Probability & statistics

Macrostates and Microstates

Let's start with an example:
If we toss up two coins, the possible outcomes are:


 coin 1
 
 coin 2
H
H
H
T
T
H
T
T


With: H = heads and T = tails.

The all four possible outcomes of the set
of two coins  with 2 states H and T are the 
microstates: H H, H T, T H, T T. But the two coins 
are exactly the same; that is  indistiguishable.
The two microstates  1 Head  HT and TH constitutes one macrostate.

We have:
1 Head : 1 microstate: 
2 Heads: 2 microstates
0 Heads: 1 microstate

 
The number of microstates corresponding to a given macrostate 
is the multiplicity Ω.


It follows:
Ω(1) = 1, Ω(2) = 2, and Ω(0) = 1. We assume that the coins 
are fair; then the all the 4 microstates are equally likely. 
The probability for the four microstates are: P(HH) = P(HT) = P(TH) = P(TT) = 1/4.
The probability for the the three macrostates : P(2H) = P(0H) = 1/4, and 
P(1H) = 2/4 = 1/2 ( the most probable). Generally, the probability of 
n heads is equal to Ω(n)/Ω. Ω is the total number  of microstates.

If we toss up 20 coins, the total number of microstates is 
220 = 1,048,576 and the number of macrostates (0 H, 1 H, 2 H, ..., 20 H) 
is (20 + 2 - 1)!/20! (2 - 1)! = 21!/20!1! = 21.
 
 
 
 The number of ways to place N particles distinguishables within n places 
 is nN. If they are indistiguishable , the numer of ways, in this 
 case is: (N + n - 1)!/n!(n - 1)!


The related multiplicities for 1 Head and 2 Heads are: 
Ω(0) = 1, Ω(1) = 20!/1!0! = 20

What is he multiplicity for n Heads: Ω(n)? It is just the number of 
combinations of n among N. In other words, how many ways of getting n heads 
among the existent N heads?. It is then equal to: C(n,N) = N!/n!(N - n)!

For n = 17, we have Ω(11) = 20!/17! 3! = 18 x 19 x 20/6 = 1140
 
 
 
 The number of ways to arrange n objects among N, that is 
 the combination of n objects among N is C(n,N) = N!/n!(N - n)! 
 if the objects are indistinguishable;  otherwise, if 
 they are distinguishable the number of ways is equal 
 to N!/(N - n)!, that is the number of arrangements.