Abstract
We derive an expression for the minimal rate of entropy that sustains two reservoirs at different temperatures T_0T0 and T_\ellTℓ. The law displays an intuitive \ell^{-1}ℓ−1 dependency on the relative distance and a characterisic \log^2 (T_\ell/T_0)log2(Tℓ/T0) 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 ballistic behaviour, whose incompatibility with the FL posed a long-time challenge.
Highlights
Temperature differences and gradients are a common motif in the physics of systems out of equilibrium
They serve as fixed boundary conditions for studying how energy flows within a system [1,2,3] and couples to other currents, e.g. electric or matter ones [4,5,6]
We first provide a simple heuristic argument based on the Fourier Law (FL) of conduction, and rederive our results in a stochastic model of a linear chain of harmonic oscillators coupled to heat reservoirs, analyzed in the light of the First and Second laws of thermodynamics [7]
Summary
Temperature differences and gradients are a common motif in the physics of systems out of equilibrium. If we think of a refrigerator with T0 and T respectively the temperatures inside and Figure 1: Illustration of constructive vs self-consistent approaches to heat conduction in temperature gradients In the former heat flows from the hot reservoir to the cold one through the oscillators. It former draws its motivation in the derivation of the physics of open systems from that of isolated systems (micro-to-meso/scopic); its dissipative observables are the temperatures of the bulk oscillators. In the latter heat flows to and from each reservoir. Notice that for fixed T , letting T0 → 0 the EPR diverges: reaching zero temperature requires ever-increasing dissipated power, providing a self-consistent formulation of the Third Law of thermodynamics – that is not independent of the First and Second
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