Nonequilibrium Steady-State Fluctuations in Actively Cooled Resonators

We analyze heat and work fluctuations in the gravitational wave detector AURIGA, modeled as a macroscopic electromechanical oscillator in contact with a thermostat and cooled by an active feedback system. The oscillator is driven to a steady state by the feedback cooling, equivalent to a viscous force. The experimentally measured fluctuations are in agreement with our theoretical analysis based on a stochastically driven Langevin system. The asymmetry of the fluctuations of the absorbed heat characterizes the oscillator’s nonequilibrium steady state and reveals the extent to which a feedback cooled system departs from equilibrium in a statistical mechanics perspective.

Figure 1

The normal mode is approximated, around its resonance frequency, by a series-RLC circuit. The dc SQUID is represented as a current amplifier. The observable is the current $I_s$, and the electronic feedback cooling is obtained by sending back a current $I_d$ which is a delayed copy of $I_s$ reduced by $G\ll 1$. The SQUID output voltage is $V_{\mathrm{out}}=A{I}_s$ with $A=2.6\times {10}^6\Omega$.

Figure 2

PDF [units of $s/(k_B{T}_{\mathrm{eff}})$] of (a) the time averaged energy difference $\Delta U_{\tau }$, (b) the work $W_{\tau }$, and (c) the heat $Q_{\tau }$ averaged at increasing values of the ratio $\tau /{\tau }_{\mathrm{eff}}$. Data were collected by AURIGA in a 10 day time span. The dashed vertical line at $0.84k_B{T}_{\mathrm{eff}}/\mathrm{s}$ corresponds to the mean value of $W_{\tau }$. The dotted lines are obtained by numerical simulation of Eq. (3) for a 50 day time span. The discrepancies observed at short $\tau$ between experimental and numerical data are within the uncertainty due to the experimental error in ${\tau }_{\mathrm{eff}}$.

Figure 3

Comparison between theory (dashed lines) and experimental data at increasing values of the ratio $\tau /{\tau }_{\mathrm{eff}}=1.2$, 2.0, 2.9, 3.7: the data set is, respectively, 828, 768, 729, 457 days long. (a) Plot of second derivative $D_2({\widetilde{ϵ}}_{\tau })$; the experimental data of $D_2$ for $\tau /{\tau }_{\mathrm{eff}}=2.0$ are not shown for clarity. (b) Plot of $\rho ({\widetilde{ϵ}}_{\tau })$. Vertical error bars on experimental points of both plots come from statistical uncertainty. The shaded areas on the theoretical curves represent the uncertainty due to the experimental errors on the parameters, ${\tau }_{\mathrm{eff}}$, $T_{\mathrm{eff}}$, and $T_0$.