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I was going to stay out of this (hah!) but as long as I have my oar
in, let me stir it a bit more. When Joule was twirling his oar he was
increasing the entropy of the water. It is true that he was not
heating the water.
Change in entropy is not associated with an actual
process;
..... associated with *any* hypothetical reversible process
capable of taking an identical system from the same initial to the
same final state as the system in question. All such processes will
yield the same path integrals of dQ/T, and the meaning of dQ is well
defined for the hypothetical process.