Consider a quantum system in thermal equilibrium with a heat bath. In determining the density operator of the system, the usual procedure is to maximize the von Neumann entropy subject to the constraint that the ensemble average of the Hamiltonian has a fixed value.
What justifies this assumption?
Sakurai, in his QM text, writes
To justify this assumption would involve us in a delicate discussion of how equilibrium is established as a result of interactions with the environment.
I'd appreciate if someone could shed some light on this. References are welcome as well.
I've heard the suggestion that a thermal equilibrium ensemble is simply defined by that density operator which solves the constrained optimization problem above. If this is the case, then why are real physical systems that are in weak contact with heat baths for long periods well described by such mathematical ensembles, and how would one identify the Lagrange multiplier $\beta$ that arises with inverse temperature of the heat bath?
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