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

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+by] , y(') =d [ax+(1-b) y] (x> or =y) , where (0< or =) a,b (< or =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 not = 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 not = d and a+b not = 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 compared with the theory. Good agreement is achieved for both gamma and s.