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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 $2$?

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  • $\begingroup$ Yes, that's how you find the probability for being in state 1 or state 2. $\endgroup$ – knzhou Apr 3 '18 at 22:53
  • $\begingroup$ @knzhou What about the continuous case? Can we do the integral between states 1 and 2? $\endgroup$ – Drew Apr 4 '18 at 0:49
  • $\begingroup$ 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). $\endgroup$ – user8153 Apr 8 '18 at 10:30

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