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Sustaining a temperature difference
by Matteo Polettini, Alberto Garilli
This Submission thread is now published as
Submission summary
| Ontological classification |
|---|
| 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.
List of changes
- We added some additional references [15,23,29]
- 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
