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We analyze the time-dependent free energy functionals of the semiclassicalone-dimensional Bose-Hubbard chain. We first review the weakly chaotic dynamicsand the consequent early-time anomalous diffusion in the system. The anomalousdiffusion is robust, appears with strictly quantized coefficients, and persistseven for very long chains (more than hundred sites), crossing over to normaldiffusion at late times. We identify fast (angle) and slow (action) variablesand thus consider annealed and quenched partition functions, corresponding tofixing the actions and integrating over the actions, respectively. We observethe leading quantum effects in the annealed free energy, whereas the quenchedenergy is undefined in the thermodynamic limit, signaling the absence ofthermodynamic equilibrium in the quenched regime. But already the leadingcorrection away from the quenched regime reproduces the annealed partitionfunction exactly. This encapsulates the fact that in both slow- and fast-chaosregime both the anomalous and the normal diffusion can be seen (though atdifferent times).

Thermodynamic formalism and anomalous transport in 1D semiclassical Bose-Hubbard chain