MICROSCOPIC DERIVATION OF AN ADIABATIC THERMODYNAMIC TRANSFORMATION

Abstract. We obtain macroscopic adiabatic thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to ran-dom collisions with the environment. The microscopic dynamics is given by a chain of oscillators subject to a varying tension (external force) and to col-lisions with external independent particles of “infinite mass”. The effect of each collision is to change the sign of the velocity without changing the mod-ulus. This way the energy is conserved by the resulting dynamics. After a diffusive space-time scaling and cross-graining, the profiles of volume and en-ergy converge to the solution of a deterministic diffusive system of equations with boundary conditions given by the applied tension. This defines an irre-versible thermodynamic transformation from an initial equilibrium to a new equilibrium given by the final tension applied. Quasi-static reversible adiabatic transformations are then obtained by a further time scaling. Then we prove that the relations between the limit work, internal energy and thermodynamic entropy agree with the first and second principle of thermodynamics. 1