Abstract
In this paper, we discuss the large-time behavior of solution of a simple kinetic model of Boltzmann–Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time-monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis.
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Pareschi, L., Toscani, G. Self-Similarity and Power-Like Tails in Nonconservative Kinetic Models. J Stat Phys 124, 747–779 (2006). https://doi.org/10.1007/s10955-006-9025-y
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DOI: https://doi.org/10.1007/s10955-006-9025-y