# Does the N! Gibbs factor SAVE or DESTROY extensivity?

1. Jan 11, 2012

### nonequilibrium

So we all know the argument that shows that a factor $\frac{1}{N!}$ makes entropy extensive: it makes the entropy the same for system A and system B where system A is a box of indistinguishable/indistinguished particles and where B is system A is split into two isolated parts.

But regard this slight variation, where system A is the same as above, and where system B is again a splitting of system A, but now we let the two parts interchange energy: i.e. instead of isolated, the two parts are simply closed (= interchanging energy but not particles).

Wouldn't we expect the relation $\Omega(E) = \int \mathrm d E_A \Omega_A(E_A) \Omega_B(E-E_A)$ to hold for the latter A and B (a form of extensivity)? However, this formula is only true if we do not include the Gibbs factor!