Stochastic thermodynamics of Langevin systems under time-delayed feedback control. II. Nonequilibrium steady-state fluctuations

This paper is the second in a series devoted to the study of Langevin systems subjected to a continuous time-delayed feedback control. The goal of our previous paper [Phys. Rev. E 91, 042114 (2015)] was to derive second-law-like inequalities that provide bounds to the average extracted work. Here we study stochastic fluctuations of time-integrated observables such as the heat exchanged with the environment, the extracted work, or the (apparent) entropy production. We use a path-integral formalism and focus on the long-time behavior in the stationary cooling regime, stressing the role of rare events. This is illustrated by a detailed analytical and numerical study of a Langevin harmonic oscillator driven by a linear feedback.

Figure 1

Time-reversed paths ${\mathbf{X}}^{†}$ and ${\mathbf{Y}}^{†}$. When the dynamics is governed by the acausal Langevin equation (21), the feedback force depends on the future state of the system, as schematically represented by the arrowed line.

Figure 2

Stability diagram of the feedback-controlled oscillator for $Q_0=34.2$ (the time unit is the inverse angular resonance frequency ${\omega }_0^{-1}$). The oscillator is unstable inside the shaded regions. The acausal response function $\tilde{\chi }\left(s\right)$ has all its poles located in the r.h.s. of the complex $s$ plane inside the regions delimited by the dashed red lines and two poles in the l.h.s. outside these regions.

Figure 3

Stochastic fluctuations of $w=\beta {\mathcal{W}}_t/t$ (solid black line), $q=\beta {\mathcal{Q}}_t/t$ (dotted blue line), and $\sigma ={\mathrm{\Sigma }}_t/t$ (dashed red line) for $Q_0=34.2g/Q_0=0.25,\tau =7.6$ (a) and $\tau =8.4$ (b). The figure shows the results obtained for an observation time $t=100$ and 75 independent noise realizations. Lines are a guide for the eyes.

Figure 4

Probability distribution functions $P\left(\beta {\mathcal{W}}_t\right)$ (black circles), $P\left(\beta {\mathcal{Q}}_t\right)$ (blue stars), and $P\left({\mathrm{\Sigma }}_t\right)$ (red squares) plotted against the scaled variable $a={\mathcal{A}}_t/t$ ($a=w,q$, or $\sigma$) for $Q_0=34.2,g/Q_0=0.25$, and $\tau =7.6$. The observation time is $t=100$. Symbols represent numerical data obtained by solving the Langevin equation (39) for $2\times {10}^6$ realizations of the noise.

Figure 5

Same as Fig. 4 for $\tau =8.4$.

Figure 6

Numerical estimates of ${\mu }_A\left(\lambda \right)$ using $t=100$ and $2\times {10}^6$ realizations of the noise for $\tau =7.6$ (a) and $\tau =8.4$ (b): ${\mu }_W$ (black circles), ${\mu }_Q$ (blue stars), and ${\mu }_{\mathrm{\Sigma }}$ (red squares). The solid black line represents the theoretical SCGF $\mu \left(\lambda \right)$ given by Eq. (53) in the interval $({\lambda }_{\min },{\lambda }_{\max })$ in which this quantity is real.

Figure 7

Numerical estimates of ${\mu }_A\left(1\right)={lim}_{t\to \infty }(1/t)ln\langle e^{-{\mathcal{A}}_t}\rangle$ as a function of $\tau$ in the second stability lobe: ${\mu }_W\left(1\right)$ (black circles), ${\mu }_Q\left(1\right)$ (blue stars), and ${\mu }_{\mathrm{\Sigma }}\left(1\right)$ (red squares). The dashed-dotted black line is the average extracted work rate $\langle \beta {\dot{\mathcal{W}}}_{\mathrm{ext}}\rangle =-\langle \beta \dot{\mathcal{Q}}\rangle$, which is positive for $7.26\le \tau \le 8.43$.

Figure 8

Comparison between the numerical estimates of ${\mu }_A\left(1\right)={lim}_{t\to \infty }(1/t)ln\langle e^{-{\mathcal{A}}_t}\rangle$ displayed in Fig. 7 and the values of $\mu \left(1\right)$ (solid black line) and ${\dot{\mathcal{S}}}_J$ (dashed red line) computed from Eqs. (60) and (59), respectively. One has $\mu \left(1\right)={\dot{\mathcal{S}}}_J\le 1/Q_0$ for $\tau \le {\tau }_{p,1}\approx 7.37$ and $\tau \ge {\tau }_{p,2}\approx 8.32$, and $\mu \left(1\right)=1/Q_0\le {\dot{\mathcal{S}}}_J$ for ${\tau }_{p,1}\le \tau \le {\tau }_{p,2}$. Note that ${\dot{\mathcal{S}}}_{\mathcal{J}}$ is a tighter bound to the extracted work rate (dashed-dotted black line) than $1/Q_0$ for $\tau <{\tau }_{p,1}$ and $\tau >{\tau }_{p,2}$.

Figure 9

Probability distribution functions $P({\mathcal{A}}_t=at)$ for $\tau =7.6$. The symbols are the data obtained from the numerical simulations (see Fig. 4), the dashed black line is the large-deviation form $e^{-I\left(w\right)t}$, and the solid black line is the semiempirical asymptotic expression given by Eq. (71). The dashed red line on the l.h.s. for $\sigma ⪅-0.048$ is the curve $e^{-I_1\left(\sigma \right)t}$ obtained from Eq. (73).

Figure 10

Same as Fig. 9 for $\tau =8.4$. The dashed blue line on the l.h.s. for $q⪅-0.042$ is the curve $e^{-I_1\left(q\right)t}$ obtained from Eq. (73).

Figure 11

Symmetry function for the work fluctuations $f\left(w\right)=I(-w)-I\left(w\right)$ for $\tau =7.6$ (a) and $\tau =8.4$ (b). The dashed red lines represent the asymptotic regime of large fluctuations $f\left(w\right)\sim ({\lambda }_{\min }+{\lambda }_{\max })w$ (see text).

Figure 12

Verification of the SSFTs (78a) and (79). Here $P(\beta {\mathcal{W}}_t=wt)e^{-wt}$ (black circles) is compared with (a) $\widehat{P}({\mathcal{W}}_t=wt)e^{t/Q_0}$ (red squares) for $Q_0=34.2,g/Q_0=0.25,\tau =7.6$, and (b) $\tilde{P}(\beta {\mathcal{W}}_t=wt)e^{{\dot{\mathcal{S}}}_{\mathcal{J}}t}$ (red squares) for $Q_0=2,g/Q_0=0.55,\tau =2.5$ (${\dot{S}}_{\mathcal{J}}\approx 0.1$). The observation time is $t=100$.

Figure 13

Acausal response function $\tilde{\chi }\left(t\right)$ for $Q_0=2,g/Q_0=0.55$ and $\tau =2.5$. $\tilde{\chi }\left(s\right)$ has two poles on the l.h.s. of the complex $s$ plane and an infinite number of poles on the r.h.s. The poles $s\approx -0.200\pm 1.12i$ and $s\approx 0.339$ control the behavior of $\tilde{\chi }\left(t\right)$ for $t\ge 0$ and for $t\to -\infty$, respectively.

Figure 14

Autocorrelation function of the atypical colored noise ${\xi }_{\mathrm{atyp}}\left(t\right)$ for $Q_0=2,g/Q_0=0.55$ and $\tau =2.5$.

Figure 15

Kinetic temperatures computed from Eq. (A3b): the black solid line is ${\widehat{T}}_v/T$ for $7.37<\tau <8.32$ and ${\tilde{T}}_v/T$ for $\tau <7.37$ and $\tau >8.32$. The dashed red line is the kinetic temperature $T_v/T$ of the original dynamics, which diverges at the boundaries of the stability region.

Figure 16

Behavior of $(1/t)ln{Z}_Q(\lambda ,t)$ as a function of $\lambda$ in the vicinity of $\lambda =1$ for $\gamma =2$ and ${\gamma }^{\prime }=1$ ($m=1,\overline{k}=1,T=1$). From top to bottom: $t=3,5,10,25$. Observe that $(1/t)ln{Z}_Q(1,t)=\gamma /m=2$ for all values of $t$. The solid black line shows the theoretical SCGF $\mu \left(\lambda \right)$ given by Eq. (B15).

Figure 17

Behavior of $(1/t)ln{Z}_Q(\lambda ,t)$ as a function of $t/{\tau }_0$ for $\gamma =2,{\gamma }^{\prime }=1$, and $1-\lambda ={10}^{-6}$ ($m=1,\overline{k}=1,T=1$). Note the crossover from $\gamma$ to ${\gamma }^{\prime }$ around $t/{\tau }_0=\frac{\gamma }{\gamma -{\gamma }^{\prime }}ln\frac{1}{1-\lambda }\approx 28$. The crossover time decreases as $\lambda$ moves away from 1.