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

Question

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 $2$?

Answer 1

• 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: $$ p_{i\in\mathcal R} = \sum_ {i\in\mathcal R} \frac{e^{-\beta E_i}}{Z} $$

$~$

• 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 $\mathcal R$.