How does the fundamental assumption of statistical physics make sense?

Question

Consider two systems A and B in thermal contact. System A has $N_A=3$ simple harmonic oscillators and the system B has $N_B=3$ simple harmonic oscillators as well. Each system has a number of energy units $q$ (macrostates of the statistical system) such that $q=q_A+q_B=6$.

By applying the formula for the multiplicity of the systems, $$\Omega(N,q)=\frac{(q+(N-1))!}{q! (N-1)!}$$ and the fact that the total number of multiplicities is given by $\Omega_{total}=\Omega_A\Omega_B$ we obtain the following combinations of energy units {$q_A,q_B$}:

{0,6} = 28,

{1,5} = 63,

{2,4} = 90,

{3,3} = 100,

{4,2} = 90,

{5,1} = 63,

{6,0} = 28.

As far as my understanding goes, this then implies that the most likely outcome for the interacting systems is each having 3 energy units. However, the fundamental assumption of statistical physics states the following:

In an isolated system in thermal equilibrium, all accessible microstates are equally probable.

Surely this would then imply that {0,6} is no less likely to occur than {3,3}, since all accessible microstates are equally probable, thereby making q=0 and q=6 just as likely to occur as q=3 and q=3.

Can someone please explain to me where my misunderstanding lies?

Answer 1

You have to distinguish between "different states" and "number of states" - or, in the words of @Numrok, between "macrostates" and "microstates".

The fundamental theorem refers to "accessible micro states". If I have three white balls and two buckets to put them in, I could put two balls in one and one in the other (that is a macro state); there are in fact many ways (microstates) in which I could achieve that distribution. If I number the balls 1-2-3, the six ways are

bucket
#1 #2
 1  23
 2  13
 3  12
12   3
13   2
23   1

On the other hand, "three balls in one bucket" would only have two states:

bucket
#1  #2
-    123
123    -

So "two in one and one in the other" is more likely - there are more ways in which that distribution can be achieved. It doesn't mean that the other arrangement will never happen; just that it's less likely to be observed if you randomly look at the system.

Answer 2

I know it's late but maybe someone will find it useful someday. I agree, this can be a bit confusing. But try to think it that way. You have 10 coins, and imagine that you toss them in a way that they make a line. Then a microstate HHHHHHHHHH is as probable as microstate HTTTHHTHTT - think of an order, what is the chance that you get exactly HTTTHHTHTT sequence?? But when you count heads and tails, you 'reduce' it to a macrostate. Now 10 heads is not as probable as 4 heads and 6 tails, because if you just count them there are many microstates corresponding to a 4 heads and 6 tails macrostate, but only one microstate correstponding to 10 heads macrostate. Hope this helps.