Boltzmann Distribution Problem about Defining Energy
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The solution discusses the Boltzmann distribution problem about defining energy.
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We have N objects and we need to choose objects to put in box 1.
There is no importance to the order, hence the number of possible choices are:
(1.1)
Now we are left with a total of particles and we need to choose from group objects to put in box b.
the number of possible configurations to this is:
(1.2)
We can continue this process for box k-1, and the number of ways to put objects in this box is:
(1.3)
And for box k:
(1.4)
And so forth and so on.
These are the number of ways we can arrange individual boxes.
Hence the number of ways we can distribute the objects in all the boxes is:
(1.5)
If we write it explicitly we see that factors in the numerator and denominator of successive terms cancel each other out:
(1.6)
However, if we distribute all the objects then
(1.7)
And:
(1.8)
And we are left with:
(1.9)
This is the number of possible ways we can distribute N objects in k boxes. where box i can contain Ni objects.
Before we go on let's understand the idea behind degeneracy.
For that we look at a quantum one dimensional infinite potential well.
(1.10)
The one dimensional ...
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