%0 Book Section %1 statphys23_0814 %A Fujihara, A. %A Tanimoto, S. %A Ohtsuki, T. %A Yamamoto, H. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K distribution economic kinetic log-normal power-law processes social statphys23 stochastic systems tail theory topic-1 %T Effects of integral kernel in multiplicatively interacting stochastic processes %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=814 %X 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.

@incollection{statphys23_0814, 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 $x_i' = \alpha x_i + \beta x_j, x_j' = \beta x_i + \alpha x_j$, where $x_i', x_j'$ are the post-exchanged quantities of particles $x_i, x_j$, respectively. In $N \to \infty$, the processes are described by a master equation with the integral kernel $K(x,y) \propto x_i^{w_1} x_j^{w_2} + x_i^{w_2} x_j^{w_1}$ 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 $w_1 \ge w_2$ 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 $w_1 = 0, w_2 \le 0$, 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.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Fujihara, A. and Tanimoto, S. and Ohtsuki, T. and Yamamoto, H.}, biburl = {https://www.bibsonomy.org/bibtex/231126fca55c9775ba26b0082207495a6/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {3d046757b81701e6fffdc3e9dfb56a20}, intrahash = {31126fca55c9775ba26b0082207495a6}, keywords = {distribution economic kinetic log-normal power-law processes social statphys23 stochastic systems tail theory topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:30.000+0200}, title = {Effects of integral kernel in multiplicatively interacting stochastic processes}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=814}, year = 2007 }