Understanding Energy States in Complex Systems

[Wledig]

I'm having trouble picturing the energy states for some systems. For instance, I was reading Reif's Fundamentals of Statistical and Thermal Physics, and at some point he talks about the energy states of a pool acting as a heat reservoir interacting with a bottle of wine. The problem is that this is clear for me in something like an Einstein solid, how you can have many states associated to a certain energy but I can't see it how it is possible in this example. When Reif draws a bar graph and says that we could have for example 100 000 states for an energy of 1000 J it doesn't feel right. Maybe something like an Einstein solid is implicit here, and we're taking states to correspond to vibrations of the atoms of the pool?

 

[Henryk]

I don't have that book, but let me try to explain.
It might be a bit easier to understand a much simpler system, say only three particle, each one can have energy of 1,2,3,4,5,6 J (like dice).
If the particles are classical, there are the following states with the total energy of 7 J (using the notation (x,y,z) where x, y, z, are, respectively, energies of the first, second and third particle):
(1,1,5), (1,5,1), (5,1,1), (2,2,3), (2,3,2), (3,2,2), (1,3,3), (3,1,3), (3,3,1), (1,2,4), (1,4,2), (2,1,4), (2,4,1), (4,1,2), (4,2,1)
Therefore, we have 15 possible states of the three particle system and each of them gives us a total energy of 7 J.
This is the case when the particles are distinguishable. If these are idential quantum particles, they are not distinguishable. That means that the state with two particles of energy 1 J and the third with energy of 5 J is the same as the state with the first particle of energy 5 J and the remaining 2 of energy of 1 J ( you can't tell which is the first and which is the third particle!).
So, in quantum statistics, there are only the following states:
(1,1,5), (2,2,3), (1,3,3), (1,2,4), only 4 states.

 

[Wledig]

You see that's the thing, your example is also very nice and neat, I understand it perfectly. It's easy to picture the states in it, like for an Einstein solid or a spin system. It's the pool, the bottle of wine, or rather how we can generalize this for any system and say that there will be many states corresponding to a certain energy that's my problem. I'm guessing we're assuming that there will be internal vibrations in any system and therefore it'll function somewhat like an Einstein solid, allowing many states of energy, but it could be something more subtle than that. I just don't know.

 

[crossword.bob]

Yes, pretty much, though it depends on state. Molecules/atoms can have a number of degrees of freedom within a system. For example, in a gas made up of diatomic molecules, each molecule will have six degrees of freedom:

1. Three degrees of freedom from translational velocity of its centre of mass,
2. Two degrees of freedom from rotational frequency about axes perpendicular to the molecule's primary axis,
3. One degree of freedom from vibrations between the two atoms.

For something like a solid crystal, the only degrees of freedom would be translation and rotation of the whole, plus vibrational modes between atoms.

Each degree of freedom can accept certain quanta of energy; the number of states is the total number of ways you can assign these quanta to individual components.

 

[Wledig]

I see, thanks for confirming that.