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.