Spin and the Thermal Equilibrium Distribution of Wave Functions

Consider a quantum system $S$ weakly interacting with a very large but finite system $B$ called the heat bath, and suppose that the composite $S\cup B$ is in a pure state $Ψ$ with participating energies between $E$ and $E+δ$ with small $δ$. Then, it is known that for most $Ψ$ the reduced density matrix of $S$ is (approximately) equal to the canonical density matrix. That is, the reduced density matrix is universal in the sense that it depends only on $S$'s Hamiltonian and the temperature but not on $B$'s Hamiltonian, on the interaction Hamiltonian, or on the details of $Ψ$. It has also been pointed out that $S$ can also be attributed a random wave function $ψ$ whose probability distribution is universal in the same sense. This distribution is known as the "Scrooge measure" or "Gaussian adjusted projected (GAP) measure"; we regard it as the thermal equilibrium distribution of wave functions. The relevant concept of the wave function of a subsystem is known as the "conditional wave function". In this paper, we develop analogous considerations for particles with spin. One can either use some kind of conditional wave function or, more naturally, the "conditional density matrix", which is in general different from the reduced density matrix. We ask what the thermal equilibrium distribution of the conditional density matrix is, and find the answer that for most $Ψ$ the conditional density matrix is (approximately) deterministic, in fact (approximately) equal to the canonical density matrix.

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