In this Wikipedia page: https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)

.. the total sum of energy in an ideal gas is given as:

$$\langle E \rangle = \sum_s E_s P_s $$

where $s$ runs over all states.

But isn't this just the expected value of the energy for a single particle? Shouldn't the sum of energy be just:

$$ E = \sum_s E_s $$

Why do we take the expected value?

  • $\begingroup$ What is $s$? Tell us $\endgroup$ – FGSUZ Feb 11 at 17:47

Reading carefully the Wikipedia page, one finds that the internal energy is the "ensemble average energy, which is the sum of the microstate energies weighted by their probabilities". Therefore, $E_s$ is the energy of the s-th microstate, where a microstate is the microscopic state of $N$ particles. Once the equilibrium ensemble has been fixed, the probability of each microstate is a function of the energy of the microstate and the ensemble average must be $$ \left<E\right>= \sum_s P_s E_s $$


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