Probability distributions for Maxwell-Boltzmann, B-E, F-D

[george743]

Homework Statement

From Arfken Math Methods, chapter 23 (prob and stats), problem 23.1.6:

Determine directly or by mathematical induction the probability of a distribution of N (Maxwell-Boltzmann) particles in k boxes with $N_1$ in Box 1, $N_2$ in Box 2, . . . , $N_k$ in the kth box for any numbers $N_j$ ≥ 1 with $N_1$ + $N_2$ + · · · + $N_k$ = N , k < N . Repeat this for Fermi-Dirac and Bose-Einstein particles.

Relevant Equations

N/A

I don't even understand what question is being posed here. The answers given by the author are as follows:

These are numbers, potentially very large ones.

 

[Philip Koeck]

Homework Statement: From Arfken Math Methods, chapter 23 (prob and stats), problem 23.1.6:

Determine directly or by mathematical induction the probability of a distribution of N (Maxwell-Boltzmann) particles in k boxes with $N_1$ in Box 1, $N_2$ in Box 2, . . . , $N_k$ in the kth box for any numbers $N_j$ ≥ 1 with $N_1$ + $N_2$ + · · · + $N_k$ = N , k < N . Repeat this for Fermi-Dirac and Bose-Einstein particles.
Relevant Equations: N/A

I don't even understand what question is being posed here. The answers given by the author are as follows:

View attachment 347872

These are numbers, potentially very large ones.

The answer definitely doesn't give probabilities.

To me it looks like the number of possible arrangements for distinguishable particles at the top and then indistinguishable fermions and bosons.

 

[Philip Koeck]

Homework Statement: From Arfken Math Methods, chapter 23 (prob and stats), problem 23.1.6:

Determine directly or by mathematical induction the probability of a distribution of N (Maxwell-Boltzmann) particles in k boxes with $N_1$ in Box 1, $N_2$ in Box 2, . . . , $N_k$ in the kth box for any numbers $N_j$ ≥ 1 with $N_1$ + $N_2$ + · · · + $N_k$ = N , k < N . Repeat this for Fermi-Dirac and Bose-Einstein particles.
Relevant Equations: N/A

I don't even understand what question is being posed here. The answers given by the author are as follows:

View attachment 347872

These are numbers, potentially very large ones.

There's a lot that's strange here.

k < N is not possible for fermions.

The answer for BE is problematic if k < N_i.

The first answer simplifies to k^N, which is the total number of arrangements of N distinguishable particles among k boxes.

My impression is that the question is about how many ways there are to place N₁
particles in box 1 etc.. This is not really a probability, but it's proportional to one.

For indistinguishable particles I would say there is always only one way.
For example if you have 2 boxes and 5 bosons there is only one way to put 3 particles in box 1 and 2 in box 2. The answer given doesn't make sense to me. Maybe they're not asking about that after all.