# Microscopic Derivation of an Isothermal Thermodynamic Transformation

## Abstract

We obtain macroscopic isothermal thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics in contact with a heat bath. The microscopic dynamics is given by a chain of anharmonic oscillators subject to a varying tension (external force) and the contact with the heat bath is modeled by independent Langevin dynamics acting on each particle. After a diffusive space-time scaling and cross-graining, the profile of volume converges to the solution of a deterministic diffusive equation 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. Quasistatic reversible isothermal transformations are then obtained by a further time scaling. Heat is defined as the total flux of energy exchanged between the system and the heat bath. Then we prove that the relation between the limit heat, work, free energy and thermodynamic entropy agree with the first and second principle of thermodynamics.

## Keywords

Gibbs Measure Heat Bath Thermal Bath Isothermal Transformation Anharmonic Oscillator## Notes

### Acknowledgements

This work has been partially supported by the European Advanced Grant *Macroscopic Laws and Dynamical Systems* (MALADY) (ERC AdG 246953). I thank Claudio Landim for stimulating conversations on quasi static limits and for the help in the proof of Proposition 1.

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