Lyapunov Functionals in Nonlinear Diffusion and Exchange Problems

This paper discusses the kinetic exchange model, in which all the agents mutually interact with each other, leaving a "lambda" part of their wealth in the stock, and the rest is distributed randomly among the participants in proportions "epsilon" and 1-"epsilon". Distribution function is found in the equation of the Boltzmann type for the stationary case under the conditions of uniform distribution "epsilon" in the interval [0, 1], and values of "lambda" close to 1. Then we propose the method of Lyapunov functional searching for nonlinear equations of the Fokker-Planck equation with the diffusion coefficient, which depends on the distribution function itself. The method uses an inequality for entropy production, which includes instantaneous equilibrium distribution function. For example, a simple equation, with the diffusion coefficient that depends on the variance of the distribution, shows how inequality is obtained from said Lyapunov functional. Unfortunately, the diffusion coefficient in the equation for the exchange model also depends on the coordinate and the proposed method generally does not work. Therefore, the only thing that was able to do - is to get the Lyapunov functional for the case when the distribution is localized close to the mean value, where x can be considered close to <x>.