Instead of calculating exact state of the gas at every time the demon observes, one may be able to make a more efficient, but less direct, model of a gas which uses statistical probability to determine whether or not the there are molecules in the tunnel when the demon looks. In this way the effects of the demon waiting for very long times between observations may be determined without calculating the state of the gas at any time between the observations.
If the demon's door is on the right side of the tunnel, and the gas is at equilibrium, the each molecule on the left of the door has a probability of being in the tunnel:
where is the area of the tunnel and is the area of the left chamber.
Likewise, if the door is on the left, then each molecule has a chance of being in the tunnel:
Our program proceeds as follows.
This program has the advantages of being considerably faster and able to allow the system to come to equilibrium between each demon observation. In fact, this is simply assumed to be the case. This is at the expense of a more direct correspondence to physical gases.
The program was run with initial conditions approximating those of the tunnel demon. The results of this program had many of the same features of the hard disk model. Again, the three steps in the transfer of a molecule were visible on the graph. The memory used by the demon again increases linearly, as the entropy of the gas decreased irregularly. At later time steps, when the gas was further from equilibrium, the drops in entropy were large when molecules were transferred from the one chamber to the other, but these drops occurred less frequently. See Figure 15.
There is a great deal of variation between runs. There are some sections of some runs during the demon actually seems to maintain the system at a near-constant entropy while reducing the entropy of the gas, the best performance allowed by the second law.