Figure 1

Modeling the chamber of gas by a one-dimensional infinite square well. (a) The chamber is modeled as a one-dimensional infinite square well with width $L$ where the particles are confined. The inner lines in the well represent the single-particle energy levels of the well. (b) After the insertion, the chamber is split into two chambers, which are modeled as two 1D infinite square wells with widths $l$ and $L-l$, respectively. The inner lines represent the single-particle energy levels.Reuse & Permissions

Figure 2

The states of the chamber and the demon (a) before and (b) after the expansion process. (a) After the measurement, the demon has revealed the number of the particles in the right compartment $i$ and recorded this information in its memory. If the demon is error free, then its inner state (or memory) will be $|i\rangle$. However, when the demon contains some errors, there would be some mistakes in the demon's memory. For example, subfigure (2) shows a situation when $i=0$ and the demon is in state $|1\rangle$. Thus, there are total ${(N+1)}^2$ situations after the measurement for a system with $N$ particles. Here the total of nine situations are listed for $N=2$. (d) After the measurement, the partition acts as a piston under the control of the demon. If the demon is in the state $|i\rangle$, the piston will move to a new position $l_i$. Such movement is called controlled expansion. Here the results of the expansion for all nine situations are listed for $N=2$.Reuse & Permissions

Figure 3

The removing process. (a) Before the removing, the state of the working substance may not be an equilibrium state because of the “impenetrability” of the partition (see Sec. 3c). (b) When the partition is lifted a little, it is no longer impenetrable, and the particles can fly between the two sides of the chamber freely to reach the equilibrium state. (c) The partition keeps being lifted and finally can be removed isothermally.Reuse & Permissions

Figure 4

Erasing the memory of a three-energy-level demon reversibly. Suppose that the populations of the three levels after the removing process the partition are $p_0^{\left(1\right)}=0.3$, $p_1^{\left(1\right)}=0.5$, and $p_2^{\left(1\right)}=0.2$ as in (a) and the initial state's populations of the three levels are $p_0^{\left(0\right)}=0.7$, $p_1^{\left(0\right)}=0.2$, and $p_2^{\left(0\right)}=0.1$ as in (d). In step 1 (a),(b), the demon is isolated from the heat bath, and the energy-level spacings are adjusted from ${\Delta }_i$ to ${\Delta }_i^{\prime }$ for $i=1,2$. During this process, the populations of the levels remain constant. The new level spacings are chosen so that the demon has an effective temperature $T_2=1$ after the adjusting, which implies ${\Delta }_1^{\prime }=-0.5108$ and ${\Delta }_2^{\prime }=0.4055$. In (b) the minus sign in front of ${\Delta }_1^{\prime }$ is due to the negative value of ${\Delta }_1^{\prime }$. In step 2 (b),(c), the demon is in contact with a real heat bath at temperature $T_2$, and the energies of the levels are adjusted. In this step the populations of the levels change while its energy spacings change. The goal of this step is to let the populations of the levels return to be $p_0^{\left(0\right)}$, $p_1^{\left(0\right)}$, and $p_2^{\left(0\right)}$. Thus the new level spacings are ${\Delta }_1^{\prime \prime }=1.2528$ and ${\Delta }_2^{\prime \prime }=1.9459$. In step 3 (c),(d), the demon is isolated from the heat bath again, and its level spacings are adjusted to return to its initial values. The isolation guarantees that the populations of the levels are not changed. Thus the state of the demon has returned to its initial state.Reuse & Permissions

Figure 5

Work extracted $W$ as the function of insertion position $l/L$ and the working temperature $T_1$. (a) Bose case with $N=4$. (b) Fermi case with $N=4$.Reuse & Permissions

Figure 6

Asymptotic erasure scheme for $T_2=0$.Reuse & Permissions

Figure 7

Degenerate ground states for a Fermi gas. When the Fermi surface contains more than one single-particle states, there is a chance that the ground state of the Fermi gas is degenerate. For example, the states in (a) and (b) are two ground states of the system.Reuse & Permissions

Figure 8

$W/T_1$ as the function of the insertion position $l/L$ and the working temperature $T_1$. (a) A Bose system with $N=2$. (b) A Bose system with $N=3$. (c) A Bose system with $N=4$. (d) A Fermi system with $N=2$. (e) A Fermi system with $N=3$. (f) A Fermi system with $N=4$.Reuse & Permissions