Statistical properties of the heat flux between two nonequilibrium steady-state thermostats

We address the question of transport of heat in out-of-equilibrium systems. The experimental setup consists in two coupled granular gas nonequilibrium steady-state (NESS) heat baths, in which Brownian-like rotors are imbedded. These rotors are electromechanically coupled, thanks to DC micromotors, through a resistor $R$ such that energy flows between them. The average flux depends linearly in the difference in the baths' temperature. Varying $R$ allows extrapolation in the nondissipative coupling limit ($R\to 0$). We show that in this limit the heat flux obeys the fluctuation theorem in a form proposed by Jarzynski and Wójcik in 2004 [C. Jarzynski and D. K. Wójcik, Phys. Rev. Lett. 92, 230602 (2004)] for the fluctuations of the flux between finite size equilibrium heat baths.

Figure 1

The granular gas is excited in a cylindrical vessel by the vertical acceleration from an electromagnetic shaker. A small blade is fixed on the vertical shaft of a DC micromotor set on the immobile cover of the cell. Its rotation is caused by the random collisions with the gas.

Figure 2

The two motors, coupled to baths 1 and 2, are represented in dashed rectangles by ideal voltage sources of EMF, $e_1$ and $e_2$ (the instantaneous values of which are linked to the velocity of the rotors), and their internal resistances $r_1$ and $r_2$. The coupling resistor is $R$.

Figure 3

Histograms of the coarse-grained heat flux ${\varphi }_{\tau }$ for several values of $\tau$. (From the widest to the narrowest histograms: $\tau \simeq 50\mathrm{ms}, 100\mathrm{ms}, 300\mathrm{ms}, 1.2\mathrm{s}$.) Here, the average flux is $\overline{\varphi }\simeq 113\mathrm{nW}$, for $\mathrm{\Delta }\beta \simeq 9.37\times {10}^4{\mathrm{J}}^{-1}$.

Figure 4

The slope of the asymmetry function for a 1-h-time sample, obtained from Eq. (11), is plotted for various times $\tau$ ($\circ$) during relaxation. The red curve ($+$) shows the analysis of the same sample by the best linear fit of the asymmetry function. Here $R=100.23\mathrm{\Omega }$, and $\mathrm{\Delta }\beta \simeq 9.37\times {10}^4{\mathrm{J}}^{-1}$.

Figure 5

The slope of the asymmetry function for a few time series, calculated thanks to Eq. (11), is plotted for various time lags $\tau$ ($+$). The red curves show best exponential fits. The dashed black line represents the average of these limits at large $\tau$ over dozens of time series. The rms of $\mu$ is in black. Here $R=23.28\mathrm{\Omega }$, while $\mathrm{\Delta }\beta \simeq 1.5\times {10}^5{\mathrm{J}}^{-1}$.

Figure 6

The relation between $\mu$ and $\mathrm{\Delta }\beta$ is linear for $R=22\mathrm{\Omega }$, and the slope is $\frac{\mu }{\mathrm{\Delta }\beta }\simeq 5.69$. This figure is taken from Ref. [7].

Figure 7

The slope $\mu$ is plotted against $\mathrm{\Delta }\beta$ for a few values of the coupling resistance $R=23.27\mathrm{\Omega }$(solid), 100.23 $\mathrm{\Omega }$ (dash-dot), $242.6\mathrm{\Omega }$(dash), and $617.4\mathrm{\Omega }$(dot). The red lines represent in each case the best linear fits through zero.

Figure 8

The ratio $\frac{\mu }{\mathrm{\Delta }\beta }$ is plotted against the resistance $R_{\mathrm{tot}}$. A second-order polynomial fitting is performed $y=p_1{x}^2+p_{2}x+p_3$ (in red). The coefficients obtained are $p_1\simeq 5.63\times {10}^{-5},p_2\simeq 0.30,p_3\simeq 0.85$. In the zero-resistance limit, $\frac{\gamma }{\mathrm{\Delta }\beta }\to p_3\simeq 0.85$.