# Does the partition function define probability of being in multiple states?

The partition function is defined as a sum over all microstates $j$ as:

$Z=\sum_{j}exp(-\beta E_j)$

or

$Z=\int_{-\infty}^{\infty} exp(-\beta E)dE$

if the states are continuous.

We can use $Z$ to get the probability $p_i$ that a system has microstate $i$ via:

$p_i=\frac{exp(-\beta E_i)}{Z}$

Now, does this work for finding the probability that a system is in multiple states?

For example, does

$p_{1,2}=\frac{exp(-\beta E_1)+exp(-\beta E_2)}{Z}$

define the probability of the system being in states $1$ and $2$?

If we have continuous states between $E_1$ and $E_2$, does

$p_{1,2}=\frac{\int_1^2 exp(-\beta E)dE}{Z}$

define the probability that the system has a microstate between states $1$ and $2$?

• Yes, that's how you find the probability for being in state 1 or state 2. – knzhou Apr 3 '18 at 22:53
• @knzhou What about the continuous case? Can we do the integral between states 1 and 2? – Drew Apr 4 '18 at 0:49
• Your equations for the continuous case are not correct: they are missing the density of states (the part that makes the partition function dependent on the system you are describing). – user8153 Apr 8 '18 at 10:30