Microscopic derivation of an adiabatic thermodynamic transformation

TL;DR

This paper derives macroscopic adiabatic thermodynamic transformations from microscopic Hamiltonian dynamics with random collisions, showing convergence to diffusive equations and consistency with thermodynamic principles.

Contribution

It provides a microscopic derivation of adiabatic transformations using space-time scaling and collision dynamics, linking microscopic models to macroscopic thermodynamics.

Findings

Profiles of volume and energy converge to diffusive equations

Relations between work, energy, and entropy match thermodynamic principles

Reversible adiabatic transformations obtained through additional time scaling

Abstract

We obtain macroscopic adiabatic thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment. The microscopic dynamics is given by a chain of oscillators subject to a varying tension (external force) and to collisions with external independent particles of "infinite mass". The effect of each collision is to change the sign of the velocity without changing the modulus. This way the energy is conserved by the resulting dynamics. After a diffusive space-time scaling and cross-graining, the profiles of volume and energy converge to the solution of a deterministic diffusive system of equations with boundary conditions given by the applied tension. This defines an irreversible 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.