Maxwell's demon and data compression

In an asymmetric Szilard engine model of Maxwell's demon, we show the equivalence between information theoretical and thermodynamic entropies when the demon erases information optimally. The work gain by the engine can be exactly canceled out by the work necessary to reset the demon's memory after optimal data compression in the manner of Shannon before the erasure.

Figure 1

A protocol of the symmetric Szilard engine (left side) and demon's memory (right side). This is an example in which the molecule was found in the right-hand side of the cylinder. In the demon's memory, the state after removing the partition is denoted by an asterisk (*).

Figure 2

The model of an asymmetric Szilard engine. The position of the partition in the demon's memory is the same as that in the case of the symmetric version.

Figure 3

The protocol of data compression for the demon's memory. The dashed blocks express the trivial initial memory state 0 after data compression.

Figure 4

The $p$-$V$ diagram of an alternative erasing process with a heat bath with a lower temperature. The erasure is realized with any path from the asterisk (*) state to the 0 state, which are denoted by a diamond and a circle, respectively. The solid black line represents the standard erasure by an isothermal compression (at temperature $T_H$). Although the demon may want to use a colder heat bath of temperature $T_L$, the entropy increase due to the erasure cannot be smaller than $k_B\ln 2$. The path shown, consisting of two adiabatic processes (red dashed lines) and one isothermal compression (blue dot-dashed line), is the optimal one in terms of entropy increase and attains the Landauer limit of $k_B\ln 2$. Naturally, the entropy cost is the same even if the temperature of the memory is always $T_L$, while the Szilard engine is operated at $T_H$.