Ten initial state files, each with fifty molecules with random momenta and positions within the chambers, were generated. Each of these initial states were then run for one thousand collisions so that they could thermalize, because the random state generator did not generate initial states with the proper velocity distributions. The state file was then saved.

Each of the thermalized states were then used as initial conditions for demon runs of ten thousand collisions. The demon checked the tunnel every one hundred time units. (There were approximately two collisions per time unit.) After each run, the program recorded the state of the system, the demon's memory, and the entropy at each of the demon observations. Another program was then run to estimate the actual of information content per bit of the demon's memory, using the Markov chain approximation.

The results of individual demon runs have some interesting features. As expected, the entropy of the gas drops as the demon operates. As time goes on, the decreases in the entropy of the gas become larger. After the demon has been operating for a while, the gas is far from equilibrium, and the shift of an individual molecule has a larger affect on the entropy. One can follow the demon's operation on the graph of the entropy of the gas. Each transfer of a molecule involves two changes in entropy, each corresponding to a time at which the demon moves the doors. The first change in entropy occurs when the demon observes a molecule in the tunnel switches the doors. At this point the molecule stops being counted in the left chamber and is counted instead in the right chamber; this results in a drop in the entropy. (This drop is not quite as large as it would be if we merely pushed the molecule from one side to the other, because the area of the right chamber increases by the volume of the tunnel.) When the molecule exits the tunnel and the demon is able to return the doors to their original positions, the entropy drops again as the chamber with the larger number of molecules is reduced in area. (See Figures 10, 11, 12, 13. Demon run 3 is omitted because this run failed to record all relevant information.)

Because of random fluctuations, the gas does not begin exactly at equilibrium. In cases in which more molecules are in the left chamber than the right to begin with, the demon may actually increase the entropy of the gas before it can reduce it. This can be seen in runs four and five, for example.

Because the demon checks the gas at regular time intervals, the entropy of the demon increases linearly with time. The entropy of the system as a whole is therefore the addition of the gas entropy and the linearly-increasing demon entropy. The slope of the linear term depends on the amount of information gained in each demon observation. We find that the entropy of the system, even after the demon's memory is corrected for correlation, rises with time. There are fluctuations in the entropy of the system where the entropy will decrease over a small amount of time, but these decreases are unpredictable and cannot be used to do work. Over time, the operation of the demon increases the entropy of the system, even though the gas entropy decreases.

The equilibrium states generated were used again as the initial conditions of a demon which observes the tunnel every two time steps instead of every one hundred. Because of the increased frequency of observations of the entropy, the double drop which occurs when the demon moves the molecule is more clearly apparent. With the increased number of observations, the demon is able to reduce the entropy of the gas more rapidly, because it does not miss as many of the molecules that enter the tunnel and exit it quickly, possibly before the demon has observed it there. Despite this, the entropy of the system, including the demon's memory, increases at a faster rate than before. See Figure 14.

As the demon observes more often, it uses more of its memory a given time interval. Successive bits in this memory, however, are far more correlated. Each time the tunnel is either empty or occupied, the demon observes it many times in that state. This correlation is roughly taken into account by the Markov chain approximation, and does result in a smaller amount of information per bit. In spite of this reduction, the drop in the amount of information per bit is not enough to prevent the entropy of the system from rising even faster than it did when the observations were made less frequently.

One possible reason for this is that the demon, by retaining the number of
times it observes a molecule before it leaves the tunnel, is essentially
gaining information about the *x* velocity of the molecule. This information
is never used by the demon to reduce the entropy of the gas, and is
therefore wasted.

To be as efficient as possible, a demon should gather only the information it can use to reduce the entropy of the gas. In this case, the demon should wait a very long time between observations so it gathers as little information about the velocity of the molecules as possible. This is impractical with the program described so far, so a less cumbersome model may be useful.

Mon Jun 16 13:53:44 EDT 1997