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

TL;DR

This paper generalizes Szilard's engine to multiple particles and partitions, showing how the maximum work extractable relates to mutual information and how optimal partitioning depends on particle number.

Contribution

It introduces a model for work extraction with multiple particles and partitions, optimizing partition placement to maximize work based on mutual information.

Findings

Maximum work is proportional to mutual information.

Optimal partition configuration depends on particle number.

Asymptotic behavior analyzed for large particle numbers.

Abstract

A 1929 Gedankenexperiment proposed by Szil\'ard, often referred to as "Szil\'ard's engine", has served as a foundation for computing fundamental thermodynamic bounds to information processing. While Szil\'ard'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 $N^{\star}(q)$ 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 $\hat{N}(q)<N^{\star}(q)$, with a symmetric partition. We calculate asymptotic values for $N\rightarrow \infty$.