From dynamics to thermodynamics: Linear response and statistical mechanics

We illustrate a dynamic approach to statistical mechanics based on a procedure which reverses the standard approach to the Fokker-Planck equation (FPE). Rather than using statistical mechanics, derived from thermodynamics along the lines pioneered by Boltzmann [L. Boltzmann, Wiss. Ber. 58, 517 (1868)], we derive a FPE for a set of variables of interest interacting with a booster, i.e., a dynamical system mimicking the action of an ideal thermostat with no need of ad hoc statistical assumptions. We derive a mechanical expression for the temperature of the system of interest, which is proven to be a generalization of the one proposed by Boltzmann; in the case of boosters with a limited number of degrees of freedom, it is shown that our definition depends also on dynamic properties, which are not accounted for by the ``standard'' approach.