Metabolic Dynamics of Ecosystems Realizing Steady Log-Uniform Distributions: The Case of Commodities in Shops

From the viewpoint of statistical physics, ecosystems in the real world are very attractive targets of research as examples of far-from thermal equilibrium systems where various kinds of components are coming in and out continuously while keeping the whole systems quasi-stationary. As a fortunate example of a fully-observable ecosystem, we analyzed the comprehensive data of convenience stores where approximately 5% of the commodity species is replaced by new ones daily. The share of stores for each species fluctuates significantly; however, the entire distribution of shares is fairly stationary and follows the log-uniform distribution, that is, the power law distribution with exponent 0. We introduce an empirical time evolution model of shares and firstly deduce that the key mechanism of realizing this stationary distribution is random multiplicative diffusion in finite size spaces. Our model based on the general stochastic process is expected to be applicable to various dynamic systems, especially complex systems with highly nonlinear interactions.