# Mathematical Physics

The group members are : Dario Camiola, Paolo Falsaperla, Andrea Giacobbe, Orazio Muscato, Giovanni Nastasi, Vittorio Romano, Rita Tracinà, Massimo Trovato. All group members investigate mathematical models in applied sciences. Some of the investigation arguments are:

**Mathematical models for transport of charges in low-dimensional structures:** it is dealt with models and numerical simulations of transport of charges in electronic devices of nanometric dimensions such as MOSFET, double-gate and nanofili. The approach is based on the introduction of a subbands structure obtained resolving a Schroedinger-Poisson system. Under appropriate conditions on the wave function, coupled with a system of semiclassical Boltzmann equations in the non-quantized directions. From such system it is possible to obtain *physics-based* hydrodynamical models that use the Maximum Entropy Principle. Such models keep under consideration the heating of the device, and include phonons trasport. Boltzmann equation is not reasonable if the device contains highly variable potentials that can induce tunneling effects. In such case one has to use other models, that include Wigner transport equations, which is the natural generalisation of Boltzmann equations. The numerical solution of Wigner equations is non-local, and can be complicate to obtain. Monte Carlo simulations based on “signed –particle” schemes have been recently introduced from the point of view of accuracy calculus.

Another argument is transport of charges in graphene, an innovative material which will probably be the base of new generation electronic devices. There exist efficient solvers, that use mathematical models with different degrees of complexity for the simulation of semiconductors built with classical materials, such as silicium or GaAs, the same cannot be found for graphene. We are contributing to such lack of solvers formulating appropriate mathematical models for mesoscopic and macroscopic transport – drift-diffusion, energy-transport and hydrodynamical -- starting at a kinetically level from Boltzmann equations and applying moments method, then closed using the method of maximal entropy. We investigate appropriate numerical schemes for such models that will be validated with Monte Carlo simulations and with direct simulation of Boltzmann equations by means of finite-difference schemes and with discontinuous finite elements of Garerkin type. (Dario Camiola, Orazio Muscato, Giovanni Nastasi, Vittorio Romano)

**Construction of kinetic and hydrodynamic models for the description of transport phenomena in gas-dynamics and in solid state physics. ****Stability-instability problems in fluid dynamics: **One of the objectives that we wanted to achieve with the present research activity is to provide an advanced theoretical "framework" on the application of the Maximum Entropy Principle and modern Extended Thermodynamics (ET), for classical and quantum physical systems of wide interest for the scientific community. In this framework, classical systems (local theories) have been studied, both in gas dynamics and in solid state physics, satisfying Boltzman statistics, Bose and Fermi statistics and more generally "Fractional exclusion" statistics. In particular, hyperbolic, closed and self-consistent hydrodynamic models (HD) have been built, using the ET with the aim of studying the ballistic and/or dissipative processes of the "hot-carriers", determining the Response Functions, the Correlation's functions, the Differential Mobilities associated with charge transport in semiconductors and latest generation 2D materials such as graphene (https://doi.org/10.1393/ncr/i2012-10075-8 ; https://doi.org/10.1063/1.5088809 ).

A second class of problems, which has been addressed in the context of the present research activity, is in fact a modern thematic reference point for the development of innovative models, both in the framework of quantum kinetic theory (Wignerian formulation of statistical quantum mechanics) than in the framework of a non-local theory of the quantum hydrodynamic systems (quantum formulation of ET).In recent years, using the reduced density matrix formalism, a non-local formulation of the Quantum Maximum Entropy Principle (QMEP) has been proposed for the determination of closed hydrodynamic models for a system of identical particles subject to different statistics (Fermi, Bose, and Fractional statistics) and for different degeneracy levels ( https://doi.org/10.1103/PhysRevE.84.061147 ; https://doi.org/10.1103/PhysRevLett.110.020404 ).

Also in this case, besides providing an advanced theoretical "framework" on the QMEP, the aim is the application of these new theories to the study of: hot carriers transport in nano-scale electronic devices; elementary excitations obeying fractional statistics; advanced materials of next generation (graphene), with the relevant advantage of a reduced computational effort compared to the usual kinetic simulation methods. Finally, by using a non-local mean-field theory (in second quantization), we are trying to use the QMEP procedure to include not only different types of statistics, but also the spin variables, in order to formulate a systematic theory also for the spintronic systems.

__In the context of stability-instability problems__: (i) A method was sought for the construction of optimal Lyapunov functions using a *canonical reduction method*. This method has been applied to systems of ordinary differential equations, to systems of reaction-diffusion equations and to systems of equations in fluid dynamics. (ii) A study of the linear stability of the steady states achieved by *hot carriers* has been done, for various materials, using the small-signal analysis. (M. Trovato, P. Falsaperla)

**Mathematical Epidemiology:** we investigate the asymptotic of ODE and PDE in which the kinetic of the equations is based on the way diseases of behaviours do propagate in populations. Some applications are made also to social phenomena such as drug abuse. (Paolo Falsaperla, Andrea Giacobbe)

**Qualitative methods of mechanics in fluids and porous media:**we consider a base flow, solutions to Navier-Stokes equations or their generalizations to non-isotermal fluids and/or magneto-fluids, and we investigate the evolution of the generic perturbation as the initial data are varied. In particular we use spectral methods (by means of theoretical or numerical methods) and we determine the critical thresholds of instability. We also use rigorous Lyapunov methods to investigate sufficient stability conditions and to determine the relative critical values of parameters. The application to porous media in inclined layers are particularly interesting for landslides. (Paolo Falsaperla, Andrea Giacobbe)

##### Contacts

prof. Andrea Giacobbe