Maxwell’s demon and Smoluchowski’s trap door

A simulation has been performed to reveal the detailed dynamics and statistical behavior of a Maxwell demon of the simplest kind, a trap door held over by a spring inside a box filled with gas molecules. The role of such a demon can be controlled by tuning Smoluchowski’s fluctuations. When the demon is in thermal equilibrium with the rest of the system, it fails to function as designed, and when it is separately subjected to a thermal bath with a different temperature, it creates a temperature or density gradient between the two chambers of the box it divides. As a Maxwell demon, the trap-door device creates more readily a density gradient than that of temperature.

Figure 1

(a) An illustration of the trap-door Maxwell demon. The hinged door between the two compartments can only open inside the left chamber so that fast-moving gaseous particles from the right chamber can move into the left one by lifting the door with significant collision impact. The average temperature of the two chambers is $T_1$, and those for the left and right chambers are $T_L$ and $T_R$, respectively. (b) The trap door may be connected to another door in a third chamber with temperature $T_2$ such that the two doors have to move in unison.

Figure 2

Upper panel: the energy histogram of the gaseous particles in the left chamber at simulation time $t=2000$; the dashed line is the fitted Boltzmann distribution with $T=2000$. Middle panel: temperatures of the two chambers as a function of time given that the trap door is in a thermal equilibrium with the gaseous particles it separates; solid circles, left chamber; open circles, right chamber; solid and dashed lines are a guide for the eye. Lower panel: schematic illustration of the principle of microreversibility; (a) a molecule is moving from the right chamber to the left while the trap door rotates clockwise, and (b) the molecule is moving from the left to the right while the trap door moves counterclockwise. Distance (mass) is in the unit of the gas particle’s diameter (mass), time is in the arbitrary unit, and temperature is evaluated by $T=\frac{1}{2}m{v}^2$ with $k_B$ set to 1.

Figure 3

Upper panel: temperature differences of the two chambers are plotted as a function of time after averaging over an ensemble of 10 000 trajectories; (a) the trap door demon (▵), the temperatures are rescaled to keep their average constant $(T_1=2000)$, and the door temperature is rescaled to remain at $T_2=10$; (b) the logic-gate Maxwell demon (◻), no temperature rescaling is needed. Lower panel: the number of particles in the left chamber as a function of time; (a) the trap door demon (▵); (b) the logic-gate Maxwell demon (◻) with a threshold velocity $v_c=14.1$. Each chamber initially contains 30 particles. The units are the same as in Fig. 2.

Figure 4

Upper panel: the rate of change of the temperature difference between the left and right chambers in the trap door device, $d(T_L-T_R)∕d\mathrm{t}$, as a function of ${\log }_{10}(T_1∕T_2)$; dots, simulation results; solid line, a smooth fit to the simulation results that employs an expression taking into account both the regular heat transfer in the direction of temperature gradient, $C(T_2-T_1)$, and the Feynman term $(e^{-ϵ∕T_1}-e^{-ϵ∕T_2})∕\tau$, where $C$ is a proportionality constant, $\tau$ is the characteristic time between two consecutive collisions, and $ϵ$ is the energy required to lift the trap door [5]. Lower panel: the rate of change of the particle-number difference between the left and right chambers in the trap door device, $d(N_L-N_R)∕d\mathrm{t}$, as a function of ${\log }_{10}(T_1∕T_2)$; dots, simulation results; solid line, a smooth fit to the simulation results with Feynman’s expression $D(e^{-ϵ∕T_1}-e^{-ϵ∕T_2})∕\tau$. $C=0.0033$, $\tau =0.104$, $ϵ=70$, and $D=0.0152$. Data displayed are averaged by an ensemble of 2000 trajectories. The units are the same as in Fig. 2.

Figure 5

A collision occurs when the center of mass of a particle reaches the area labeled by $a$, $b$, $c$, $d$, $e$, or $f$.

Figure 6

The range of the angle $\theta$ would be $(0,\pi )$.