I am reading an articles introducing the Nobel Price on Bose-Einstein condensates from where I have further reading on Bosonic and Fermionic statistics on some texts. I know one of the mathematical difference is the +/- 1 term in the denominator of the distribution function as below(adsbygoogle = window.adsbygoogle || []).push({});

##f_{BE} = \dfrac{1}{A\exp(E/k_BT) - 1}##

##f_{FD} = \dfrac{1}{B\exp((E-E_f)/k_BT) + 1}##

for the Fermi-Dirac distribution, again the term "+1" arise from the fact of indistinguishable particles. But since there is a Fermi energy when the total energy lower than the Fermi energy (so ##E-E_f<0##), at extreme low temperature #\exp((E-E_f)/k_BT) \to 0# so the probably becomes 1. So the "+1" becomes important.

However, the similar reasoning applied on the Bose-Einstein statistics is misleading me. In some book, it is said that the "-1" term for BE distribution arise from the fact that the particles are indistinguishable so to increase the likelihood of multiple occupation of one energy state. Well, if we subtract a positive number from the denominator of a distribution, you get bigger output for ##f_{BE}##, I think that's why they said "increasing the likelihood"? But in the case when T approaches extreme low temperature so ##\exp(E/k_BT)## become a really big number, can we ignore that "-1" so to conclude that #f_{BE} \to 0#. But it seems opposite to what I learn from the article where it said at very low temperature, all particles occupying the same state so it is almost 100% to see all particles in ground state. Am I missing or misunderstanding something here?

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# I Does the +/- 1 term in bosonic and fermionic statistics matter

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