A Boltzmann-like equation for choice formation

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March 2009, 2(1): 135-149. doi: 10.3934/krm.2009.2.135

A Boltzmann-like equation for choice formation

Valeriano Comincioli ^1, , Lucia Della Croce ^1, and Giuseppe Toscani ^2,

1.	Department of Mathematics at the University of Pavia, via Ferrata 1, 27100 Pavia, Italy, Italy
2.	Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia

Received September 2008 Revised October 2008 Published January 2009

Abstract
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We describe here a possible approach to the formation of choice in a society by methods borrowed from the kinetic theory of rarefied gases. It is shown that the evolution of the continuous density of opinions obeys a linear Boltzmann equation where the background density represents the fixed distribution of possible choices. The binary interactions between individuals are in general non-local, and take into account both the compromise propensity and the self-thinking. In particular regimes, the linear Boltzmann equation is well described by a Fokker-Planck type equation, for which in some cases the steady states (distribution of choices) can be obtained in analytical form. This Fokker-Planck type equation generalizes analogous one obtained by mean field approximation of the voter model in [27]. Numerical examples illustrate the influence of different model parameters in the description both of the shape of the distribution of choices, and in its mean value.

Keywords: linear Boltzmann equation, Fokker-Planck equation, Sociodynamics, opinion formation..

Mathematics Subject Classification: Primary: 35B40, 91C20; Secondary: 82B21, 60K3.

Citation: Valeriano Comincioli, Lucia Della Croce, Giuseppe Toscani. A Boltzmann-like equation for choice formation. Kinetic & Related Models, 2009, 2 (1) : 135-149. doi: 10.3934/krm.2009.2.135

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