## 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.

Original language | English |
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Pages (from-to) | 747-779 |

Number of pages | 33 |

Journal | Journal of Statistical Physics |

Volume | 124 |

Issue number | 2-4 |

DOIs | |

State | Published - Aug 2006 |

Externally published | Yes |

## Keywords

- Boltzmann equation
- Granular gases
- Overpopulated tails
- Pareto distribution
- Wealth and income distributions