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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

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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