Micro-canonical thermodynamics: Why does heat flow from hot to cold

TL;DR

This paper uses a central limit approximation to model energy transfer in non-equilibrium thermodynamics, explaining why heat flows from hot to cold and re-deriving the second law violation in thermostatted systems without relying on time-reversal symmetry.

Contribution

It introduces a stochastic model based on central limit approximation for energy transfer, applicable to both Hamiltonian and non-Hamiltonian systems, providing new insights into heat flow and second law violations.

Findings

Heat flows from hot to cold on average.

Fluctuations in energy are linearly related to heat transfer.

Re-derivation of second law violation in thermostatted systems.

Abstract

We show how to use a central limit approximation for additive co-cycles to describe non-equilibrium and far from equilibrium thermodynamic behavior. We consider first two weakly coupled Hamiltonian dynamical systems initially at different micro-canonical temperatures. We describe a stochastic model where the energy-transfer between the two systems is considered as a random variable satisfying a central limit approximation. We show that fluctuations in energy observables are linearly related to the heat-transfer (dissipation). As a result, on average, heat flows from hot to cold. We also consider the far from equilibrium situation of a non-Hamiltonian thermostatted system as in Evans et al. {\em Phys.\ Rev.\ Lett.} {\bf 71}, 2401 (1993). Applying the same central limit approximation we re-derive their relation for the violation of the 2nd law of thermodynamics. We note that time-reversal symmetry is not used in our derivation.