# Why is the internal energy the expected value of energies of individual particles?

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?

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

## 1 Answer

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 $$\left= \sum_s P_s E_s$$