Counting problems and Statistical Physics

 In this post we will analyze four variants of a counting problem and introduce it's relation with statistical physics.

 The problem

We have N balls inside an urn, we want to count the number of possible outcomes when we draw M balls.

 The variants

We have to choose if the sequence of the balls that we draw matters and if every time we draw a ball we replace it or not.

Case 1: Maxwell-Boltzmann statistics

Ordered sample with replacement

(1, 1)	(1, 2)	(1, 3)	(1, 4)
(2, 1)	(2, 2)	(2, 3)	(2, 4)
(3, 1)	(3, 2)	(3, 3)	(3, 4)
(4, 1)	(4, 2)	(4, 3)	(4, 4)

Table 1: The 16 possibilities for N = 4, M = 2

For a generic value of N and M, the number of possibilities is:

N^M

This case is equivalent to arrange M distinguishable particles inside N cells where the cells can contain multiple particles. For example classical particles, witch we assume that are distinguishable.

Case 2

Ordered sample without replacement

(1, 1)	(1, 2)	(1, 3)	(1, 4)
(2, 1)	(2, 2)	(2, 3)	(2, 4)
(3, 1)	(3, 2)	(3, 3)	(3, 4)
(4, 1)	(4, 2)	(4, 3)	(4, 4)

Table 2: The 12 possibilities for N = 4, M = 2

For a generic value of N and M, the number of possibilities is:

N! / (N - M)!

This case is equivalent to arrange M distinguishable particles inside N cells where each cell can contain only one particle.

Case 3: Bose-Einstein statistics

Unordered sample with replacement

(1, 1)	(1, 2)	(1, 3)	(1, 4)
(2, 1)	(2, 2)	(2, 3)	(2, 4)
(3, 1)	(3, 2)	(3, 3)	(3, 4)
(4, 1)	(4, 2)	(4, 3)	(4, 4)

Table 3: The 10 possibilities for N = 4, M = 2

For a generic value of N and M, the number of possibilities is:

(N + M - 1)! / M! (N - 1)!

This case is equivalent to arrange M indistinguishable particles inside N cells where the cells can contain multiple particles. For example bosons (proton) are the indistinguishable particle witch do not obey Pauli exclusion principle. This is the so called Bose-Einstein statistics.

Case 4: Fermi-Dirac statistics

Unordered sample without replacement

(1, 1)	(1, 2)	(1, 3)	(1, 4)
(2, 1)	(2, 2)	(2, 3)	(2, 4)
(3, 1)	(3, 2)	(3, 3)	(3, 4)
(4, 1)	(4, 2)	(4, 3)	(4, 4)

Table 4: The 6 possibilities for N = 4, M = 2

For a generic value of N and M, the number of possibilities is:

N! / M! (N - M)!

This case is equivalent to arrange M indistinguishable particles inside N cells where each cell can contain only one particle. For example fermions (electron) are the indistinguishable particle witch obey Pauli exclusion principle. This is the so called Fermi-Dirac statistics.