On Rigorous Derivation of the Enskog Kinetic Equation

We develop a rigorous formalism for the description of the kinetic evolution of infinitely many hard spheres. On the basis of the kinetic cluster expansions of cumulants of groups of operators of finitely many hard spheres the nonlinear kinetic Enskog equation and its generalizations are justified. It is established that for initial states which are specified in terms of one-particle distribution functions the description of the evolution by the Cauchy problem of the BBGKY hierarchy and by the Cauchy problem of the generalized Enskog kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.Comment: 28 page

Discrete kinetic and stochastic game theory for vehicular traffic: Modeling and mathematical problems

n this thesis we are concerned with the mathematical modeling of vehicular traffic atthe kinetic scale. In more detail, starting from the general structures proposed by Arlottiet al. and by Bellomo, we develop a discrete kinetic framework in which thevelocity of the vehicles is not regarded as a continuous variable but can take a finite number of values only. Discrete kinetic models have historically been conceived in connection with the celebrated Boltzmann equation, primarily as mathematical tools to reduce the analytical complexity of the latter (see e.g., Bellomo and Gatignol, Gatignol): The Boltzmann’s integro-differential equation is converted into a set of partial differential equations in time and space, which share with the former some good mathematical properties being at the same time easier to deal with. In the present context, however, the discretization of the velocity plays a specific role in modeling the system rather than being simply a mathematical simplification, because it allows one to relax the continuum hypothesis for the velocity variable and to include, at least partially, the strongly granular nature of the flow of cars in the kinetic theory of vehicular traffic. The discrete velocity framework also gives rise to an interesting structure of the interaction terms of the kinetic equations, which are inspired by the stochastic game theory

Mathematical topics in nonlinear kinetic theory II : the Enskog equation

xii, 207 p. : ill. ; 23 cm

Mathematical topics in nonlinear kinetic theory II : the Enskog equation

xii, 207 p. : ill. ; 23 cm