Being not an expert in these fields, I wonder whether the quantum mechanical entanglement, e.g., of electrons in an electron gas, is already taken into account in the statistical mechanics and thermodynamics of such a system and what impact (if any) entanglement has on its physical behavior.
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There is an entanglement entropy between two systems $A$ and $B$ that is $$ S(A,B)~=~S(A)~+~S(B)~-~S(A|B). $$ This is a fine grained entropy where the observer has access to everything about the system. For thermodynamics the thermal entropy drops the $S(A|B)$ which results in a larger entropy. This is because $S(A|B)$ is a conditional entropy for the entanglement, which in a complex system will migrate throughout. The entanglement phase, or equivalently the overlap phase for a superposition, will enter in to a larger number of states and it is not practical to account for this. Coarse graining of the system results in the thermal entropy that is larger than the entanglement entropy
Lawrence B. Crowell
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