Deriving thermodynamics from linear dissipativity theory

Complete and rigorous foundations for basic thermodynamic laws from the statistical description of microscopic systems has been a long-standing goal for mathematicians and physicists alike since Boltzmann. In this paper, we show how Willems´s dissipativity theory provides a convenient framework to study a physical system at both microscopic and macroscopic level, and suggests a natural storage function different from the usual free energy to derive the theorem of energy equipartition of energy for linear systems. In this setup, we introduce a simple and general definition for temperature defined also out of equilibrium which allows to test the limits of validity of Fourier´s law describing the transfer of heat from hot systems to cold systems. In particular under time-scale separation conditions, we derive the Maxwell-Cattaneo law, allowing for instantaneous flow of energy from cold to hot systems, which should be considered instead of Fourier´s law for a proper description of energy exchanges between interconnected linear systems.