Statistical Mechanics/The Two-State Problem/Multiplicity

Standing alone from the Two-State problem, the multiplicity is a very important concept (which also allows us to calculate entropy (albeit, a very hard way to calculate it)).

Given certain Parameters of a system, the multiplicity function will tell us how many microstates a system can have (i.e., for a gas, there's one possible microstate where all the gas in your room is shoved in one corner, another microstate where it's shoved in another corner, and so on) (by comparison, a macrostate tells us general properties of a system, the system is at a state with energy U0, even though many possible microstates have energy U0).

so, g(X1,...)...

The Meaning of Entropy
An alternate definition of Entropy is the following (if we accept that each state has an equal probability of occurring, this assumption is called The Fundamental Assumption of Thermal Physics):

$$ S = ln (g) $$

This is where the common explanation of entropy being the 'disorder of a system' comes from. If the 'disorder' is large, then the system can be in many possible states, and the entropy increases to allow this information.

In Terms of the Two-State Problem
With the up or down spin statistics we can readily see the connection between micro and macro states. say we have ten sites, then the following are possible microstates (u is up, d is down):

uuuuuddddd

uuuududddd

duuuuuuuuu

And so on...

However, a macrostate might be one where the 2m (where m is the magnetization of a single site), notice that for this single macrostate, we have a number of microstates that would give this outcome:

uuuuuudddd

uuudduuudd

ududududuu

And so on...

And with this example, for any given number of sites (N) we can create the following equation for the multiplicity of the two-state system using combinatorics:

$$ g(N) = 2^N$$

And so, we could find the entropy of the system dependent only on the number of sites:

$$ S = Nln2 $$

A much more involved example would be to construct the multiplicity function in terms of not only the site-number, but the excess spin number. Then, due to our postulates, when the entropied is maximized (which is its equilibrium state), the system is at zero excess spin (in the absence of a magnetic field).

Another interesting example is to consider a two-orbital system, with only two states accessible: a ground and excited state. Using the maximation principle we can automatically determine that this system's equilibrium entropy is $$ln2$$, because the multiplicity at such a point will be either one of two states, making g=2.