Kinetic theoretical foundation of Lorentzian statistical mechanics

A new kinetic theory Boltzmann-like collision term including correlations is proposed. In equilibrium it yields the one-particle distribution function in the form of a generalised-Lorentzian resembling but not being identical with the so-called $\kappa$ distribution frequently found in collisionless turbulent systems like space plasmas. We show that this distribution function satisfies a generalised $H$-theorem, yields an expression for the entropy that is concave. Thus, the distribution is a `true' thermodynamic equilibrium distribution, presumably valid for turbulent systems. In equilibrium it is possible to construct the fundamental thermodynamic quantities. This is done for an ideal gas only. The new kinetic equation can form the basis for obtaining a set of hydrodynamic conservation laws and construction of a generalised transport theory for strongly correlated states of a system.