Bit reset protocols that obey activity-constrained speed limits do not minimize work for a given speed

Abstract

The goal of thermodynamic optimal control theory is to find protocols to change the state of a system from an initial to a desired final distribution, within a finite time, with the least possible expenditure of work. The optimal protocol is closely linked to the intrinsic dynamics of the system at hand. The fact that these dynamics can vary widely has made a general solution elusive. Recent years have seen great progress by recasting the question in terms of a quantity called total activity, i.e. the average number of jumps between states of the system, rather than the time that the operation is allowed to take. This perspective has allowed for general expressions for the minimal work as a function of the total activity, and the minimal total activity required for a given work. The expression for minimal total activity can be recast as an apparent minimal operation time or speed limit. However, it is unclear whether protocols optimized under a constrained activity actually require the lowest work input for a given operation time. In the context of bit reset, we show that directly minimizing work for a given operation time leads to protocols that require significantly less work to perform the operation than the activity-constrained protocol of the same duration. We show how the resulting protocols differ. One reason for the difference is the fact that the activity rate is not constant over the course of the protocol: it depends on both the transition rates and the distribution of the bit, both of which change during the copy operation. In the limit of long protocol duration, we find an expression for the difference between the resulting minimal work for both optimization schemes, for a general class of dynamics. The time-constrained approach always outperforms the activity-constrained approach for a given constrained duration, and the difference in work can be arbitrarily large.

Figures (3)

• Figure 1: Thermodynamics of bit reset. (a) A bit system with corresponding rates $k_{01}$ and $k_{10}$. The energy difference determines the ratio of the rates through the detailed balance condition. (b) Different relations between the relaxation rate (the sum $k_{01}+k_{10}$) and energy difference are possible, despite the ratio being fixed by the detailed balance condition, e.g. fixing $k_{01}$ (pink) or $k_{10}$ (green), a combination of both which keeps the sum of the two rates equal (black), or their product ($k_{01} \sim e^{\Delta E/ 2 k_B T}$ and $k_{10} \sim e^{-\Delta E/ 2 k_B T}$, orange). All relations are normalized such that $k_{01}=k_{10}=1$ if $\Delta E = 0$ (c) The minimal work to perform a bit reset of an initially symmetric bit with accuracy $s=0.99$ for a given value of $A_T$ equals $W^\text{min}$ given by Eq. (\ref{['eq:Wmin']}), plotted as a red line. This minimal work leads to a forbidden region under the plot. In the limit $A_T \to \infty$, the work reduces to the quasistatic work (horizontal grid line). The minimal total activity $A_T$ at which the work diverges equals $(s-1/2)$ (vertical grid line).
• Figure 2: The protocol that minimizes the work for a given duration $\tau$ does not lead to the minimal work for the corresponding value of $A_T$, and vice versa: the activity-constrained protocol is not optimal for a given duration $\tau$. The two optimization schemes lead to markedly different optimal protocols. We study a reset that takes the bit from a symmetric initial state to state $0$ with accuracy $s=0.99$, and dynamics with a fixed forward rate $k_{01}=1$. Panel (a) shows, as a function of the total activty $A_T$, the minimal irreversible work for that $A_T$, $W_\text{irr}^\text{min}|_{A_T}$ (in red), as well as the irreversible work for a protocol that minimizes the work for a given, yet different duration $\tau$, $W_\text{irr}^\text{min}|_{\tau}$ (in blue), yet with the same total activity. Below the x-axis, we plot the corresponding value of $\tau$ for both protocols. The protocol optimized for a given value of $\tau$ has a higher work than the one optimized for $A_T$. However, for the same value of $A_T$, it performs the operation in a shorter time $\tau$. The inset shows the ratio between the two required amounts of work, $W_\text{irr}^\text{min}|_{\tau}/W_\text{irr}^\text{min}|_{A_T}$. Conversely, in (b), we show $W_\text{irr}^\text{min}|_{A_T}$ (again in red) and $W_\text{irr}^\text{min}|_{\tau}$ (again in blue) as a function of $\tau$, with the corresponding value of $A_T$ plotted below the $x$-axis. Note that this is essentially the same plot as (a), but with the axes for $\tau$ and $A_T$ switched. For a given value of $\tau$, the protocol optimized for a given value of $A_T$ has a higher work than the one optimized for $\tau$, but performs the operation with a lower total activity $A_T$. The ratio $W_\text{irr}^\text{min}|_{A_T}/W_\text{irr}^\text{min}|_{\tau}$ is shown as an inset. The points 1-4 correspond to the same protocols in panels (a) and (b), respectively. Panels (c)-(g) show different performance-related quantities as a function of time $t$, over the course of the protocol. We compare the $\tau$-constrained protocol corresponding to point 1 in panel (a) and (b) ($\tau=6$, plotted in blue) with $A_T$-constrained protocols that have the same duration $\tau$ (point 2 in panel (a) and (b), dashed red lines), the same work (point 3, dot-dashed red lines) and the same total activity (point 4, dotted red lines). The endpoints of the protocols are marked with grey vertical lines. The behaviour of the protocols is discussed in the main text. (c) Energy difference $\Delta E$. (d) Activity rate $a(t)$ on a logarithmic scale. The black line gives the equilibrium activity rate (defined in Eq. (\ref{['eq:eqA']})) for the $\tau$-constrained protocol. (e) Flux $\dot{p}_0(t)$ on a logarithmic scale. (f) Generalized force $\ln (K)$, with $K$ given as in Eq. (\ref{['eq:KA']}). (g) Irreversible entropy production $\dot{W}_\text{irr}$. The area under the blue line and dot-dashed red line is the same since both have the same work.
• Figure 3: The change in activity rate over the course of the protocol $q$ determines the ratio between the minimal irreversible work for the two optimization schemes $R$, in the limit $\tau \to \infty$. Panel (a) shows $\sqrt{R-1}$, with $R = W_\text{irr}^\text{min}|_{A_T}/W_\text{irr}^\text{min}|_{\tau}$ (see Eq. (\ref{['eq:R']})) as a function of $\tau$. Colors correspond to values of $q$, the ratio between the maximal and minimal value of equilibrium activity rate over the course of the protocol (Eq. (\ref{['eq:eqA']})). For dynamics with $k_{01}=1$, starting symmetrically from $p_0(0)=1/2$, the values $q=1.25$, $q=1.923$ and $q=5$ correspond to an accuracy of $s=0.6$, $s=0.74$ and $s=0.9$, respectively. In the limit $\tau \to \tau_\text{min} = -\log(2(1-s)) = \log(q)$ the two protocols are the same and the ratio becomes 1. For larger values of $\tau$, the ratio increases, and the higher $q$, the larger $\lim_{\tau \to \infty} R$. When the reset is started asymmetrically from $p_0(0) =0.25$ (lines with circles), the results are practically indistinguishable from the symmetric case, when compared for the same value of $q$; the values of $q$ now correspond to $s=0.4$, $s=0.605$ and $s=0.85$, respectively. For an energy-rate relation where $k_{10}$ is kept fixed (dashed lines), $\tau_\text{min}=0$ (see main text) and the maximal value of $q$ is $2$. For $q<2$, $s$ is chosen such that $q$ takes the same values as before ($s=0.625$ and $s=0.95$). For the same value of $q$, $\lim_{\tau \to \infty} R$ is again the same. For a class of dynamics which encompasses both $k_{01}=1$ and $k_{10}=1$, $\lim_{\tau \to \infty} R$ can be expressed as a function of $q$ (Eq. (\ref{['eq:limR']})). Panel (b) shows this relation as a function of $\ln(q)$. The difference between the two optimization schemes can increase indefinitely with $q$.