Similarity and Probability Distribution Functions in Many-body Stochastic Processes with Multiplicative Interactions

TL;DR

This paper reviews analytical and numerical methods for studying many-body stochastic processes with multiplicative interactions, focusing on distribution functions, growth rates, and power-law tails, with validation through simulations.

Contribution

It introduces a method using moment relations to analyze effects of asymmetry and randomness, providing new insights into distribution tails and growth dynamics.

Findings

Distribution functions exhibit similarity solutions with power-law tails.

Growth rates and tail exponents are derived from transcendental equations.

Analytical results agree well with Monte Carlo simulations.

Abstract

Analytical and numerical studies on many-body stochastic processes with multiplicative interactions are reviewed. The method of moment relations is used to investigate effects of asymmetry and randomness in interactions. Probability distribution functions of the processes generally have similarity solutions with power-law tails. Growth rates of the system and power-law exponents of the tails are determined via transcendental equations. Good agreement is achieved between analytical calculations and Monte Carlo simulations.