Non-Clausius heat transfer: The method of the nonstationary Langevin equation

TL;DR

This paper explores scenarios where the classical Clausius statement of the second law of thermodynamics does not hold, using a non-stationary Langevin equation to analyze heat transfer in non-equilibrium and non-ergodic systems.

Contribution

It introduces a novel analysis of heat transfer anomalies in non-ergodic and non-equilibrium systems using a non-stationary Langevin equation framework.

Findings

Heat can flow from cold to hot in short time intervals in ergodic systems.

Non-ergodic systems can violate the Clausius statement at all time scales due to localized vibrational modes.

The non-stationary Langevin equation captures anomalous heat transfer behaviors.

Abstract

Compared to other formulations of the second law of thermodynamics, the Clausius statement that heat does not spontaneously flow from cold to hot concerns a system in non-equilibrium states, and in that respect is more ambitious but also more ambiguous. We discuss two scenarios when the Clausius statement in its plain form does not hold. First, for ergodic systems, the energy transfer may be consistent with the statement on a coarse-grained time scale, but be anomalously directed during time intervals shorter than the thermalization time. In particular, when an initially colder system is brought in contact to a hotter bath, the internal energy of the former increases with time in a long run but not monotonically. Second, the heat transfer may not respect the Clausius statement on any time-scale in non-ergodic systems due to the formation of localized vibrational modes. We illustrate the two scenarios with a familiar model of an isotope atom attached to a semi-infinite harmonic atomic chain. Technically, the discussion is based on a Langevin equation for the isotope, using the initial condition when the isotope and chain are initially prepared in uncorrelated canonical states under the constraint that the boundary atom between the isotope and chain is initially fixed and later released. In such setting, the noise in the Langevin equation is non-stationary, and the fluctuation-dissipation relation has a non-standard form.