a useful tool in modelling offshore structures, plates in turbulent flows, and other fluid-structure interaction phenomena.

Generalized Birth-Death Stochastic Process in Nonequilibrium Open System

Classification of statistics for a generalized birth-death stochastic process is studied with the use of the system size expansion method. Clarified is the physical and mathematical nonlinear structure which gives sub-Poissonian statistics in nonequilibrium open systems. It is also shown that sub-Poissonian statistics is ubiquitous in nonequilibrium open systems.

Fokker-Planck equation for a single two-level atom: Applications in the bad-cavity limit.

A generalized second-order Fokker-Planck equation is derived for a single two-level atom in a cavity driven by an external classical field without using system size expansion or any truncation. Effects of detuning, as well as atomic and cavity decays, are incorporated in this Fokker-Planck equation. This equation is used to study atomic inversion and the second-order intensity correlation function of light in the bad-cavity limit by adiabatically eliminating the field variables. Several novel features in atomic inversion and intensity correlations that arise due to atomic and cavity detuning are discussed. Curves are presented to illustrate the behavior.

The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator excited by colored noise

In Rodriguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity α of the forcing function. In Weinstein and Benaroya, the effect of the order of expansion is investigated by carrying the expansion to a higher order. The effect of varying the system parameters is also investigated. All results are verified by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is modified and applied to the case of a Duffing oscillator excited by colored noise. The effect of the correlation time is investigated. Again the results are compared to those of Monte Carlo experiments. It is found that the expansion compares closely with those of the Monte Carlo experiments as the correlation time τc is varied from 0.001 to 10 sec. Examination of the results reveals that the colored noise can be categorized in one of four ways: (1) for $$\tau _c< \mathcal{O}(0.01{\text{ }}\sec )$$ the noise can be considered as white for all intents and purposes, (2) for $$\tau _c = \mathcal{O}(0.1{\text{ }}\sec )$$ the noise can be considered white for some purposes, (3) for $$\tau _c = \mathcal{O}(1.0{\text{ }}\sec )$$ the correlated nature of the noise must be considered in an analysis, and (4) for $$\mathcal{O}(1.0{\text{ }}\sec )< \tau _c $$ the noise can be considered as deterministic.

Stochastic modeling of adsorption in a batch system

The adsorption of molecules of organic or inorganic compounds in an aqueous solution onto granular activated carbon involves a sequence of steps including: transfer of the molecules from the bulk phase of solution through the relatively stagnant, layer of solution adjacent to the external surface and solution in macropores and micropores of the pellet of activated carbon and occupation of the active sites on the inside of the pellet by the molecules. The present work analyzes and models this sequence of steps by resorting to the stochastic population balance of the numbers of molecules of adsorbate in the three states. The first comprises the bulk phase of the solution; the second, the layer of solution adjacent to the external surface and the solution in the macropores; and the third, the active sites on the inside surfaces of the micropores. The master equation has been derived for the case of a single adsorbate compound. The equations for the means, variances, and covariances of the random variables have been obtained through the system size expansion of the master equation. At equilibrium, the equations for the means reduce to the equation of the Langmuir isotherm. The unknown parameters in the equations for the means have been estimated by comparing the calculated results with the experimental data. These parameters have been adopted to predict the evolution of variances and covariances of the numbers of adsorbate molecules in the three states.

Stochastic dynamics of learning with momentum in neural networks

We study on-line learning with a momentum term for nonlinear learning rules. Through introduction of auxiliary variables, we show that the learning process can be described by a Markov process. For small learning parameters eta and momentum parameters alpha close to 1, such that gamma = eta /(1- alpha )2 is finite, the time-scales for the evolution of the weights and the auxiliary variables are the same. In this case Van Kampen's expansion can be applied in a straightforward manner. We obtain evolution equations for the average network state and the fluctuations around this average. These evolution equations depend (after rescaling a of the time and fluctuations) only on gamma : all combinations ( eta , alpha ) with the same value of gamma give rise to similar behaviour. The case with alpha constant and eta small requires a completely different analysis. There are two different time-scales: a fast time-scale on which the auxiliary variables equilibrate and a slow time-scale for the change of the weights. By projection on the space of slow variables the fast variables can be eliminated. We find that, for small learning parameters eta and finite momentum parameters alpha , learning with momentum is equivalent to learning without a momentum term with a rescaled learning parameter eta = eta /(1- alpha ). Simulations with the nonlinear Oja learning rule confirm the theoretical results.

Fluctuation effects on wave propagation in a reaction-diffusion process

The reaction-diffusion process corresponding to the Fisher-Kolmogorov equation is studied by means of a discrete multivariate master equation. For travelling wave fronts the stability criterion necessary for the applicability of a system-size expansion is shown to be violated due to the existence of a zero mode of the first variational equation. This zero mode is connected to the translational invariance of the system. Performing stochastic simulations of the master equation in a wide range of parameters it is demonstrated that for finite size of the system (up to about 10 7 particles in the frontal region) a rather large fluctuation effect on the wave propagation speed results: in general, the asymptotic wave speed lies below the stable, minimal speed which is given by a theorem of Kolmogorov for the macroscopic equation. The wave front position exhibits a diffusion-type behaviour associated with translative fluctuations along the propagation direction.

The reversible reaction A+B = C in solution. A system-size expansion approach on the base of reactive many-particle diffusion equations

The diffusion-influenced reaction A+B■C is reconsidered by using an approach which starts directly from the reactive many-particle diffusion equations which govern the change in time of system states with a defined number of reactive particles. The classical problem is transformed into a more compact ‘‘quantum’’ one by using a second quantization procedure. In this way, by straightforward operator manipulations, exact state-specific evolution equations can be derived. To prove the conditions for an approximate deterministic description of macroscopic systems, a system-size expansion in the sense of van Kampen is applied to these equations. By approximating the triplet and quadruplet terms in the evolution equations, a rate equation, a Fokker–Planck equation for the particle number fluctuations, and an evolution equation for the AB-pair distribution function can be derived which are consistent with one another. The results of this approach are compared with those of other recent studies including the stochastic approach I used in [Chem. Phys. 150, 187 (1991)].

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.

Nonlinear phase-space diffusion approach to turbulent organized structures

The spatial evolution of the velocity in a turbulent fluid is shown to consist of a mean strain rate plus a noise term whose mean-square amplitude is dependent upon the local velocity. Assuming the independence of different length scales allows one to eliminate the effect of cubic and higher moments of the velocity upon the evolution of the two-point probability density function for the velocity, analogous to the system-size expansion of the master equation (N. G. van Kampen. Adv. Chem. Phys. 34, 245 (1976); R. Kubo, K. Matsuo, and K. Kitahara. J. Stat. Phys. 9, 51 (1973); M. Suzuki. Adv. Chem. Phys. 46, 198 (1981)). Furthermore, for the velocity structures that exhibit the most internal correlation (i.e., the flow fields that are most likely to be observed) the mean strain rate drops out, leaving only the nonlinear diffusion term. In other words, the dependence of the two-point velocity statistics upon length scale or separation is found to be governed by the gradients of the stress (rather than of the velocity, as is usually assumed.) This provides a new approach for predicting the dominant patterns observed in turbulent flows and a new means of characterizing or classifying different structures, namely, by their nonlinear diffusion coefficient. As a physically relevant example, the isotropic case with algebraic nonlinearity is considered.

Fluctuations of a charged Brownian particle in a plasma.

The system-size expansion technique, is used to study the fluctuations both near and far from equilibrium of a charged Brownian particle in a two-component plasma. The method is based on the master equation that governs the conditional probability density function of the velocity. This method yields a macroscopic equation for the slowing down of the heavy charged particle; and in the linear noise approximation, equations for the mean and variance of the fluctuations around the macroscopic behavior are found and a fluctuation-dissipation theorem is demonstrated. The velocity-velocity correlation function and the diffusion coefficient of these particles are also determined in this approximation, both near and far from equilibrium. Numerical examples are used to illustrate the results. The generalization to multispecies plasma is apparent.

Stochastic analyses of the dynamics of generalized Little-Hopfield-Hemmen type neural networks

Stochastic analyses are conducted of model neural networks of the generalized Little-Hopfield Hemmen type, in which the synaptic connections with linearly embedded p sets of patterns are free of symmetric ones, and a Glauber dynamics of a Markovian type is assumed. Two kinds of approaches are taken to study the stochastic dynamical behavior of the network system. First, by developing the method of the nonlinear master equation in the thermodynamic limit N--* oe, an exact self-consistent equation is derived for the time evolultion of the pattern overlaps which play the role of the order parameters of the system. The self-consistent equation is shown to describe almost completely the macroscopic dynamical behavior of the network system. Second, conducting the system-size expansion of the master equation for the N-body probability distribution of the Glauber dynamics makes it possible to analyze the fluctuations. In the course of the analysis, the self-consistent equation for the pattern overlaps is derived again. The main result of the rigorous fluctuation analysis is that as far as the fluctuations are concerned, the time course of the pattern overlap fluctuations behaves independently of the fluctuations in the remaining modes of the system's macrovariables, in accordance with the self-determining property of the macroscopic motion of the pattern overlaps for neural networks with linear synaptic couplings.

Foundations and Extensions of the Cahn-Hilliard-Cook Theory

Abstract In this paper we examine the foundations of the Cahn-Hilliard-Cook (CHC) theory of phase separation, which has been used routinely in recent years to interpret dynamic scattering experiments on binary polymer blends, and discuss its extension to multi-component mixtures. The CHC-theory is based on the following nonlinear Langevin equation1: where Φ(q,t) is the Fourier transform of the local volume fraction Φr,t) of one of the components of the binary mixture which is assumed to be incompressible; λ(q) is the q-dependent onsager coefficient; μ(q,t) is the Fourier transform of the local chemical potential difference μ(r,t); and η(q,t) is the random force which is added to account for the thermal fluctuations. The latter is assumed to be a stationary white noise process with zero mean and an autocovariance (η(q,t) q(-q,t′)) = λ(q) q2 δ(t - t′), which is determined from fluctuation-dissipation theorem. The use of the Langevin equation method to describe fluctuations in nonlinear macroscopic systems, in the way done in the CHC theory, was questioned by van Kampen2,3 for reasons which we discuss in this paper. The proper way of using the Langevin equation method in nonlinear systems was provided by Akcasu4 in 1977 through van Kampen's system size expansion, which we sketch next.

Non-equilibrium fluctuations of a Rayleigh particle in a two-component gas

A double system size expansion technique is used to study the fluctuations of a Rayleigh piston in a two-component gas. The macroscopic law for the most probable path of the velocity of the particle is obtained; and using the linear noise approximation the characteristics of the fluctuations around this path are determined, for both, fluctuations near equilibrium and far from equilibrium. The mean and variance of the fluctuations, the velocity-velocity correlation function and the diffusion coefficient in the configuration space are calculated. The generalization to the case where the gas is composed by more than two different types of molecules is straightforward.

Nonlinear fluctuations, separation of procedures, and linearization of processes

A concept of separation of procedures is introduced to study cooperative phenomena theoretically. Some typical important examples of this concept are presented to clarify its usefulness; Kubo's stochastic Liouville equation, some generalized diffusionlike equations, van Kampen's expansion, Kubo's extensivity, Prigogine's entropy production, the scaling theory of transient phenomena based on the Lie algebra, and Suzuki's CAM theory of cooperative phenomena are discussed from the new viewpoint of separation of procedures.

STOCHASTIC MODELING OF NON-LINEAR SIEVING KINETICS

The feed to a sieve is classified as oversize particles, undersize particles, and near mesh size particles. The sizes of the near mesh size particles vary around that of the sieve opening; the passage of these particles through the sieve effects the separation. A stochastic approach is employed for analyzing and modeling the sieving kinetics of the near mesh size particles, usually constituting the bulk of the material required to be separated. The master equation of the process is formulated based on probabilistic considerations, describing the passage of the particles in the presence of sieve blinding. However, an analytical solution cannot be obtained directly and hence a rational approximation technique, the system size expansion, is utilized to solve the master equation. This results in a system of non-linear, coupled, ordinary differential equations for the various statistical quantities characterizing the number of particles retained on the sieve and the number of blinded apertures. These equations...

New approach to Bayesian estimation through partial observation of nonlinearly interacting stochastic variables.

A theory of Bayesian estimation of the behavior of a nonlinear stochastic system through observing a part of the system (partial observation) is developed. A new method to solve the evolution equation for the a posteriori probability density of the unobserved variable is studied with the extended use of the system-size expansion method. The resultant equations are of a computationally feasible form and enable us to estimate the unobserved variable with the use of the data on the observed variable. The method is shown to be effectively applied to the Lotka-Volterra population system with fluctuating forces to estimate the population of predators upon knowing that of the prey.

Relaxation of an unstable system driven by a colored noise

The scaling solution of an unstable system driven by a Ornstein-Uhlenbeck noise is derived from the two-variable Fokker-Planck equation. A quasiprobability distribution for the joint process (system parameter and noise) is then introduced and the system-size expansion is shown to yield a description of the relaxation process in the entire time domain.