This volume aims to provide an overview of some recent developments of mathematical kinetic theory focused on its application in modelling complex systems in various ?elds of applied sciences. Mathematical kinetic theory is essentially based on the Boltzmann eq- tion, which describes the evolution, possibly far from equilibrium, of a class of particles modelled as point masses. The equation de?nes the evolution in time and space of the distribution function over the possible microscopic states of the test particle, classically position and velocity. The test particle is subject to pair collisions with the ?eld particles. The interested reader can ?nd in the book, Theory and Application of the Boltzmann Equation, by C. Cercignani, R. Illner, and M. Pulvirenti, Springer, Heidelberg, 1993, all necessary knowledge of the physics and mathematical topics related to this celebrated model of non-equilibrium statistical mech- ics. Another important model of mathematical kinetic theory is the Vlasov equation, where interactions between particles are not speci?cally collisions, but mean ?eld actions of the ?eld particles over the test particle. The model de?nes again an evolution equation for the one-particle distribution function over the microscopic state of the test particle. The two models brie?y mentioned above can be regarded as the fun- mental models of mathematical kinetic theory and the essential background o?ered from the kinetic theory for classical particles towards the modelling of large systems of several particles undergoing non classical interactions.
Cuprins
Analytic Aspects of the Boltzmann Equation.- Rigorous results for conservation equations and trend to equilibrium in space-inhomogeneous kinetic theory.- Results on optimal rate of convergence to equilibrium for spatially homogeneous Maxwellian gases.- Nonresonant velocity averaging and the Vlasov-Maxwell system.- Modeling Applications, Inverse and Computational Problems in Quantum Kinetic Theory.- Multiband quantum transport models for semiconductor devices.- Optimization models for semiconductor dopant profiling.- Inverse problems for semiconductors: models and methods.- Deterministic kinetic solvers for charged particle transport in semiconductor devices.- Miscellaneous Applications in Physics and Natural Sciences.- Methods and tools of mathematical kinetic theory towards modelling complex biological systems.- Kinetic modelling of late stages of phase separation.- Ground states and dynamics of rotating Bose-Einstein condensates.- Two inverse problems in photon transport theory: evaluation of a time-dependent source and of a time-dependent cross section.
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