Entropic balance with feedback control: information equalities and tight inequalities

Abstract

We consider overdamped physical systems evolving under a feedback-controlled fluctuating potential and in contact with a thermal bath at temperature $T$. A Markovian description of the dynamics, which keeps only the last value of the control action, is advantageous -- both from the theoretical and the practical side -- for the entropy balance. Novel second-law equalities and bounds for the extractable work are obtained, the latter being both tighter and easier to evaluate than those in the literature based on the whole chain of controller actions. The Markovian framework also allows us to prove that the bound for the extractable work that incorporates the unavailable information is saturated in a wide class of physical systems, for error-free measurements. These results are illustrated in model systems. For imperfect measurements, there appears an interval of measurement uncertainty, including the point at which work ceases to be extracted, where the new Markovian bound is tighter than the unavailable information bound.

Figures (6)

• Figure 1: Sketch of the system dynamics. The feedback controller measures the system state at regular times $t_n=n\Delta t_{\text{m}}$, $x_n\equiv x(t_n)$, $n=0,1,\ldots$ The value of the controller variable $c_{n}$ in the time interval $\tau_{\text{FP}}^{(n+1)}\equiv (t_{n}^+,t_{n+1}^-)$ is drawn from a probability distribution $\Theta(c_{n}|x_{n})$, conditioned to the measured $x_{n}$. The system follows overdamped Fokker-Planck (FP) evolution under a fluctuating potential $V(x,c_{n})$, different for each value of $c_n$, in the same interval.
• Figure 2: (Left) Sketch of an information engine. The potential may be inverted by the feedback controller after measuring the position of the particle $x_n$ at times $t_n=n\Delta t_{\text{m}}$. In the error-free case, the inversion is done---as shown by the arrows---if the potential energy decreases, so work is always extracted. For the sake of clarity, more than one period is shown. (Right) Introduction of error in the measurement. The particle position is measured with precision $\Delta x$. The function $\Theta(1|x)$, which gives the conditional probability of having the potential $V(x,c=1)\textcolor{black}{=+U(x)}$ at $t_n^+$ is plotted for different values of $\Delta x$, from $\Delta x=0$ (error-free, perfect closed-loop control) to $\Delta x=1$ (maximum error, open-loop control).
• Figure 3: (Left) Average work and bounds as a function of the uncertainty in the measurement. The data shown correspond to non-equilibrium situation with $f\ne 0$, specifically $f=0.7$. The behaviour for $f=0$ is qualitatively similar, eliminating the curves with the vanishing housekeeping contribution $\Sigma_{xc}^{\text{hk}}$. (Right) Phase diagram of the information engine in the $(\Delta x, f)$ plane. The regions $\left\langle W \right\rangle<0$ and $\left\langle W \right\rangle>0$ are separated by a line $\Delta x_0(f)$ (solid), accurately predicted by the Markovian bounds (dashed black) but not by the chain ones (blue). Additional parameters are $\Delta t_{\text{m}}=0.5$ and $V_0=5$.
• Figure 4: (Left) Numerical check of Eq. \ref{['eq:W-Ix-Iu-equality']}. Both $\beta\left\langle W \right\rangle$ (solid line) and $\overline{\mathcal{I}_u-\mathcal{I}_{\vec{x}}}$ (red circles) are plotted as a function of $\Delta t_{\text{m}}$, for $\Delta x=0$. The agreement is excellent for all the values of $V_0$ considered, from $V_0=0.1$ ($V_0\ll T$, high temperature) to $V_0=10$ ($V_0\gg T$, low temperature). (Right) Average work and bounds as a function of $\Delta x$. Specifically, the bounds shown are the unavailable information's $\overline{\mathcal{I}_u-\mathcal{I}_{\vec{x}}}$ (red circles) and the Markovian's $-\Delta_{\text{m}} I$. The region in which the Markovian bound improves the unavailable information bound is highlighted in grey. Both panels correspond to $f=0$.
• Figure 5: (Left) Sketch of the harmonic information engine. The centre of the trap may be switched between $-L$ and $+L$ by the feedback controller after measuring the position of the particle $x_n$ at times $t_n=n\Delta t_{\text{m}}$. (Right) Average work and bounds as a function of the uncertainty in the measurement $\Delta x$. Note that $\Delta x$ has no upper bound here because $x\in(-\infty,+\infty)$. Additional parameters are $\Delta t_{\text{m}}=0.5$ and $k=1$.