From Particle Systems to Partial Differential Equations

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 thermo-dynamic 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.

Microscopic derivation of an adiabatic thermodynamic transformation

“…The corresponding solution is denote by (r ε , u ε ). Then Proposition 3.1 of [10] can be applied and it follows that…”

“…This deterministic evolution of the profiles describes an irreversible adiabatic trasformation, and, as shown in section 4, it increases the thermodynamic entropy of the system. The reversible or quasi-static transformations are then obtained by a further rescaling of time, see subsection 4.2, similar as proposed in [1,2,10]. It should be possible to obtain these quasi-static transformation in a direct limit at a larger (subdiffusive) time scale, this will be object of further investigation.…”

“…Isothermal transformations in this model have been deduced in [10] in the nonlinear case, where the heat bath is modelled by Langevin thermostats. In this evolution only the volume evolves macroscopically.…”

“…In this evolution only the volume evolves macroscopically. In [10] these heat baths act on the bulk of the chain, at every point. If we want to make them act only at the boundaries of the chain, then we should obtain the same macroscopic equations as in the present article, but with boundary conditions corresponding to the thermostat temperature (this will be object of further investigation).…”