Boltzmann

Maxwell-Boltzmann distribution

When you throw two dice the number of configurations, W(E)  belonging to a given dice value E  is given by the following table. For example for E = 4, we get W(4) = 3, This is because there are two  (1,3) set and one (2,2) possible outcome.  This means that for E= 4,  the odd (1,3) to occur compared to (2,2) combination 2 to 1.

 T 2 3 4 5 6 7 8 9 10 11 12 W 1 2 3 4 5 6 5 4 3 2 1

If you throw 3 dice and ask the similar question. For E=6, what is the odd that (1,1,4) , (1,2,3), (2,2,2)  combination to occur? Simple calculation shows that  the ratio of the odd is 3:6:1.

For 4 dice and E = 8, ask the probability that (1,1,1,5), (1,1,3,3),(1,1,2,4),(1,2,2,3), (2,2,2,2) combinations to occur. The answer in this case is that the odds are  4:6:12:12:1.

If you throw very large number of dice what combination is most likely to occur? That is what is most probable occupation numbers? This program demonstrate that if your throw sufficiently large number of dice the distribution of occupation numbers follow the Maxwell-Boltzmann distribution given by

(13 )

This is because if we concentrate single die at the top left corner and ask what is probability that this die show number r . This will be proportional to the number of configurations of rest of dice (N-1  of them) to have the total value E-r, which is eqaul to WN-1(E-r).  This number, WN-1(E-r)  can be approximated by following way for E>>r, N>>1 as

(14 )

This is the simple derivation of the Maxwell-Boltzmann distribution and students can easily follow the derivation if students could understand simulations performed so far.   Since every die is equivalent  we can simulate the distribution by simply examining the distribution of occupation numbers { n0,  n1,  n2,  n3,  n4,  n5} in the equilibrium state instead of counting relative frequency of occurrence of r  of single die which would be time consuming. This is actually a demonstration of the ergodic hypothesis that the time average is the ensemble average!

The simulation program, Applet 4 we trace down the distribution of occupation numbers from the highly nonequilibrium state to equilibrium state.

The curve drawn using cross mark ("x")  in the middle rectangle is the entropy defined by the equation (8).

The temperature used for the Maxwell-Boltzmann distribution curve is the canonical temperature defined in the equation (12).

Points to watch in this simulation program.

(A) Perform the experiments for systems of small number of dice and increase the size gradually to see from what size the M-B distribution is applicable.

(B) Watch the fluctuation subsides as the size of the system grows.  In equilibrium the entropy of the equilibrium state is represented by  a single term of set of occupation number which is the MB distribution in the eq. (9).