Statistical properties of the energy flux between two NESS thermostats

We address the question of transport of heat, in out-of-equilibrium systems. The experimental set-up consists in two coupled granular gas Non-Equilibrium Steady State (NESS) heat baths, in which Brownian-like rotors are imbedded. These rotors are electro-mechanically coupled thanks to DC micro-motors, through a resistor $R$, such that energy flows between them. The average flux depends linearly in the difference of the baths' temperature. Varying $R$ allows to extrapolate in the non-dissipative coupling limit ($R\rightarrow0$). We show that, in this limit, the heat flux obeys the Fluctuation Theorem, in a form proposed by Jarzynski and W\'ojcik in $2004$ for the fluctuations of the flux between finite size equilibrium heat baths.