# Difference between energy level, energy of a system and energy of microstate

In statistical mechanics we recently took and studied the microcanonical/canonical and grand canonical ensemble. Because of the different terminology I am having a hard time understanding some concepts, their difference and how they relate to each other. For quite sometime now even though I kind of understand the the above concepts, I feel like there is something missing in my understanding of the things. Initially I want to describe how I understand these concepts. If there is a mistake please let me know:

Lets say we have a system made out of N particles, then:

$$E_r=\Sigma_i n_i \epsilon_i$$ is the energy of a microstate, meaning each particle, has a certain value of position $$\vec r$$ and momentum $$\vec p$$.

$$\epsilon_i$$ is the energy value that $$n_i$$ particles can have.

The energy of the system E should be:

$$E = =\Sigma_r P_rE_r$$.
Explanation : The microstates in which the system can be found at a moment in time can have the same energy value or an energy value which is between : [E - dE,E]. This difference should be big enough to include a certain nr of microstates but small enough so the energy of the system can be accurate till a certain value. Then the mean of the different energies of different microstates which belong in this energy interval gives us the energy of the system. I also have to point out that by considering an arbitrary energy interval, all the other microstates in which the system can be found but have an energy that doesn't belong in this interval, all these microstates are excluded from our calculations.

This is how I understand these different concepts and how they are related to each other. Am I correct in this or is there something wrong with the way I interpret them.

And as for my questions, I have the following:

1. Are the above concepts the same for all 3 ensembles ? (micro,canonical and grand canonical)?

2. Often we talk in our class about the energy level of the system. As i pointed above we have different energies, i.e the energy of the whole system, the energy of a microstate, and the energy $$\epsilon_i$$ that a certain nr. of particles can have. To which of these three is the energy level referred to?

3. When we have a quasi-static heat exchange, is it correct to say that the energy of the microstates does not change, but the number of microstates (we add more with a higher/lower energy value) changes?$$\delta Q_{qs}=\Sigma_r dP_rE_r$$. The change in probability (for a microstate) changes because the nr. of them changes, and therefore the probability for each must change, is that correct?

4. For quasi-static work, we have a change in the energy value of each microstate, because we change external parameters like Volume, or Number of particles, which affect the energy of the microstate. After the change the microstate whose energy before the work was $$E_r$$ now it's $$E_r'$$. Is it correct to say that this is what happens for quasi-static work?

5. For a moment of time t, is it correct to say that a macrostate= a microstate ?

Thanks!

## 1 Answer

Your concepts are quite right in general. I'd like to point out that "each particle has a certain position and a certain momentum" is a very classical point of view. You can also apply it to quantum systems (in fact, they are probably more usual than classical ones). In quantum systems, a particle has a cerain energy for the fact of being in that state, jsut like electrons in Bohr's model, which have a certain energy for the fact of being in an orbit.

That said,

1. Yes, the concepts are the same for those 3 ensembles, but not only for those ones... there are many more ensembles.

2. Indeed, the energy level of a system doesn't make much sense to me... It's either "the (total) energy of a system" or "one energy level in a system"...

3. Yes, the energy can vary either because more particles rise to higher levels, or because the energy levels themselves change. So Points 3 and 4 are correct.

4. No, a macrostate is not a fixed photograph of a microstate. They are different things, so it doesn't make sense to equate them. A microstate contains all information (position and momentum of all particles in a certain time t). However, a macrostate is defined by macroscopic variables, such as pressure, volume, temperature... Those cquantities are not microscopical, they are averaged-over-time quantities, and they do not provide exact information of all particles, but only mean values of some properties, and remember that systems usually have about $$10^{23}$$ particles...