Carlo Bianca

This monograph is devoted to the recently developed thermostatted kinetic theory and its application to the modeling of complex systems in a variety of fields. In the first part, the frameworks of the theory, which are obtained by coupling the kinetic theory for active particles with the thermostats coming from non-equilibrium statistical mechanics (Gaussian isokinetic thermostat), are presented. Among the theoretical results discussed are existence and uniqueness of the solution of the different mathematical frameworks, existence and uniqueness of the nonequilibrium stationary state, convergence to the nonequilibrium stationary state, derivation of macroscopic equations by asymptotic limits, and the construction of solutions to inverse problems based on the thermostatted kinetic theory. The second part deals with the derivation of mathematical models for nonequilibrium complex systems. Among those described within this new framework are immune systems and competition with tumors, keloid formation and therapy, dynamics of train crowds, modeling of a smart grid composed by different energy sources, and financial markets.

This monograph has the ambitious aim of developing a mathematical theory of complex biological systems with special attention to the phenomena of ageing, degeneration and repair of biological tissues under individual self-repair actions that may have good potential in medical therapy. The approach to mathematically modeling biological systems needs to tackle the additional difficulties generated by the peculiarities of living matter. These include the lack of invariance principles, abilities to express strategies for individual fitness, heterogeneous behaviors, competition up to proliferative and/or destructive actions, mutations, learning ability, evolution and many others. Applied mathematicians in the field of living systems, especially biological systems, will appreciate the special class of integro-differential equations offered here for modeling at the molecular, cellular and tissue scales. A unique perspective is also presented with a number of case studies in biological modeling.