Scientific Interests

- Mathematical Modeling of Complex Living Systems by Generalized Kinetic Theory Methods

The kinetic theory for active particles, briefly KTAP, is a mathematical method that has been developed to model the dynamics of complex systems constituted by living entities, called active particles, interacting in space or along networks. Active particles have the ability to develop behaviour that cannot only be explained by the classical mechanics laws, and, in some cases, can generate proliferative and/or destructive processes. Moreover active particles communicate each other either directly or through media. As a consequence each entity interacts with all the other entities that are in a domain where they are able to communicate each other. In some cases, such a domain is identified by the visibility zone, while in other cases by a communication network. The microscopic state of the active particles is described not only by the classical mechanical variables, but also by a continuous or discrete variable, called activity, which expresses a biological or social function or purpose. The mathematical method describes the system under consideration by means of a continuous or discrete distribution function over the microscopic state. After modeling the microscopic interactions, one derives an integro-differential evolution equation for the distribution function or a system of (ordinary or partial) integro-differential equations, suitable for modeling the evolution of a discrete distribution function.

The mathematical KTAP frameworks are equilibrium models, because there is no dissipation of energy. Indeed in complex systems composed of a large number of identical individuals, where external effects are neglected, the random interactions among individuals will eventually move the system towards equilibrium. If, on the other hand, an external force field acts on the system, the applied field does work on the system thereby moving it away from equilibrium. Such situations necessitate the modeling of an infinite dimensional thermal reservoir that is able to continuously absorb energy in order to prevent a subsystem from heating up. The dissipation of energy into a thermal reservoir thus properly counterbalances the pumping of energy into the system by external field and enables the system to evolve into a nonequilibrium steady state, namely the statistical physical parameters describing the system on macroscopic scales are constant in time, despite the fact that the system in no longer in thermal equilibrium. A popular deterministic and time-reversible modeling of a thermal reservoir is known as the deterministic thermostat. The use of deterministic thermostats consists of introducing a damping term into the equations and amounts to projecting the force field onto the tangent plane to the energy surface. The damping term is adjusted so as to keep the energy constant (Gaussian thermostat). The Gaussian thermostat is based on the Gauss’ principle of least constraint, which states that a system subject to constraints will follow trajectories which, in the least-square sense, differ minimally from their unconstrained Newtonian counterparts. A Gaussian thermostat is introduced into KTAP frameworks in order to keep constant the density and the energy of the system. This new framework, called Thermostatted KTAP, led to a new class of dynamical systems contemplating stochastic interactions, expressed in the form of systems of partial differential equations or, in particular cases, of ordinary differential equations.

Qualitative and quantitative analysis of specific models is the main interest in this field. At present, the KTAP methods and the Thermostatted KTAP methods have been profitably applied to model complex living systems such as biological systems including the immune system, traffic flow, crowds dynamics, social and economic systems.

- Dynamical Systems: Billiard Theory, Chaos, and Anomalous Transport in Microporous Media

The theory of billiards subjected to external forces and to deterministic thermostats, i.e., nonequilibrium billiard models of transport of matter, is much less developed than the theory of equilibrium billiards, i.e., billiards whose particles move under the action of no forces, except for the collisions with scatterers. Indeed, the phenomenology of nonequilibrium billiards has been little investigated. In particular, nondispersing nonequilibrium billiards have been studied only in a few works, and it is not known whether nonequilibrium billiards with flat obstacles are chaotic. Collisions with the obstacles do not defocus trajectories, while the external fields and the corresponding dissipative forces have a focusing effect. Exponential separation of phase space trajectories, if any, may then only be produced by the singularities of the dynamics, i.e., by a set of zero Liouville measure.

- Numerical Methods for Kinetic Equations

A new approach was proposed by Motta and Wick in the MWF method and a new formulation, oriented to implementation purpose, was suggested by Motta. The basic idea of the method consists in rewriting the collision kernel as the divergence of a flux and formally transform the kinetic equation with collision into a system of a collisionless kinetic equation (Vlasov equation) and the divergence equation for the flux with appropriate boundary conditions. At each time step the flux is computed at a finite set of particle points which are the quadrature points. Then the collision induced velocity vector is computed and added to the Vlasov equation which is solved numerically with an upwind scheme. For this reason the method is referred to as the MWF method (Motta-Wick Flux method). When the density function is approximated with a finite set of points with equal weight (moving particles) using Dirac functions, and these points are chosen as quadrature nodes, then the Vlasov equation can be used to compute the particle motion and, consequently, the evolution of the density distribution. In this form, the method belongs to the class of the meshfree methods which have been widely used mostly in fluid dynamics and solid mechanics. The method was tested in different simple scenarios like the case of semiconductors.