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Statistical mechanics of Floquet systems: the pervasive problem of near degeneracies

The statistical mechanics of periodically driven ("Floquet") systems in contact with a heat bath exhibits some radical differences from the traditional statistical mechanics of undriven systems. In Floquet systems all quasienergies can be placed in a finite frequency interval, and the number of near degeneracies in this interval grows without limit as the dimension N of the Hilbert space increases. This leads to pathologies, including drastic changes in the Floquet states, as N increases. In earlier work these difficulties were put aside by fixing N, while taking the coupling to the bath to be smaller than any quasienergy difference. This led to a simple explicit theory for the reduced density matrix, but with some major differences from the usual time independent statistical mechanics. We show that, for weak but finite coupling between system and heat bath, the accuracy of a calculation within the truncated Hilbert space spanned by the N lowest energy eigenstates of the undriven system is limited, as N increases indefinitely, only by the usual neglect of bath memory effects within the Born and Markov approximations. As we seek higher accuracy by increasing N, we inevitably encounter quasienergy differences smaller than the system-bath coupling. We therefore derive the steady state reduced density matrix without restriction on the size of quasienergy splittings. In general, it is no longer diagonal in the Floquet states. We analyze, in particular, the behavior near a weakly avoided crossing, where quasienergy near degeneracies routinely appear. The explicit form of our results for the denisty matrix gives a consistent prescription for the statistical mechanics for many periodically driven systems with N infinite, in spite of the Floquet state pathologies.