On persistent energy currents at equilibrium in non-reciprocal systems

TL;DR

This work shows that the mean Poynting vector in global thermal equilibrium remains divergence-free even in non-reciprocal media, i.e. $∇·⟨S⟩_{eq}=0$, by deriving a general fluctuation–dissipation framework based on dyadic Green's functions and the spectral function $ψ^{EH}$. It proves this for arbitrary non-reciprocal environments and confirms the result in concrete cases: a normal-mode expansion, a planar non-reciprocal substrate, and dipolar many-body assemblies. The analysis clarifies that a persistent energy flux can exist in equilibrium (nonzero $⟨S⟩_{eq}$) without producing net heating, while it cannot be detected via out-of-equilibrium heat-transfer measurements, distinguishing it from the photon thermal Hall effect. Collectively, the results provide a thermodynamically consistent foundation for persistent currents in non-reciprocal systems and delineate the prospects and limits for experimental observation.

Abstract

We investigate the properties of the mean Poynting vector in global thermal equilibrium, which can be non-zero in non-reciprocal electromagnetic systems. Using dyadic Green's functions and the fluctuation-dissipation theorem, we provide a general proof that the mean Poynting vector is divergence-free under equilibrium conditions. Relying on this proof, we explicitly demonstrate that for systems where a normal mode expansion of the Green's function is applicable, the divergence of the equilibrium mean Poynting vector vanishes. As concrete examples, we also examine the equilibrium mean Poynting vector near a planar non-reciprocal substrate and in configurations involving an arbitrary number of dipolar non-reciprocal objects in free space. Finally, we argue that the so-called persistent heat current, while present in equilibrium, cannot be detected through out-of-equilibrium heat transfer measurements.

Figures (1)

• Figure 1: (a) Persistent heat current: The spherical nanoparticles $i = 1,2,3$ are in thermal equilibrium with the environment. A transferred power $P_{i \rightarrow j}$ from particle $i$ to particle $j$ can be defined even in equilibrium. When a magnetic field is applied one finds $P_{i \rightarrow j} \neq P_{j \rightarrow i}$, but since for the heat current in clockwise direction we have $P_{1 \rightarrow 2} = P_{2 \rightarrow 3} = P_{3 \rightarrow 1}$ and for the anti-clockwise direction $P_{1 \rightarrow 3} = P_{3 \rightarrow 2} = P_{2 \rightarrow 1}$, there is no net heat transfer ZhuFan. (b) Hall effect for thermal radiation: The temperatures $T_4 > T_2$ are fixed. In steady state, due to the exchanged heat by thermal radiation one finds $T_1 \neq T_3$ when a magnetic field is applied. Again, this is due to the fact that $P_{i \rightarrow j} \neq P_{j \rightarrow i}$ in that case PBAHall.