# Probability of a state containing n particles (Maxwell-Boltzmann)

1. Jun 12, 2014

### akoe

1. The problem statement, all variables and given/known data

For a system of particles at room temperature (use kT = 0.026 eV), the energy of a single particle state is 0.01 eV greater than its chemical potential, so that ε-μ = 0.01 eV. Computer the average occupancy of this single particle state, as well as the probability of the state containing 0, 1, or 2 particles, assuming the particles obey Maxwell-Boltzmann statistics.

2. Relevant equations
Ztot=$\frac{1}{N!}$Z1N

3. The attempt at a solution

I've already solved the first part of the problem - the part about the average occupancy, but I'm having trouble calculating the probability of a state containing n particles. I am pretty sure that it should be P(n)=$\frac{1}{Z}$e$^{-nε/kT}$, because if we view the single particle states as the energies allowed, then if the single particle state has n particles in it, the energy of the state (total) is nε. But, I'm not really sure which Z to use here... is it Zint, Ztot, or Z1? Also, I don't know ε, just ε-μ, so I think that my expression needs to involve those variables...

2. Jun 13, 2014

### vanhees71

Since the give a chemical potential, you are working in the grand-canonical ensemble, i.e., your distribution function is
$$f(\vec{p})=\frac{1}{(2 \pi \hbar)^3}\exp \left [-\beta \left (\frac{\vec{p}^2}{2m}-\mu \right ) \right] \quad \text{with} \quad \beta=\frac{1}{T}.$$