Log-normal distribution in growing systems with weighted multiplicative interactions

Akihiro Fujihara; Satoshi Tanimoto; Toshiya Ohtsuki; Hiroshi Yamamoto

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Log-normal distribution in growing systems with weighted multiplicative interactions

Akihiro Fujihara·Satoshi Tanimoto·Toshiya Ohtsuki·Hiroshi Yamamoto

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Abstract

Many-body stochastic processes with weighted multiplicative interactions are investigated analytically and numerically. An interaction rate between particles with quantities $x, y$ is controlled by a homogeneous symmetric kernel $K(x, y) \propto x^{w} y^{w}$ with a weight parameter $w$. When $w<0$, a method of moment inequalities is used to derive log-normal type tails in probability distribution functions. The variance of log-normal distributions is expressed in terms of the weight $w$ and interaction parameters. When interactions are weak and a growth rate of systems is small, in particular, the variance is in proportion to the growth rate. This behavior is totally different from that of one-body stochastic processes, where the variance is independent of the growth rate. At $w>0$, Monte Carlo simulations show that the processes end up with a winner-take-all state.

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Akihiro Fujihara, Satoshi Tanimoto, Toshiya Ohtsuki, Hiroshi Yamamoto. 2006-05-10. Log-normal distribution in growing systems with weighted multiplicative interactions. https://arxiv.org/abs/cond-mat/0511625

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