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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, 2018 at 22:53
  • $\begingroup$ @knzhou What about the continuous case? Can we do the integral between states 1 and 2? $\endgroup$ Apr 4, 2018 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, 2018 at 10:30

1 Answer 1

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  • Discrete case The probability of canonical microstate $i$ is $$ p_i = \frac{e^{-\beta E_i}}{Z} $$ The probability to find the system in a region $\mathcal R$ of microstates is the sum over all these probabilities: $$ p_{i\in\mathcal R} = \sum_ {i\in\mathcal R} \frac{e^{-\beta E_i}}{Z} $$

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  • Continuous case The probability to find the canonical microstate in the region $(\Gamma,\Gamma+d\Gamma)$ of phase space is $$ P(\Gamma) d\Gamma = \frac{e^{-\beta E(\Gamma)}}{Z} d\Gamma $$ where $\Gamma = (\mathbf r_1,\cdots; \mathbf q_1\cdots)$ is the vector with the positions and momenta of all particles. The probability to find the system in a region $\mathcal R$ of microstates is the sum over all these probabilities: $$ P(\Gamma \in \mathcal R) = \frac{1}{Z} \int_{\Gamma\in\mathbf R} e^{-\beta E(\Gamma)} d\Gamma. $$

In both cases the result can be expressed in the form $$ p(\mathcal R) = \frac{Z_{\mathcal R}}{Z} $$ with the denominator calculated by adding/integrating the factor $e^{-\beta E}$ over all phase space and the numerator is the summation/integral of the same factor in region $\mathcal R$.

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