Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non-Markovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The modified relation includes a contribution arising from the acausal character of the reverse process. This, in turn, leads to another second-law-like inequality. We illustrate the general formalism with a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.

Figure 1

“Phase” diagram of the stationary state in the plane $(Q_0,g/Q_0)$. The regions labeled 1–4 are each characterized by a different behavior of the effective temperatures $T_x$ and $T_v$ as a function of $\tau$ (region 1 extends from $g/Q_0=-1$ to $-\infty$ but is arbitrarily truncated at $g/Q_0=-1.5$).

Figure 2

“Christmas-tree” stability diagram in the $(\tau ,g/Q_0)$ plane for $Q_0=2$. The horizontal dashed line $g/Q_0=(1/Q_0)\sqrt{1-1/\left(4Q_0^2\right)}\approx 0.484$ separates regions 3 and 4 in Fig. 1 (see Appendix pp3). In the multistability region 4, the feedback-controlled oscillator reaches a stable stationary state in the zones located outside the shaded area.

Figure 3

Asymptotic behavior in the large-$Q_0$ regime for $g=10$ as a function of $\tau$. A stationary state exists when the effective temperatures are positive. (a) $T_x$ or $T_v$ (they are equal when $Q_0\to \infty$). (b) $(Q_0/g){\dot{\mathcal{W}}}_{\mathrm{ext}}/T$ (solid black line) and $(Q_0/g){\dot{\mathcal{S}}}_{\mathrm{pump}}^v$ (dashed red line). (c) ${\mathcal{I}}^{v;y}$. (d) ${\dot{\mathcal{I}}}_{\mathrm{flow},v}^{xv;y}$ (long-dashed red line) and $(Q_0/g)[{\dot{\mathcal{S}}}_{\mathrm{pump}}^{v;y}+{\dot{\mathcal{I}}}_{\mathrm{flow},v}^{v;y}]$ (dotted red line).

Figure 4

Schematic distribution of the branch points of $ln[\tilde{\chi }\left(s\right)/{\chi }_{g=0}\left(s\right)]$ in the complex $s$ plane. The black dots are the two poles $s_0^{\pm }$ of ${\chi }_{g=0}\left(s\right)$ and the stars are the poles ${\tilde{s}}^{\pm }$ of type 0 (blue) and of type 1 (red) of $\tilde{\chi }\left(s\right)$. Only the first poles of type 1 are shown in the figure. The dashed (red) lines represent the branch cuts (on the right-hand side of the $s$ plane, they go to $+\infty$). The solid black lines, including the vertical line that represents the Bromwich contour $\text{Re}\left(s\right)=c$, form the contour used for calculating ${\dot{S}}_{\mathcal{J}}$ from Eq. (155).

Figure 5

Feedback-controlled harmonic oscillator: configurational ($T_x$) and kinetic ($T_v$) temperatures as a function of the feedback gain for $Q_0=2$ and $\tau =2.5$. One has $T_x/T=1$ for $g/Q_0\approx 0.60$ and $T_v/T=1$ for $g/Q_0\approx 0.63$ (red arrows).

Figure 6

Feedback-controlled harmonic oscillator: second-law-like inequalities as a function of the feedback gain for $Q_0=2$ and $\tau =2.5$. (a) Comparison of the rates ${\dot{\mathcal{W}}}_{\mathrm{ext}}/T$ (solid black line), ${\dot{\mathcal{S}}}_{\mathrm{pump}}^v$ (short-dashed red line), ${\dot{\mathcal{I}}}_{\mathrm{flow},v}^{xv;y}$ (long-dashed red line), ${\dot{\mathcal{I}}}_{\mathrm{flow},v}^{v;y}+{\dot{\mathcal{S}}}_{\mathrm{pump}}^{v;y}$ (dotted red line), and ${\dot{\mathcal{S}}}_{\mathcal{J}}$ (dash-dotted blue line). (b) Plot of the “EP” rate ${\dot{\mathcal{R}}}_{cg}=-{\dot{\mathcal{W}}}_{\mathrm{ext}}/T+{\dot{\mathcal{S}}}_{\mathcal{J}}$.

Figure 7

Same as Fig. 5 for $\tau =8$. The temperatures $T_x$ and $T_v$ diverge at the stability limit of the stationary state ($g/Q_0\to g_c/Q_0\approx 0.61$). One has $T_x/T=1$ for $g/Q_0\approx 0.09$ and $T_v/T=1$ for $g/Q_0\approx 0.15$ (red arrows).

Figure 8

Same as Fig. 6 for $\tau =8$. ${\dot{\mathcal{W}}}_{\mathrm{ext}}$ (solid black line) diverges as $g/Q_0\to g_c/Q_0\approx 0.61$, whereas ${\dot{\mathcal{S}}}_{\mathrm{pump}}^{xv}$ (short-dashed red line) and ${\dot{\mathcal{S}}}_{\mathcal{J}}$ (dash-dotted blue line) remain finite. The other bounds built from information flows (long-dashed and dotted red lines) also stay finite.

Figure 9

(a) Second-law-like inequalities (see caption of Fig. 6) as a function of the delay for $Q_0=2$ and $g/Q_0=0.45$. The information flow ${\dot{\mathcal{I}}}_{\mathrm{flow},v}^{xv;y}$ is not represented as it diverges for $\tau =0$. Note that ${\dot{\mathcal{S}}}_{\mathcal{J}}$ (dashed-dotted blue line) changes sign with $\tau$, in contrast with its behavior with $g/Q_0$ for $\tau =2.5$ and $\tau =8$. (b) “EP” rate ${\dot{\mathcal{R}}}_{cg}$.

Figure 10

Same as Fig. 9 for $Q_0=2$ and $g/Q_0=0.55$. The stability lobes are delimited by the dashed vertical lines; the third stability lobe (for $15.02<\tau <15.97$) is not represented. The dotted blue line represents the continuation of ${\dot{S}}_{\mathcal{J}}$ in the instability regions.

Figure 11

Same as Fig. 9 for $Q_0=34.2$ and $g/Q_0=0.25$ (in this case, there are only two stability lobes). (a) ${\dot{\mathcal{W}}}_{\mathrm{ext}}/T$ (solid black line) and various upper bounds. (b) “EP” rate ${\dot{\mathcal{R}}}_{cg}$.

Figure 12

Same as Fig. 9 for $Q_0=250$ and $g/Q_0=0.04$. There are many other stability lobes that are not represented.

Figure 13

Reduced temperatures $T_x/T$ and $T_v/T$ as a function of $\tau$ for $Q_0=2$ and $g/Q_0=-1.5$ (region 1). The two temperatures (or variances $\langle x^2\rangle$ and $\langle v^2\rangle$) diverge at the critical delay ${\tau }_{0,1}^{*}\approx 0.349$ beyond which a stationary solution no longer exists.

Figure 14

Same as Fig. 13 for $Q_0=0.25, g/Q_0=0.6$ (solid black line), and $g/Q_0=-0.6$ (dashed red line) (region 2). $T_x/T$ and $T_v/T$ tend to $\approx 1.27$ and $\approx 1.01$, respectively, as $\tau \to \infty$, as predicted by Eqs. (C5).

Figure 15

Same as Fig. 13 for $Q_0=2, g/Q_0=0.45$ (solid black line), and $g/Q_0=-0.45$ (dashed red line) (region 3). $T_x/T$ and $T_v/T$ tend to $\approx 2.95$ and $\approx 2.63$, respectively, as $\tau \to \infty$, as predicted by Eqs. (C6).

Figure 16

Same as Fig. 13 for $Q_0=2$ and $g/Q_0=0.55$ (region 4). The stationary solution disappears at the critical delays ${\tau }_{1,1}^{*}\approx 4.19,{\tau }_{2,1}^{*}\approx 10.08,{\tau }_{3,1}^{*}\approx 15.98$ and exists again at ${\tau }_{1,2}^{*}\approx 7.01,{\tau }_{2,2}^{*}\approx 15.03$. There is no stationary solution beyond ${\tau }_{3,1}^{*}$.

Figure 17

Evolution with $\tau$ of the real part of the two pairs of poles of $\tilde{\chi }\left(s\right)$ that are closest to the imaginary axis. Here $Q_0=34.2$ and $g/Q_0=0.25$ and only the second stability lobe is considered (see Fig. 11 of the main text). The poles are of type 0 (solid black lines), except for a small interval $7<\tau \lesssim 7.15$, where one pair of poles is of type 1 (dashed red line).

Figure 18

Acausal response function $\tilde{\chi }\left(t\right)$ versus time for $Q_0=2, g/Q_0=0.55$, and $\tau =1.2$. $\tilde{\chi }\left(s\right)$ has no poles on the left-hand side of the complex plane. The poles ${\tilde{s}}^{\pm }\approx 0.0394\pm 0.847i$ and ${\tilde{s}}^{\left(2\right)}\approx 1.977$ on the right-hand side control the behavior of $\tilde{\chi }\left(t\right)$ for $t\ge 0$ (hence the oscillations and the diverging time dependence) and $t\to -\infty$ (hence the rapid decay to zero), respectively. The ROC of $\tilde{\chi }\left(s\right)$ is defined by ${\sigma }_{\min }=\text{Re}\left({\tilde{s}}^{\pm }\right)<\text{Re}\left(s\right)<{\sigma }_{\max }={\tilde{s}}^{\left(2\right)}$.

Figure 19

Acausal response function $\tilde{\chi }\left(t\right)$: Same as Fig. 18 for $\tau =8$. $\tilde{\chi }\left(s\right)$ has 4 poles on the left-hand side of the complex plane. The poles ${\tilde{s}}^{\pm }\approx -0.187\pm 0.958i$ and ${\tilde{s}}_{\pm }^{\left(2\right)}\approx -0.033\pm 0.967i$ control the behavior of $\tilde{\chi }\left(t\right)$ for $t\ge 0$ (hence the oscillation and the asymptotic decay to zero) and $t\to -\infty$ (hence the oscillations and the diverging asymptotic behavior), respectively. The ROC of $\tilde{\chi }\left(s\right)$ is defined by ${\sigma }_{\min }=\text{Re}\left({\tilde{s}}^{\pm }\right)<\text{Re}\left(s\right)<{\sigma }_{\max }=\text{Re}\left({\tilde{s}}_{\pm }^{\left(2\right)}\right)$.

Figure 20

Acausal response function $\tilde{\chi }\left(t\right)$: Same as Fig. 18 for $g/Q_0=0.45$ and $\tau =10$. The poles ${\tilde{s}}^{\pm }\approx -0.257\pm 0.984i$ and ${\tilde{s}}_{\pm }^{\left(2\right)}\approx 0.0407\pm 0.692i$ control the behavior of $\tilde{\chi }\left(t\right)$ for $t\ge 0$ and $t\to -\infty$, respectively. Note the cusp behavior for $t=0$ and the weaker singularities for $-\tau ,-2\tau ,$ etc.