Random Multiplicative Processes Research Articles

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Multifractal Properties of Diffusion‐Limited Aggregates and Random Multiplicative Processes

AbstractThe multifractal properties are considered of irreversible diffusion‐limited aggregation (DLA) from the point of view of the self‐similarity of fluctuations in random multiplicative processes. In particular the breakdown of multifractal behaviour and phase transition associated with the negative moments of the growth probabilities in DLA are analysed.

Two-point statistics of multifractal measures

The relationship between the ƒ(α) function of a multifractal and the spatial correlations of its singularity strengths is examined. An expression to compute the probability of observing two different singularities α’ and α″ within a distance r is derived for measures arising from isotropic random multiplicative processes. The correlation of α's is shown to decay logarithmically with distance. Possible applications to turbulence models are discussed, and the results are illustrated for a binomial measure, where the scaling of two-point correlation functions is shown to exhibit a phase transition.

Random multiplicative processes: An elementary tutorial

An elementary discussion of the statistical properties of the product of N independent random variables is given. The motivation is to emphasize the essential differences between the asymptotic N→∞ behavior of a random product and the asymptotic behavior of a sum of random variables—a random additive process. For this latter process, it is widely appreciated that the asymptotic behavior of the sum and its distribution is provided by the central limit theorem. However, no such universal principle exists for a random multiplicative process. In this case, the ratio between the average value of the product 〈P〉 and the most probable value Pmp diverges exponentially in N as N→∞. Within a continuum approximation, the classical log-normal form is often invoked to describe the distribution of the product. It is shown, however, that the log-normal provides a poor approximation for the asymptotic behavior of the average value and, also, for the higher moments of the product. A procedure for computing the correct leading asymptotic behavior of the moments is outlined. The implications of these results for simulations of random multiplicative processes are also discussed. For such a simulation, the numerically observed ‘‘average’’ value of the product is of the order of Pmp, and it is only when the simulation is large enough to sample a finite fraction of all the states in the system that a monotonic crossover to the true average value 〈P〉 occurs. An idealized, but quantitative account for this crossover is provided.

Fractal Phenomena in Disordered Systems

Fractal phenomena in disordered systems are now so general that a review of the past year's developments covers a broad category of topics. These include the formation of particulate aggregates, with implications for electrochemical deposition, interfaces, electrodes, contacts and mem­ branes; dynamics of silica aerogels, and the first direct observation of quantized lattice vibrations of a fractal nctwork (fractons); mechanical simulations of many-body model systems, and the study of non equilibrium processes like aggregation and fracturing; multi fractals, measures with the special property of self-similarity, and their use for the study of random multiplicative processes; earthquakes and fractals and the stochastic behavior of the stress prior to an earthquake; and a newly discovered phenomenon of immense importance to all of us, the fractal mechanisms of cardiac stability. This compendium of short articles should provide the reader with a good sense of the remarkably diverse and important applications of the notion of fractal geometry. For those who wish more detailed information in a specific area, the references at the end of each article can be consulted. The articles are grouped according to rather broad areas of interest, with the last two (on earthquakes and cardiac rhythms) focusing on newly developing fields of major personal importance. Thus, articles two through six concern diffusion limited aggregation from a variety of perspectives; seven presents very new X-ray, neutron, and optical scattering evidence

Random multiplicative processes and transport in structures with correlated spatial disorder.

Random multiplicative processes and transport in structures with correlated spatial disorder.

Comment on "Self-similarity of fluctuations in random multiplicative processes"

Comment on "Self-similarity of fluctuations in random multiplicative processes"

Self-similarity of fluctuations in random multiplicative processes.

A mapping of the unit interval of all realizations of the disorder for a random multiplicative process allows us to show that (a) the flucatuations due to different realizations of the disorder are characterized by a multifractal spectrum and (b) the fluctuations in space for a given realization of the disorder do not possess properties of self-similarity. These results have relevant implications for various physical problems with a multiplicative structure.

The statistics of chemical trace concentrations in the steady state

The steady state probability distribution of chemical tracers in the atmosphere and in rain can be calculated from stochastic models, under explicit assumptions. We compare predictions from two such models, one based upon a linear stochastic equation in which random removal events … or storms … form a Poisson process, and the other based upon multiplicative random processes which produce a lognormal distribution. The two models predict different shapes for the resulting distributions of concentrations and fluxes of atmospheric tracers such as sulfates in ‘acid rain’. Consequently, estimates of the frequencies of extreme events, if based upon an assumed parametric form of a distribution function whose parameters are fitted by measurements of typical events, are model sensitive. To establish a preferred model for wet deposition, we have examined the statistics of sulfate deposition from the “European Network for the Study of Long Range Transport of Air Pollutants” (LRTAP), during the years 1973–1975. The shapes of the distributions of airborne sulfur species (SO 2+ SO 4 2−), and of raincarried sulfate fluxes to the ground, are between those predicted by the two models. Our results are of operational significance to efforts directed to establishing national standards for rain quality.

Frequency stability criteria for non-linear stochastic systems

In this paper general classes of stochastic systems described by stochastic integrals of the Volterra type are considered. Existence and uniqueness of random solutions as well as stochastic stability criteria of the Popov frequency form are established. These frequency stability criteria relate not only to the linear system dynamics (or the kernel of the Volterra integral) of the system, but also to the characteristics of the additive and multiplicative random processes that govern the system.

Coherence in Multiple Random Processes

A multivariate coherence function is described that gives a quantitative measure of the degree of linear dependence of a random process on a set of n other random processes. The properties of this function are discussed in relation to familiar properties of the coherence function for two random processes. This bivariate case constitutes a special case of the more general problem. The effect of transmission of signals through linear systems both with and without noise and the role of coherence in optimum multivariate linear filtering are discussed.