# Multiplicity of particles on a lattice

1. Jan 26, 2013

### Arjani

1. The problem statement, all variables and given/known data
A particle can exist in three microstates, with energies E0 < E1 < E2. Consider N >> 1 such particles, fixed on a lattice. There are now n0 particles with energy E0, n1 particles with energy E1 and n2 = N - n0 - n1 particles with energy E2. We have that n_j >> 1 for j = 0, 1, 2. What is the multiplicity?

3. The attempt at a solution

I've established that there are 2N + 1 possible macrostates for this system, but that doesn't really help me further. I've done this problem before, but then it was only two groundstates and the answer would be something like $\Omega = \frac{N!}{\frac{N + M}{2}!\frac{N - M}{2}!}$, so I was thinking it should be something like:

$$\Omega = \frac{N!}{\frac{N - n_1 - n_2}{3}! \frac{N - n_0 - n_2}{3}! \frac{N - n_0 - n_1}{3}!}$$

But somehow I'm guessing that's not going to be correct?