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Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Its said that the probability of having the configuration of $n$ particles in the left side is given as $P_n=C(n)/2^N$ with $C(n)$ being the total number of ways in which $n$ particles from $N$ can be placed in the left side.

Why should $P_n=C(n)/2^N$ be the probability? It should be true only if each configuration is equiprobable, but I don't think it is equiprobable. Our initial conditions (position and velocity of particles) will determine the evolution of the states of the system, so it might be that some states occur more frequently than other?

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Yes, you are right, the initial conditions basically decide the probability at any time in the future. If the initial conditions are that all particles have velocity zero and are in the left side, then the probability is 1 that all particles are on the left side all the times. However in statistical mechanics one does not exactly know the initial conditions, so one averages over all possible initial conditions, and then I think it seems logical that the probability is determined by the formula you stated.

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  • $\begingroup$ Is there a rigorous way to it? $\endgroup$
    – Kashmiri
    Commented Jan 6, 2022 at 7:53

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