# Stat Mech: distinguishable particles

1. May 7, 2014

### erogard

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

Consider a system of 2 non interacting distinguishable particles in thermal equilibrium at temperature T, and which as two possible energy states available: E1 and E2>E1.

How would you go about finding the average number of particles in energy level E1, and in hight T limit?

3. The attempt at a solution

Is this equal to the ratio of its Gibbs function to the grand partition function? How does distinguishability affect the computation of the latter?

Also how would this result change with a system at equilibrium with a reservoir at T and chemical potential μ?

This seems like a standard SM problem but get can't get my head around it. Any help would be appreciated.

2. May 7, 2014

### dauto

Use the Boltzmann factor to calculate the probabilities. Distinguishability plays a role because for identical particles there are only three available states (2,0), (1,1), and (0,2), where the first number is the occupation number of the first state and so on. If the particles are distinguishable there are four available states since each particle has to possible states.

3. May 8, 2014

### unscientific

Find the partition function $Z_1$ of one particle. If N particles are non-interacting and distinguishable, what is the system's partition function?

Since the partition function is the sum of all possible states, then probability of being in energy $\epsilon_j$ is $\frac{exp(-\beta \epsilon_j)}{Z}$.

Given total number of N particles, How can you use this to find the average number of particles with energy $\epsilon_j$?

(Hint: The answer is between 1 and 2, as the particles are more likely to be in a lower state of energy.)

Last edited: May 8, 2014