Statistical dynamics of driven systems of confined interacting particles in the overdamped-motion regime

Systems consisting of confined, interacting particles doing overdamped motion admit an effective description in terms of nonlinear Fokker–Planck equations. The behavior of these systems is closely related to the $Sq$ power-law entropies and can be interpreted in terms of the $Sq$-based thermostatistics. The connection between overdamped systems and the $Sq$ measures provides valuable insights on diverse physical problems, such as the dynamics of interacting vortices in type-II superconductors. The $Sq$-thermostatistical approach to the study of many-body systems described by nonlinear Fokker–Planck equations has been intensively explored in recent years, but most of these efforts were restricted to systems affected by time-independent external potentials. Here, we extend this treatment to systems evolving under time-dependent external forces. We establish a lower bound on the work done by these forces when they drive the system during a transformation. The bound is expressed in terms of a free energy based on the $Sq$ entropy and is satisfied even if the driving forces are not derivable from a potential function. It constitutes a generalization, for systems governed by nonlinear Fokker–Planck equations involving general time-dependent external forces, of the $H$-theorem satisfied by these systems when the external forces arise from a time-independent potential.