On the interaction domain reconstruction in the weighted thermostatted kinetic framework

This paper is devoted to the modeling of out-of-equilibrium complex living systems by means of the weighted thermostatted kinetic theory framework. The weighted mathematical framework is based on the definition and interaction of different functional subsystems each of them able to express a specific strategy. The time evolution of the functional subsystems is described by nonlinear partial integro-differential equations with quadratic type nonlinearity coupled with a thermostat in order to ensure the reaching of nonequilibrium stationary states. In particular the weighted framework is based on the definition of the weighted interactions which are modeled by introducing an interaction domain. This paper focuses on the interaction domain reconstruction by employing the methods of the inverse theory and the information theory. Specifically the solution of different inverse problems based on the knowledge of global weighted measurements related to the system is investigated. An optimization problem based on the maximum entropy principle of Shannon is analyzed and the existence of the interaction domain is proven by employing fixed-point arguments. Applications to living systems, such as social systems and crowd dynamics, and further research directions are outlined in the last section of the paper.