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.
- 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
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.
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.
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.
- 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.
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.
- 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.