The occupation number of bosons can be any number from zero to infinity, guiding us to the Bose-Einstein statistics. On the other hand, for example, a classical wave can be considered a superposition of any number of sine or cosine waves. Isn't it similar to say the occupation number of a classical wave can be any number from zero to infinity and utilizing Bose-Einstein statistics for classical waves in particular and classical fields in general?
The fundamental thing about quantum statistics is that particles are identical.
All photons have the same mass, charge and spin, the same for electrons, muons etc.
You cannot paint an electron red, so it's impossible to distinguish
them even in principle. Then you have two different kind of statistics
dependend upon spin when exchanging two particles. The same operation cannot
apply to something less fundamental like a classical object, it simply
makes no sense.
Moreover, the two quantum statistics, Bose-Einstein and Fermi-Dirac
differ because of spin, which has integer or half-integer values respectively
in units of h. In the classical limit h -> 0, so again this concept makes no sense.
In principle at least there would be another way to distinguish particles.
Suppose you have two electrons, and can identify one by its position. Then
tracking his trajectory of motion would allow its tagging.
Unfortunately in quantum mechanics the proper concept trajectory of motion does
not exist like in classical mechanics because Delta q x Delta p >= hslash.
In fact after some time the two wavefunctions will spread out and eventually
overlap.
So we can conclude that applying the concepts of quantum statistics
to classical statistics is incorrect.
Indeed, the Bose-Einstein statistics has two classical limits depending on ratio (E-\mu)/kT: the Maxwell-Boltzmann statistics in the high-energy limit, and the Rayleigh-Jeans statistics in the high-temperature limit. But all these distributions apply to well-thermalized systems in thermodynamical equilibrium. To obey the Bose-Einstein statistics (or some of its classical analogues), Your classical waves have to be in a kind of thermodynamical equilibrium as well. The fact that a classical wave can have arbitrarily large amplitude doesn't mean much: it follows from the absence of the Pauli-like restrictions.
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