Assigning temperatures to eigenstates
Abstract:
In formulating the statistical mechanics of isolated quantum systems, an inescapable issue is the definition of temperature, which is not a priori defined within closed-system quantum mechanics. We examine and compare different possible ways of assigning temperatures to energies, or equivalently, to eigenstates. The commonly used assignment of temperature is based on the canonical energy-temperature relationship, which depends only on energy eigenvalues and not on the structure of eigenstates. For eigenstates, we consider defining temperature by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. We show that for full eigenstates, the minimizing temperature depends on the distance measure chosen, and matches the canonical temperature for the trace distance; however, the two matrices are not close. With reduced density matrices, the minimizing temperature has fluctuations that scale with subsystem and system size but is apparently independent of distance measure, and in particular limits the two matrices become equivalent.
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