Molecules Distributed between Two Compartments

A box has two compartments connected by a hole. There are molecules in the left compartment and in the right.
To update the original values of and , generate a pseudorandom number between 0 and 1; if , a molecule is added to the left and one is taken away from the right. If , a molecule is added to the right and one is taken away from the left.
With a large number of molecules on one side and a small number on the other side, the system tends to equilibrium after some steps, with small fluctuations afterward. Clearly the entropy of the system increases, evolving to a more disordered state and giving an "arrow of time", making it possible to distinguish the past from the future. But experimenting with a small number of particles is not enough to distinguish past from future. This is because a system with a large number of molecules has a huge number of microstates, and a small system has just a few.

THINGS TO TRY

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The entropy is .
The number of possible configurations for a particular number of particles in the left compartment is equal to the binomial coefficient .
This experiment is quoted by Stephen Wolfram as an early inspiration that led him to write A New Kind of Science.
References:
F. Reif, Statistical Physics, Berkeley Physics Course, Vol. 5, New York: McGraw-Hill, 1967.
R. M. Eisberg and L. S. Lerner, Physics: Foundations and Applications, Vol.
2, New York: McGraw-Hill, 1981.
See Statistical Mechanics and Microstate on Wikipedia for more information.
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