Abstract
Multiplicatively interacting stochastic processes are defined on a $N-$particle system. Every particle has a positive quantity $x_i(>0) (i=1,...,N)$. At each timestep, two particles are selected randomly and exchange their quantities as $x_i' = x_i + x_j, x_j' = x_i + x_j$, where $x_i', x_j'$ are the post-exchanged quantities of particles $x_i, x_j$, respectively.
In $N ınfty$, the processes are described by a master equation with the integral kernel $K(x,y) x_i^w_1 x_j^w_2 + x_i^w_2 x_j^w_1$ representing interaction rates, where $w_1, w_2$ are weight parameters. The probability distribution function(PDF) of the quantities $f(x,t)$ is examined analytically by investigating the master equation and numerically by Monte Carlo simulations. Without loss of generality, the condition $w_1 w_2$ is imposed because of the symmetry of weight parameters.
In this presentation, the relation between the PDF and weight parameters is discussed. Consequently, it is found analytically and numerically that when $w_1, w_2 < 0$, the PDF obeys a log-normal distribution with a constant variance. It is also found that when $w_1 = 0, w_2 0$, a power-law distribution emerges at the tail of the PDF. It is suggested numerically that in $w_1 >0$, the behavior of the tail becomes unstable. In $w_1, w_2 > 0$, especially, the PDF goes into a winner-take-all state where two particular particles possesses almost all of the total sum of the quantities.
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