As mentioned in Partition Function and Density of States, we reach a paradox when mixing the many non-interacting classical particles. In classical statistical mechanics, the free energy as derived in (9) is
Imagine we have two systems, one has \(N_1\) particles and volume \(V_1\) while the other has \(N_2\) particles and volume \(V_2\). Now we mix the two systems. Our physics intuition would tell us that the free energy of this new system should be \(A = A_1 + A_2\) since they are non-interacting particles. However, the free energy shown in (12) tells us that
which is different from the result we expect, i.e.,
That is, free energy becomes neither intensive nor extensive in our theory.
To make the free energy extensive, we could choose to divide volume by the particle number. Then a new term will appear in the epression for free energy, i.e., \(N\ln N\). On the other hand, we have \(\ln N! = N\ln N -N\) from Sterling approximation. In large systems, we can define the free energy in the following way
which is equivalent to
This definition “solves” the Gibbs mixing paradox. The explanation of this modification requires quantum mechanics.
We can’t just pull out some results from statistical mechanics and apply them to a small system of a few particles. In stat mech we use a lot of approximations like Sterling approximation, many of which are only valid when particle number is huge.