Energy level  Average number Maxwell Boltzmann  Average number Bose Einstein 
0  2.143  2.269 
1  1.484  1.538 
2  0.989  0.885 
3  0.629  0.538 
4  0.378  0.269 
5  0.210  0.192 
6  0.105  0.115 
7  0.045  0.077 
8  0.015  0.038 
9  0.003  0.038 
 There are 26 possible distributions of 9 units of energy among 6 particles, and if those particles are indistinguishable and described by BoseEinstein statistics, all of the distributions have equal probability. To get a distribution function of the number of particles as a function of energy, the average population of each energy state must be taken. The average for each of the 9 states is shown below compared to the result obtained by MaxwellBoltzmann statistics.
Low energy states are more probable with BoseEinstein statistics than with the MaxwellBoltzmann statistics. While that excess is not dramatic in this example for a small number of particles, it becomes very dramatic with large numbers and low temperatures. At very low temperatures, bosons can "condense" into the lowest energy state. The phenomenon called BoseEinstein condensation is observed with liquid helium and is responsible for its remarkable behavior.
