# Sustaining a temperature difference

### Submission summary

 As Contributors: Alberto Garilli Arxiv Link: https://arxiv.org/abs/2005.06289v2 (pdf) Date accepted: 2020-08-20 Date submitted: 2020-08-04 13:21 Submitted by: Garilli, Alberto Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Statistical and Soft Matter 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.

Published as SciPost Phys. 9, 030 (2020)

### List of changes

- The typos have been corrected. In particular the definition of the thermodynamic forces $F_{k h}$ under eq. 26
- Some further explanations on some passages have been added and labelled in red
- We corrected some labelling in the main text and in the appendices
- We changed fig. 4 with a better picture so that it is clearer than before
- We added the condition $p_0$ = 0 among the conditions of non-oscillating endpoints
- We added a missing parenthesis in eq. 18

### Submission & Refereeing History

Resubmission 2005.06289v2 on 4 August 2020
Submission 2005.06289v1 on 14 May 2020

## Reports on this Submission

### Report

The authors gave satisfactory answers to my comments. I recommend the manuscript for publication.

• validity: top
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent

### Report

I am satisfied with the authors' changes and recommend the manuscript for publication.

(please note a small typo on p9, kind -> kind of)

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent