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 Bose-Einstein 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 Maxwell-Boltzmann statistics.
Low energy states are more probable with Bose-Einstein statistics than with the Maxwell-Boltzmann 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 Bose-Einstein condensation is observed with liquid helium and is responsible for its remarkable behavior.
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