Quantum Szilard Engine

The Szilard engine (SZE) is the quintessence of Maxwell’s demon, which can extract the work from a heat bath by utilizing information. We present the first complete quantum analysis of the SZE, and derive an analytic expression of the quantum-mechanical work performed by a quantum SZE containing an arbitrary number of molecules, where it is crucial to regard the process of insertion or removal of a wall as a legitimate thermodynamic process. We find that more (less) work can be extracted from the bosonic (fermionic) SZE due to the indistinguishability of identical particles.

Figure 1

Schematic diagram of the thermodynamic processes of the classical SZE. Initially a single molecule is prepared in an isolated box. (A) A wall depicted as a vertical gray bar is inserted to split the box into to two parts. The molecule is represented by the dotted circles to indicate that at this stage we do not know in which box the molecule is. (B) By the measurement, we find where the molecule is. (C) A load is attached to the wall to extract a work via an isothermal expansion at a constant temperature $T$.

Figure 2

Schematic diagram of the quantum SZE containing three molecules. (I) Three molecules are prepared in a closed box with size $L$. (II) A wall, depicted by a vertical gray bar, is isothermally inserted at location $l$. The process $(\mathrm{I})\to (\mathrm{II})$ is called “insertion.” (III) The information on the number of molecules, $m$, on the left is acquired by the measurement. The process $(\mathrm{II})\to (\mathrm{III})$ is called “measurement.” (IV) The wall moves and undergoes an isothermal expansion until it reaches its equilibrium location denoted by $l_{\mathrm{eq}}^m$. The process $(\mathrm{III})\to (\mathrm{IV})$ is called “expansion.” Finally the wall is isothermally removed to complete the cycle. The process $(\mathrm{IV})\to (\mathrm{I})$ is called “removal.”

Figure 3

$f_0$ as a function of $T$ for bosons (solid curve), fermions (dashed curve), and classical particles (dash-dotted line) in the case of the infinite potential well. The temperature is given in units of $E_1(L)/k_B$. (a) Three possible ways in which two identical bosons are assigned over two states. (b) Four possible ways in which two distinguishable particles are allocated over two places. The inset shows $W_{\mathrm{tot}}/W_c$ as a function of $T$.