My problem is arised from the calculation result of entropy in a system.There are three degenerate energy states ,and put N indistinguishable particles to these discrete energy when the temperature of system is T.(Suppose the degeneracy of each energy state is much more than the number of particles coming in.)Then calculate entropy,using canonical distribution,and I get the NlnN term,which cannot be eliminated when N get to infinite.Then I use grand canonical distribution to calculate and obtain the fine answer.So I think the problem is that indistinguishable particles here mean numerous bosons condense together at different energy states.We cannot find proper subsystem for the system which get unstable and condense together.
For the further thinking,we can say that superposition theorem only exist in stable system.There must be some symmetry corresponding to the superposition.Such as in Electromagnetism, the global gauge symmetry relates to the conservation of charge.But in general relativity, the gauge transformation corresponding to the tangent space of one spacetime point relates to conervation of mass charge.But only the local area could have lorentz symmetry but not the macroscale .There is not superposition in general relativity while Electromagnetism do have superposition theorem.And as we all know the gravity system is not stable because of its negative heat capacity.
The problem arises from not implementing the constraint that the particles are indistinguishable. This is known as the ``Gibbs' paradox'' in classical statistical mechanics and is described in any textbook on the subject. One way to see what's going on is to notice that N ln N is, in fact, part of the approximation to ln (N!) for large values of N-and ln (N!) is the entropy (in units where Boltzmann's constant is 1) of N, indisitinguishable, objects, since N! is the number of permutations of such objects, that must be identified, if the objects are indistinguishable.
This doesn't have anything to do with gravity, so it doesn't make sense to mention it in this context.
Dear Stam Nicolis,
Clearly ,I have used the constraint that the particles are indistinguishable.If you saw the document I uploaded, the system provided has finite energy states.And I make a hypothesis that every energy state has infinite degeneracies.So bosons put in this system can result in that the entropy of system becomes nonextensive.
So problem you say I know.But it really didn't relate to this problem.
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