Maxwell’s Demons with Finite Size and Response Time

Nearly all theoretical analyses of Maxwell’s demon focus on its energetic and entropic costs of operation. Here, we focus on its rate of operation. In our model, a demon’s rate limitation stems from its finite response time and gate area. We determine the rate limits of mass and energy transfer, as well as entropic reduction for four such demons: those that select particles according to (1) direction, (2) energy, (3) number, and (4) entropy. Last, we determine the optimal gate size for a demon with small, finite response time, and compare our predictions with molecular dynamics simulations with both ideal and nonideal gasses. Also, we study the conditions under which the demons are able to move both energy and particles in the chosen direction when attempting to only move one.

Figure 1

Demon heat and mass transfer rates. Two different one-dimensional systems, with heat and mass currents plotted as functions of the demon’s temporal resolution $\tau$. Theory (solid lines) agrees with simulations (dots). (a) The subsystems have a temperature difference, but with the same number density, ${\beta }_l=1$, ${\beta }_r=0.25$, ${\rho }_l={\rho }_r=100$. Note that the heat transfer rate can be negative for the number demon. (b) The change in system entropy per unit time for the same system as in the left panel. (c) The right subsystem has a lower temperature and a much higher number density than the left subsystem, ${\rho }_l=100$, ${\rho }_r=400$, ${\beta }_l=1$, ${\beta }_r=2$. Here, it can be seen that the energy demon has a negative number current for large $\tau$.

Figure 2

Critical response time. The critical $\tau$ for energy and number demons, along with estimates of the critical values below which no ${\tau }_c$ exists. Above ${\tau }_c$, the demon will transfer heat or mass from right to left instead of from left to right. In both plots, the left subsystem has $T_l=1$, ${\rho }_l=100$, only the parameters of the right subsystem are varied. Left: the energy demon’s ${\tau }_c$ for different right subsystem temperatures, varying right subsystem number density. Right: the number demon’s ${\tau }_c$, for different right subsystem number densities, varying the right subsystem’s temperature.

Figure 3

Two-dimensional demon. A two-dimensional Maxwell demon, operating on an ideal gas and a hard sphere gas. The energy (left) and number (right) rate of demons working with an ideal gas match well with prediction for all three demon types, and the hard sphere gas is qualitatively similar. The system has ${\rho }_l=1.5$, ${\rho }_r=3$, and $T_l=0.0015$, $T_r=0.001$. The dimensions of each subsystem are $30\times 30$, and the particle radius is 0.025. The packing fraction of the left and right subsystems are ${\varphi }_l=0.0029$, ${\varphi }_r=0.0059$.