Boltzmann-type models with uncertain binary interactions

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker–Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods.

Keywords

uncertainty quantification, deterministic and stochastic kinetic equations, Boltzmann and Fokker–Planck equations, structure preserving schemes

2010 Mathematics Subject Classification

35Q20, 35Q70, 35Q84

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A.T. is member of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM (Istituto Nazionale di Alta Matematica), Italy. M.Z. is member of GNCS (Gruppo Nazionale per il Calcolo Scientifico) of INdAM, Italy.

The research that led to the present paper was partially supported by the research grant Numerical methods for uncertainty quantification in hyperbolic and kinetic equations of GNCS-INdAM.

M.Z. acknowledges support from “Compagnia di San Paolo” (Torino, Italy).

Received 7 September 2017

Accepted 17 March 2018

Published 31 October 2018