Irreversibility and heat transfer in closed Hamiltonian systems

Understanding and quantitatively characterizing irreversibility and related phenomena is a central task in the study of noise-driven non-equilibrium systems. In this talk, we consider a closed Hamiltonian dynamical system consisting of two heat baths, each with many degrees of freedom, coupled to one another via a small number of macroscopic degrees of freedom. Each heat bath is modeled as a network of linearly coupled oscillators [1] initiated so that the effective temperatures of the two baths are different from one another at time t = 0. The macroscopic degrees of freedom are modeled as a mass-spring system consisting of two particles each moving in a bath at different temperature; the particles are coupled to their respective heat bath oscillator networks, and also coupled to each other by a macroscopic spring. An initial temperature difference between the baths drives heat transfer from hot to cold. Strictly speaking, statistics on the full phase space (including all the microscopic degrees of freedom) cannot be stationary. Nonetheless, we find that quasi-stationary statistics representing the steady transfer of heat is achieved asymptotically for sufficiently large baths. Typically, the projected statistics of the full Hamiltonian system settles into this quasi-stationary state for a very long time. Conversely, the lower bound time scale for quasi-stationary behavior is set by transients to decay from arbitrary initial conditions, a time scale that is many of orders of magnitude smaller. Quasi-stationary behavior is also characterized by computing stochastic line integrals [2] corresponding, e.g., to stochastic area or heat transfer. These integrals can be performed using either the macroscopic degrees of freedom or purely microscopic degrees of freedoms associated with either of the heat bath oscillator networks.

[1] R. Zwanzig, Nonequilibrium statistical mechanics. (Oxford University Press, 2001)

[2] S. Teitsworth and J. C. Neu, Phys. Rev. E 106, 024124 (2022).