Thermal response of nonequilibrium $RC$ circuits

We analyze experimental data obtained from an electrical circuit having components at different temperatures, showing how to predict its response to temperature variations. This illustrates in detail how to utilize a recent linear response theory for nonequilibrium overdamped stochastic systems. To validate these results, we introduce a reweighting procedure that mimics the actual realization of the perturbation and allows extracting the susceptibility of the system from steady-state data. This procedure is closely related to other fluctuation-response relations based on the knowledge of the steady-state probability distribution. As an example, we show that the nonequilibrium heat capacity in general does not correspond to the correlation between the energy of the system and the heat flowing into it. Rather, also nondissipative aspects are relevant in the nonequilibrium fluctuation-response relations.

Figure 1

(a) Diagram of the circuit. The resistor $R_1$ is kept at temperature $T_1$ while $R_2$ is always at room temperature $T_2=296$ K. They are coupled via the capacitor $C$. The capacitors $C_1$ and $C_2$ account for the capacitance of the wiring, etc. The voltage generators ${\eta }_1$ and ${\eta }_2$ represent thermal fluctuations of the voltage that the resistors undergo. (b) Equivalent mechanical system with two Brownian particles moving in fluids at different temperatures $T_1$ and $T_2$, but trapped and coupled by harmonic springs.

Figure 2

Response of the total energy $U$ to a change of $T_1$, for equilibrium ($T_1=296\mathrm{K}=T_2$) and for nonequilibrium ($T_1=140$ K and $T_1=88$ K). The susceptibility ${\chi }_U$ computed with the fluctuation-response relation (10) and its terms $S_1,S_2,K_1,K_2$ are shown. The susceptibility ${\chi }_U^{\theta }$ computed with the reweighting formula (24) agrees with ${\chi }_U$.