• Previous Article
    On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation
  • KRM Home
  • This Issue
  • Next Article
    Stability of the travelling wave in a 2D weakly nonlinear Stefan problem
March  2009, 2(1): 135-149. doi: 10.3934/krm.2009.2.135

A Boltzmann-like equation for choice formation


Department of Mathematics at the University of Pavia, via Ferrata 1, 27100 Pavia, Italy, Italy


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

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028


Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017


Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016


Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485


José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401


Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120


Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056


Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008


Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028


Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250


Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic and Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169


Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65


Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079


Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic and Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028


Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics and Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013


Anton Arnold, Beatrice Signorello. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022009


Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845


Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135


Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic and Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044


Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

2020 Impact Factor: 1.432


  • PDF downloads (121)
  • HTML views (0)
  • Cited by (29)

[Back to Top]