# Sustaining a temperature difference

### Submission summary

 As Contributors: Alberto Garilli Arxiv Link: https://arxiv.org/abs/2005.06289v1 Date submitted: 2020-05-14 Submitted by: Garilli, Alberto Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approaches: Theoretical, Computational

### Abstract

We derive an expression for the minimal rate of entropy that sustains two reservoirs at different temperatures $T_0$ and $T_\ell$. The law displays an intuitive $\ell^{-1}$ dependency on the relative distance and a characterisic $\log^2 (T_\ell/T_0)$ dependency on the boundary temperatures. First we give a back-of-envelope argument based on the Fourier Law (FL) of conduction, showing that the least-dissipation profile is exponential. Then we revisit a model of a chain of oscillators, each coupled to a heat reservoir. In the limit of large damping we reobtain the exponential and squared-log behaviors, providing a self-consistent derivation of the FL. For small damping "equipartition frustration" leads to a well-known balistic behaviour, whose incompatibility with the FL posed a long-time challenge.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2005.06289v1 on 14 May 2020