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There is an uniqueness theorem in electromagnetism. It says that the solution of Maxwell's Equations is determined uniquely by boundary conditions.

We can treat superconductivity as a completely diamagnetic material with magnetic susceptibility $\chi=-1$.

I think the uniqueness theorem is still correct in this case, but I am confused very much.

If we think about a superconducting ring, and we know the magnetic field at infinite distance, the boundary condition at infinity, we can not be sure about the magnetic field. We don't know whether the ring have flux.

If we cool down the material before adding the magnetic field, the ring will have a zero flux.

If we add the magnetic field before cooling down the ring, the ring will have a non-zero flux.

Does it mean that the uniqueness theorem is not correct? The solution is not determined by state of the system only, but is determined by history and state of the system.

Could you answer my question? Thank you very much!

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Thanks for this interesting question. I think it has been resolved by London in his book Superfluids, vol.1. Of course you're right, uniqueness is not verified for a ring. I think it has nothing to do with superconductivity, rather with Maxwell's equation and the geometry of the ring. London demonstrates the non-uniqueness of the solution for a superconductor ring. The discussion about the phase transition and the diamagnetic response is discussed in detail also in this book, in several sections. I remember the discussion about the Barnett effect. –  FraSchelle Oct 16 '13 at 12:15

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