I am having difficulty finding the partition function of a system with two particles, each of which can be in any of three states with energies $0, \epsilon, 3\epsilon$. The system is in contact with a heat bath at temperature $T$.

Since the system is in contact with a heat bath at temperature $T$, I believe that I'm going to need to use a canonical ensemble. Also, I think that the partition function should be a product of six terms, since there are $6$ possible ways to assign the possible energies to the energies to two particles.

Any help is appreciated. I will update this post periodically with my attempts.


1 Answer 1


Assuming the particles don't interact, there are nine possible microstates, and six unique energy levels. The energy levels and multiplicities are: (0,1), (1,2), (2,1), (3,2), (4,2), and (6,1). So for example, there are two states with energy level 3.

From there, since we are assuming a canonical ensemble, your partition function becomes:

$$ Z = \sum_{(\epsilon,\mu)}\mu\exp(-\beta \epsilon) $$

Where the sum is taken over all (energy, multiplicity) pairs.

  • $\begingroup$ Thanks. I just cannot figure out why this equals the reciprocal of the partition function and not the partition function itself? $\endgroup$
    – Julie
    Commented Sep 23, 2018 at 7:08
  • $\begingroup$ Good spotting; usually the convention would be that that would not be the reciprocal, however I'm used to writing and using it in this way. I've edited it to the more conventional form. $\endgroup$
    – Al Nejati
    Commented Sep 23, 2018 at 7:10

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