Optimal work extraction and mutual information in a generalized Szil\'{a}rd engine

A 1929 Gedankenexperiment proposed by Szil\'{a}rd, often referred to as "Szil\'{a}rd engine", has served as a foundation for computing fundamental thermodynamic bounds to information processing. While Szil\'{a}rd'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 when $N$ particles and $q$ partitions are available. For a work extraction protocol that equalizes the pressure, 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 over the locations of the dividing walls, and find that there exists a critical value $N^{\star}(q)$ below which the extracted work is maximized by a symmetric configuration of partitions and above which the optimal partitioning is an asymmetric one. Overall, the average work is maximized for $\hat{N}(q)partition. We calculate asymptotic values for $N\rightarrow \infty$.