And so I'll ask you to remember that we discussed ensembles and we had a vision

of near infinite water cooler, filled with bottles.

Where the bottles were systems, and the collection of bottles was called the

ensemble. And I worked with that ensemble in

particular, a particular flavor of ensemble, the NVT ensemble.

Which is called the canonical ensemble. That is where you are specifying a

number, a volume, and a temperature. I want to work, instead, with a somewhat

different ensemble now. Still a collection of bottles, that's the

ensemble. Still, each system is a bottle.

But I'm actually going to control N, V, and E.

Not the temperature, but rather, the internal energy, E.

I'm going to use E here instead of U. That is a micro canonical ensemble, as

oppose to a canonical ensemble. So the difference being that E is a state

variable, not T. Now, even though every system has the

same energy. That doesn't mean that their all in the

same state. Because there can be degeneracy.

So there can be many states that have the same energy.

We indicate that by this capital omega. So that is the degeneracy associated with

a given energy. And in molecular systems and macroscopic

volumes, that number is typically enormous.

Ten to very large powers. And so, I want to actually look at

multinomial statistics. And so you may have heard of binomial

statistics we're actually going to be a little more general than that but I'll

make it clear here hopefully what I'm talking about.

So let's take w, this number to be the number of ways of having a certain number

of systems in a given state call it state 1 and so I'll indicate that by a

subscript 1. The subscript tells you which state are

you in. In another number of systems in state 2,

so that's A sub 2, and so on, there can stage three, four, however many I care

about. And let's the systems be distinguishable

for a moment. So in that case the number of ways I can

do that. Well, first off, how many ways can I

order things? So if I am given a collection of things.

And the total number of things is capital A.

How many ways can I put them in order? Well, of course, I have capital a choices

for the first thing. And I have capital A minus 1 choices for

the second thing so that's a times a minus 1.

Then capital A minus 2 for the third thing and so on.

So the number of ways I can order them is a factorial.

But I'm not interested in only how might I order them But I'm going to put them

into subclasses, that's what the little a's are.

a sub1 systems in a given state, a sub2 systems in a second state.

And I don't actually care about how they're ordered in their sub states.

All I cared about was that there was a certain number there.

And so I end up dividing a factorial by the number of ways I can organize them in

their substates. Because I've overcounted those

possibilities. They're all the same possibility as far

as I'm concerned. So, the number in each individual state

factorial. That's how many ways I could reorder

those. Appears in the denominator.

And so, I'll just write that more simply as a product over all the number of

states I'm considering of the number of of systems in that state, factorial.

And just to show an example. Let's say that I have four total systems

and hence capital a is four and I asked the question well how many ways can I

arrange two of them in state one and two of them in state two and that uses up all

my possibilities that, that is four systems.

So none in state 3, and none in state 4. And I'll just explicitly draw them.

So here are 4 states. And I basically will just label them.

This one's in 1, 1, 2, 2. 1, 2, 2,1.

1, 2, 1, 2, and son on. And if you exhaust all the possibilities,

and convince yourself. You see I've got 6 written here.

And so let's just check that out for a moment.

4 factorial, that's 4 times 3 times 2 times 1 is 24.

And divided by 2 factorial is 2. So, 24 divided by 2 times 2, 24 over 4,

6. Great, this works, okay.

It had to work, that's what the statistics say.