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Kinetic Models with Randomly Perturbed Binary Collisions

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Abstract

We introduce a class of Kac-like kinetic equations on the real line, with general random collisional rules which, in some special cases, identify models for granular gases with a background heat bath (Carrillo et al. in Discrete Contin. Dyn. Syst. 24(1):59–81, 2009), and models for wealth redistribution in an agent-based market (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009). Conditions on these collisional rules which guarantee both the existence and uniqueness of equilibrium profiles and their main properties are found. The characterization of these stationary states is of independent interest, since we show that they are stationary solutions of different evolution problems, both in the kinetic theory of rarefied gases (Cercignani et al. in J. Stat. Phys. 105:337–352, 2001; Villani in J. Stat. Phys. 124:781–822, 2006) and in the econophysical context (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009).

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Federico Bassetti & Giuseppe Toscani

  2. Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133, Milano, Italy

    Lucia Ladelli

Authors
  1. Federico Bassetti
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  2. Lucia Ladelli
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  3. Giuseppe Toscani
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Correspondence to Giuseppe Toscani.

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Bassetti, F., Ladelli, L. & Toscani, G. Kinetic Models with Randomly Perturbed Binary Collisions. J Stat Phys 142, 686–709 (2011). https://doi.org/10.1007/s10955-011-0136-8

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  • DOI: https://doi.org/10.1007/s10955-011-0136-8

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