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arXiv · 2005.06289

Sustaining a temperature difference

Matteo Polettini·Alberto Garilli
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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.

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Matteo Polettini, Alberto Garilli. 2020-05-13. Sustaining a temperature difference. https://doi.org/10.21468/scipostphys.9.3.030

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