The non-linear phenomena have an important role in many physical systems and they have drawn an increasing interest of theoretical physicists. The study of the corresponding
mathematical models requires both the results of dynamical systems and stocastics processes ,and extensive numerical simulations. Moreover the recent developing of Complex Systems Physics has proposed new models whose dynamical and statistical properties have to be studied.
The aim of the BO61 INFN project is to create a bridge among Dynamical Systems Theory, complex systems and Physics. The INFN sections of Bologna, Pisa, Perugia and Cosenza
that partecipate to this project, have expertise in dynamical systems, stochastic processes ans partial differential equations. The results are applied to different physical problems from high intensity beams to solid state physics and biology. The reserach project allows the exchange of different methodological approaches and mathematical tecniques among the sections. Moreover the comparison and the validation of the models with experimental activities is one of the key point of the BO61
project.
In particular we study:

a) the dynamics and the statistical properties of many particles systems with a longe range interaction in presence of external fields,
b) the dynamical properties of nonlinear systems in presence of generalized symmetries,
c) the diffusion and trasport phenomena in stochastic nonlinear dynamical systems,
d) the mathematical models of semiconductors coupled with electric circuits.

a) We have considered the mechanism of transition towards a thermodynamics equilibrium (Maxwell-Boltzmann distribution)
of an ensamble of a long range interacting particles in presence an external force and periodic boundary conditions. The role of collisions in the relaxation processes have been studied and in the case of Coulomb interaction we have analyze the satbility properties of Poisson-Vlasov self consistent equilibria. We have also generalized the model to non-Newtonian interations (like visual intercations) to simulate the properties of an automata gas (intelligent intercating particles).Extensive numerical simulations will be performed to justify the analytical results. This research activity will be prosecuted the next year.

b) Analytical methods essentially based on suitable extensions of the notion of Lie symmetry have been used to study nonlinear differential equations relevant in physical problems (e.g., dynamical systems, evolution equations, plasma physics).


c)We consider stochastic dymanical models characterized either by bistable states or by rectifying transport properties (ratchets).
We want to study transitions between states in bistable systems with special attention to delayed noise, e.g. when the same noise is acting on particles and potential barriers at different times. Transport properties of rocked systems are investigated when two waves of very differentfrequencies act on a particle in a periodic force, giving rise to an effective potential.

d) New mathematical models are needed by Microelectronic industry since new technologies and devices are being developed in the nanoscale. Under this circumstance, the description of thermal and electric properties of semiconductor devices and of the coupling of these devices with electric circuits is of great importance.
The main results, which we have obtained in this field, are:

1. a variational formulation of the Boltzmann transport equation for semiconductors in the steady case
with an explicit expression for the mobility in Silicon;
2. a quantum multiband hyrodynamical model, derived on the basis of Kane's quantum model for two-band charged particles;
3. a complete hydrodynamical model for electrons and holes in Silicon, derived on the basis of
the Maximum Entropy Principle.
Future objectives are:
Comparison of numerical simulations of the complete hydrodynamical model with simulations of coupled
Boltzmann equations for electrons and holes.
The derivation of a completely non-linear hydrodynamical model for semiconductors based on the Maximum Entropy
Principle.
The derivation of a hydrodynamical model for semiconductors with an arbitrarily great number of variables.
The study of the relations between Naimark's theorem and the geometrical characterization of commutative
positive operator valued measures.
The extension of the quantum Boltzmann equation derived by Carruthers and Zachariasen to the multiband case.

The models considered are relevant for high intensity beamsPhysics, weak chaotic physical systems,
Biophysics, solid state Physics and microelectronics.