Fluctuations of energy flux in a simple dissipative out-of-equilibrium system

We report the statistical properties of the fluctuations of the energy flux in an electronic $RC$ circuit driven with a stochastic voltage. The fluctuations of the power injected in the circuit are measured as a function of the damping rate and the forcing parameters. We show that its distribution exhibits a cusp close to zero and two asymmetric exponential tails, with the asymmetry being driven by the mean dissipation. This simple experiment allows one to capture the qualitative features of the energy flux distribution observed in more complex dissipative systems. We also show that the large fluctuations of injected power averaged on a time lag do not verify the fluctuation theorem even for long averaging time. This is in contrast to the findings of previous experiments due to their small range of explored fluctuation amplitude. The injected power in a system of $N$ components either correlated or not is also studied to mimic systems with large number of particles, such as in a dilute granular gas.

Figure 1

Scheme of the electronic circuit as an analog of the Langevin equation.

Figure 2

(Color online) Probability density functions of the injected power $I$ for two different noise amplitudes, [(a) and (b)] $D=1.56\times {10}^{-3}{\text{V}}_{\text{rms}}^2/\text{Hz}$ and [(c) and (d)] $D=0.75\times {10}^{-3}{\text{V}}_{\text{rms}}^2/\text{Hz}$, and damping rates, [(a) and (c)] $\gamma =200\text{Hz}$ and [(b) and (d)] $\gamma =2000\text{Hz}$.

Figure 3

(Color online) Probability density functions of injected power, $I$, for $D=0.06–1.56\times {10}^{-3}{\text{V}}_{\text{rms}}^2/\text{Hz}$ (see the arrow) for $\gamma =200\text{Hz}$. Inset: probability density functions in the rescaled variable $\left(I-⟨I⟩\right)/{\sigma }_I$.

Figure 4

(Color online) Mean $⟨I⟩$ and standard deviation ${\sigma }_I$ of the injected power as a function of the noise amplitude $D$. $\gamma =200\text{Hz}$.

Figure 5

(Color online) Mean $⟨I⟩$ and standard deviation ${\sigma }_I$ of the injected power as a function of the damping rate $\gamma$. $D=0.75\times {10}^{-3}{\text{V}}_{\text{rms}}^2/\text{Hz}$. (–): linear best fits of slopes 1.9 and 1.59 V, respectively.

Figure 6

(Color online) Scaling of the PDFs of the negative values (left) and the positive values (right) of injected power $I$ for nine values of $D$ and ten values of $\gamma$.

Figure 7

(Color online) Scaling of the mean $⟨I⟩$ and standard deviation ${\sigma }_I$ with the cutoff frequency $\lambda$. (–): linear best fits of slopes 0.11 and 0.56 V, respectively.

Figure 8

(Color online) PDFs of $I/⟨I⟩$. Comparison between experiment (–) and theory [$\left(-\cdot -\right)$ from Eq. (10)] for two different values of the damping rate $\gamma =2000\text{Hz}$ $\left(r=\frac{⟨I⟩}{{\sigma }_V{\sigma }_{\zeta }}=0.45\right)$ [black line] and $\gamma =200\text{Hz}$ $\left(r=0.15\right)$ [red (light gray) line]. The cutoff frequency $\lambda$ is fixed to 10 kHz.

Figure 9

(Color online) PDF of $I_{\tau }/⟨I⟩$ for various values of $\tau /{\tau }_c=0$, 5, 10, 50, 100, and 200 at a fixed value of $\gamma =2000\text{Hz}$. The straight line (–) corresponds to $I_{\tau }/⟨I⟩=0$ and the dashed line $\left(--\right)$ to $I_{\tau }=⟨I⟩$.

Figure 10

(Color online) Same as Fig. 9 for $\gamma =200\text{Hz}$.

Figure 11

(Color online) Most probable value ${ϵ}^{\ast }$ of PDF $\left(I_{\tau }/⟨I⟩\right)$ as a function of $\tau /{\tau }_c$ for $\gamma =50$, 100, 200, 500, and 1000 Hz. For $ϵ<{ϵ}^{\ast }$, the relation $\rho \left(ϵ\right)\sim ϵ$ is verified at finite $\tau$, whereas it does not hold for $ϵ>{ϵ}^{\ast }$ (see text).

Figure 12

(Color online) Asymmetrical function $\rho \left(ϵ\right)=\frac{{\tau }_c}{\tau }\text{ln}\left[\frac{P\left(ϵ\right)}{P\left(-ϵ\right)}\right]$ as a function of $ϵ$ for different integration times $\tau /{\tau }_c=1$ $(\star )$ to 31 $(+)$ at fixed $\gamma =100\text{Hz}$ and $D=1.56{\text{mV}}_{\text{rms}}^2/\text{Hz}$.

Figure 13

(Color online) Same as Fig. 12 for $\gamma =50\text{Hz}$.

Figure 14

(Color online) (a) Computed large deviation functions $\frac{{\tau }_c}{\tau }\text{ln}\left[P\left(I_{\tau }/⟨I⟩\right)\right]$ of $\tau /{\tau }_c$ correlated variables with $\tau /{\tau }_c=3$ (◻) to 50 $(\star )$ for $\gamma =100\text{Hz}$. (b) Computed large deviation functions $\frac{1}{N}\text{ln}\left[P\left(I_{\mathcal{N}}/⟨I⟩\right)\right]$ of $\mathcal{N}$ uncorrelated variables with $\mathcal{N}=2$ (◻), 4 (○), 6 $(+)$, 8 $(\ast )$, and 10 $(▷)$ for $\gamma =100\text{Hz}$. Inset: same with $\mathcal{N}=10$ $(▷)$, 20 (○), 30 $(+)$, 40 $(\ast )$ and 50 (◻). The dashed lines show the mean injected power $⟨I⟩$.

Figure 15

(Color online) Asymmetrical function $\rho \left(I_{\mathcal{N}\tau }/⟨I⟩\right)$ of $\mathcal{N}$-independent variables for different integration times $\tau /{\tau }_c=1$ $(\star )$ to 31 $(+)$. $\mathcal{N}=10–100$ with a 10 step.