Power-law tails in nonstationary stochastic processes with asymmetrically multiplicative interactions

TL;DR

This paper analyzes power-law tails in nonstationary stochastic processes with asymmetric interactions, deriving conditions for tail exponents and growth rates, and confirming results through numerical simulations.

Contribution

It introduces a formal method to determine power-law tail exponents in asymmetric stochastic processes using moment equations and transcendental equations.

Findings

Power-law tails characterized by a transcendental equation.

Analytical solutions for symmetric cases c=d or a+b=1.

Numerical simulations confirm theoretical predictions.

Abstract

We consider stochastic processes where randomly chosen particles with positive quantities x, y (> 0) interact and exchange the quantities asymmetrically by the rule x' = c{(1-a) x + b y}, y' = d{a x + (1-b) y} (x \ge y), where (0 \le) a, b (\le 1) and c, d (> 0) are interaction parameters. Noninteger power-law tails in the probability distribution function of scaled quantities are analyzed in a similar way as in inelastic Maxwell models. A transcendental equation to determine the growth rate \gamma of the processes and the exponent s of the tails is derived formally from moment equations in Fourier space. In the case c=d or a+b=1 (a \neq 0, 1), the first-order moment equation admits a closed form solution and \gamma and s are calculated analytically from the transcendental equation. It becomes evident that at c=d, exchange rate b of small quantities is irrelevant to power-law tails. In the case c \neq d and a+b \neq 1, a closed form solution of the first-order moment equation cannot be obtained because of asymmetry of interactions. However, the moment equation for a singular term formally forms a closed solution and possibility for the presence of power-law tails is shown. Continuity of the exponent s with respect to parameters a, b, c, d is discussed. Then numerical simulations are carried out and campared with the theory. Good agreement is achieved for both \gamma and s.