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The usual statistical mechanics treats macroscopic objects containing an enormous number N of particles, at least O(10^20); furthermore, the classical thermodynamics refers to adiabatic (i.e. infinitely slow) processes. Basically in the standard statistical mechanics and thermodynamics two asymptotic limits are present: large N and very slow changes of parameters. The challenge we face nowadays is going beyond these limits, extending thermodynamics and statistical mechanics to new models and applications. In particular, we focus on a molecular engine constituted by a gas of N~100 molecules enclosed between a massive piston and a thermostat. The force acting on the piston and the temperature of the thermostat are cyclically changed with a finite period, and different regimes are observed at different values of the period. Based upon kinetic theory, we develop a three-variables coarse-grained Langevin model where the piston’s position and velocity are linearly coupled together with the internal energy of the gas. The model reproduces many of the system’s features, such as the efficiency at maximum power and the fluctuations of thermodynamic quantities as work and heat.
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