Scientific Interests

• Mathematical Modeling of Complex Living Systems by Generalized Kinetic Theory Methods
Complex phenomena arising in living systems emerge from interactions occurring in nonlinear fashion among many elements of the system that as a whole exhibits one or more properties (emerging behaviors) not obvious from the properties of the individual parts. Mathematicians and physicists have long hoped that these collective behaviors could be described using the ideas and methods of statistical mechanics. Indeed different approaches inspired to equilibrium or nonequilibrium statistical mechanics have been developed, adapted and employed in an attempt to describe collective behaviors and macroscopic features as the result of individual interactions. Generalized kinetic theory methods have been performed in order to model complex living systems.
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
Interest in understanding transport phenomena in microscopic-scale systems is increasing. On the one hand, new theoretical tools have been developed to allow the investigation of anomalous transport and of the features that distinguish the macroscopic behavior from the microscopicone. On the other hand, these studies are of interest as nanometer-scale technologies continue to grow apace such as in the controlled delivery of drugs. Transport processes on such scales pose many intriguing questions, which are interesting both from a fundamental point of view and for a number of applications. In particular, transport processes with order unity or higher Knudsen numbers the ratio of mean free paths to characteristic pore size do not satisfy the conditions required for a traditional hydrodynamical approach: a distinct theoretical framework must be developed to understand the key results. While much progress has been made, there are still many aspects of confined transport processes where the characteristic container size is of the same order of magnitude as the particle size that remain to be understood. The usual realm of application for thermodynamics and hydrodynamics is in fluids where the size of molecules  is much smaller than the mean free path, which in turn is much smaller than the characteristic size of the container. The behavior of such systems can be determined almost completely from the nature of particle interactions, the container generally provides boundary conditions, but little more. In confined systems, however, particle-wall interactions contribute significantly to fluid properties, and must be explicitly incorporated into any successful theory. For dense confined fluids, hydrodynamics can been extended to explicitly include particle-wall contributions, with varying degrees of rigor. At lower densities, the hydrodynamic approach fails altogether because transport is dominated by particle-wall interactions. Models of such transport systems were pioneered by Knudsen, and applied to transport of rarefied gases.
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
Both analytical and numerical methods are frequently proposed to approximate the solutions of the Boltzmann equation, and in general kinetic equations. In the past three decades, the particle methods represent a class of numerical methods widely used for Vlasov equation. Their application in treating the Boltzmann collision integral has been done mainly with two different approaches: stochastic and deterministic. Classical stochastic particle methods are based on the well-known Monte Carlo method and simulate collision probabilities using stochastic events. New approaches in this direction have been proposed in recent years. Deterministic particle methods use particles as quadrature nodes for computing an approximate solution of the collision integral. In classical deterministic particle methods, particles are kept fixed in the velocity space and the evolution is reflected in changing their weights in time. This is a well-established technique used in many applications.
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.