Optimizing energetic cost of uncertainty in a driven system with and without feedback

Many biological functions require dynamics to be necessarily driven out of equilibrium. In contrast, in various contexts, a nonequilibrium dynamics at fast timescales can be described by an effective equilibrium dynamics at a slower timescale. In this work, we study two different aspects: (i) the energy-efficiency tradeoff for a specific nonequilibrium linear dynamics of two variables with feedback and (ii) the cost of effective parameters in a coarse-grained theory as given by the “hidden” dissipation and entropy production rate in the effective equilibrium limit of the dynamics. To meaningfully discuss the tradeoff between energy consumption and the efficiency of the desired function, a one-to-one mapping between function(s) and energy input is required. The function considered in this work is the variance of one of the variables. We get a one-to-one mapping by considering the minimum variance obtained for a fixed entropy production rate and vice versa. We find that this minimum achievable variance is a monotonically decreasing function of the given entropy production rate. When there is a timescale separation, in the effective equilibrium limit, the cost of the effective potential and temperature is the associated “hidden” entropy production rate.

Figure 1

(a) Plot of the stable region in the $k$ and $\alpha$ parameter space [the dynamics is stable for $k\alpha <1$ (shown in gray)] and the schematic of the feedback circuit in different quadrants. The feedback is positive for $k\alpha >0$ and negative for $k\alpha <0$. (b) Plot of the variance of $x$ as function of $k$ and $\alpha$. When the feedback is positive, ${\langle x^2\rangle }_{\mathrm{fb}}>{\langle x^2\rangle }_{\mathrm{ff}}$, and when the feedback is negative,${\langle x^2\rangle }_{\mathrm{fb}}<{\langle x^2\rangle }_{\mathrm{ff}}$. In the region between the line $\alpha =0$ and $\alpha T_y+k{T}_x=0$, the variance of $x$ is less that that without its coupling with $y$, i.e., ${\langle x^2\rangle }_{\mathrm{fb}}<T_x$.

Figure 2

The plot shows the sign of the heat flow into the temperature baths as function of the feedback parameters. When the feedback is negative, the heat dissipation rate for heat baths associated with $x$ and $y$ is positive. For positive feedback, in the region between the lines $\alpha T_y=k{T}_x$ and $\alpha =0$ ($k=0$) $h_y^{\mathrm{fb}}<0$ ($h_x^{\mathrm{fb}}<0$), and in the region between the lines $k=\alpha$ and $\alpha T_y=k{T}_x$ the total heat flow is negative ($h^{\mathrm{fb}}<0$).

Figure 3

Plot of the entropy production rate and the variance as given by Eqs. (33) and (32) of the main text, for $d=1, r=2$, along different section of the parameter space. For (a) $\alpha T_y{\mu }_x=-k{T}_x{\mu }_y$, the variance is constant and equal to that without the coupling with $y$, i.e., ${\langle x^2\rangle }_{\mathrm{fb}}=T_x$, whereas the EPR depend upon the feedback parameters. In contrast, along (b) $\alpha T_y=k{T}_x$, the dynamics is at an effective equilibrium ($\sigma =0$) and the variance depends upon the parameter values. For (c) conservative coupling ($\alpha =k$) and when the coupling is (f) feedforward ($k=0$), both EPR and variance are correlated and depend upon the parameter values. In general, as show in panels (d) and (e), the variance and EPR can show divergent trends.

Figure 4

(a) Plot of the minimum value of the variance for a fixed EPR and timescale as function of the given EPR and timescale, for $d=1$ and $r=2$, and (b) the plot showing the curves with constant value of variance (blue) and constant entropy production rate (red). The minimum value of the variance for a given EPR and vice versa is given by the intersection of the two curves at ${\mathrm{\Lambda }}^{*}$ and $S^{*}$.