Optimal work extraction and mutual information in a generalized Szilárd engine

A 1929 Gedankenexperiment proposed by Szilárd, often referred to as “Szilárd's engine”, has served as a foundation for computing fundamental thermodynamic bounds to information processing. While Szilárd's original box could be partitioned into two halves and contains one gas molecule, we calculate here the maximal average work that can be extracted in a system with $N$ particles and $q$ partitions, given an observer which counts the molecules in each partition, and given a work extraction mechanism that is limited to pressure equalization. We find that the average extracted work is proportional to the mutual information between the one-particle position and the vector containing the counts of how many particles are in each partition. We optimize this quantity over the initial locations of the dividing walls, and find that there exists a critical number of particles $N^{★}\left(q\right)$ below which the extracted work is maximized by a symmetric configuration of the $q$ partitions, and above which the optimal partitioning is asymmetric. Overall, the average extracted work is maximized for a number of particles $\widehat{N}\left(q\right)<N^{★}\left(q\right)$, with a symmetric partition. We calculate asymptotic values for $N\to \infty$.

Figure 1

Schematic representation of a Szilard engine with $q=5$ partitions and $N=20$ particles. Panel (a) depicts the initial state, where the length of each partition is given by ${\ell }_i$ and the red arrows indicate in which direction each wall is going to move as the engine performs work. The final state, which is assumed to be reached quasistatically is shown in panel (b), where the length of each partition is given by ${\ell }_i^{\left(\mathrm{eq}\right)}=k_{i}L/N$ and thus the pressure is the same in all of them. The values indicated in the lower panel corresponds to the counts obtained by setting $L=1$.

Figure 2

The mutual information $NI[X,\mathit{K}]$ as a function of the position of the wall $p=\ell /L$, for a Szilárd's engine containing $N$ particles. The different curves correspond to different values of $N$, while the dotted line is the asymptotic limit $\frac{1}{2ln\left(2\right)}\simeq 0.7213$ bits.

Figure 3

${\mathcal{F}}_N\left(n\right)/ln\left(2\right)$ [in bits] as a function of $n$, plotted for various values of $N$. In the limit of an infinite number of particles the maximum of ${\mathcal{F}}_N\left(n\right)$ is achieved at $\widehat{n}\simeq 1.338$ and has a value of ${\mathcal{F}}_{\infty }\left(\widehat{n}\right)\simeq 0.8371$ bits.

Figure 4

Plot of ${\mathcal{F}}_N^{\prime }\left(n\right)$ (solid blue line) and ${\mathcal{F}}_N^{\prime \prime }\left(n\right)$ (solid red line) as a function of $n$, for $N=10$. As we can see here, for a given value of $\lambda$ (dashed orange line), there are two solutions of ${\mathcal{F}}_N^{\prime }\left(n\right)=\lambda$ (represented here with black circles). These, denoted as $n^{-}$ and $n^{+}$ are such that ${\mathcal{F}}_N^{\prime \prime }\left(n^{-}\right)<0$ and ${\mathcal{F}}_N^{\prime \prime }\left(n^{+}\right)>0$.

Figure 5

Plot of the mutual information for the symmetric partition $q{\mathcal{F}}_N(N/q)$ (orange rhomboid markers) and the optimal mutual information ${\mathcal{I}}_q\left(N\right)$ (brown triangle markers) as a function of $N$, in units of nats. Top panel: Plots for a particular value on the number of partitions ($q=6$). As we can see, there exists a critical value $N^{★}\left(q\right)$ (vertical solid red line) on the number of particles, below which the optimal partition is the symmetric one, while above it the optimal partition corresponds to an asymmetric partition. Moreover the optimal mutual information becomes maximal for a particular number of particles $\widehat{N}\left(q\right)<N^{★}\left(q\right)$ ($\widehat{N}\left(s\right)$ is shown by the vertical solid blue line). Bottom panel: plot for various values of $q$, shown the same general features.