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:

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

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 $$ \left<E\right>= \sum_s P_s E_s $$


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.