11.11 The Big Lie of Distinguishable Particles

If you try to find the entropy of the system of distinguishable particles that produces the Maxwell-Boltzmann distribution, you are in for an unpleasant surprise. It just cannot be done. The problem is that the number of eigenfunctions for $I$ distinguishable particles is typically roughly $I!$ larger than for $I$ identical bosons or fermions. If the typical number of states becomes larger by a factor $I!$, the logarithm of the number of states increases by $I\ln{I}$, (using the Stirling formula), which is no longer proportional to the size of the system $I$, but much larger than that. The specific entropy would blow up with system size.

What gives? Now the truth must be revealed. The entire notion of distinguishable particles is a blatant lie. You are simply not going to have 10$\POW9,{23}$ distinguishable particles in a box. Assume they would be 10$\POW9,{23}$ different molecules. It would a take a chemistry handbook of 10$\POW9,{21}$ pages to list them, one line for each. Make your system size 1,000 times as big, and the handbook gets 1,000 times thicker still. That would be really messy! When identical bosons or fermions are far enough apart that their wave functions do no longer overlap, the symmetrization requirements are no longer important for most practical purposes. But if you start counting energy eigenfunctions, as entropy does, it is a different story. Then there is no escaping the fact that the particles really are, after all, indistinguishable forever.