Statistical weight for a degenerate system

[Andrew_]

Homework Statement

Find an expression of the statistical weight for a system of N particles of corresponding energy levels with 'gi' degeneracy.

Homework Equations

W = N! / N_o! N₁!₂! ...N_i!

The Attempt at a Solution

The expression is then :

W = N! g_o^N_o g₁^N₁ g₂^N₂ ... g_i^N_i / N_o! N₁! N₂! ...N_i!

However, I can't prove it. Can you help ?

 

[Jhero]

I would like to offer some assistance in proving this expression for the statistical weight of a system of N particles with corresponding energy levels and degeneracy.

First, let's define some terms. The statistical weight, denoted by W, is a measure of the number of ways in which a given state can be realized in a system. In this case, we are considering a system of N particles with corresponding energy levels, each with a degeneracy of gi. This means that there are gi possible states for each energy level.

Next, let's consider the total number of particles, N, and the number of particles in each energy level, denoted by Ni. We can express the total number of particles as the sum of particles in each energy level, so N = N0 + N1 + N2 + ... + Ni.

Now, to prove the expression for the statistical weight, we can use the definition of permutation, which is the number of ways in which a set of objects can be arranged. In this case, the objects are the particles and the arrangements represent the different states of the system.

Using the definition of permutation, we can express the statistical weight as follows:

W = (N0 + N1 + N2 + ... + Ni)! / (N0! N1! N2! ... Ni!)

However, we can also express the total number of particles in terms of the degeneracy of each energy level, since N = goNo + g1N1 + g2N2 + ... + giNi.

Substituting this into the expression for W, we get:

W = (goNo + g1N1 + g2N2 + ... + giNi)! / (N0! N1! N2! ... Ni!)

Using the property of factorial, which states that (a + b)! = a! b!, we can simplify this expression to:

W = (N!) / (N0! N1! N2! ... Ni!)

And finally, using the definition of factorial, we can rewrite this expression as:

W = N! / (No! N1! N2! ... Ni!)

This proves the expression for the statistical weight of a system of N particles with corresponding energy levels and degeneracy. I hope this helps!