Adiabatic Expansion in Case of Vanishing Increments by Barus C.

By Barus C.

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Flach Fig. 14 Left plot: evolution of the averaged norm density < nl . / > in the case without ( D 0) and with ramping ( D 0:3) in log scale for the DNLS model. Right plot: left column: the second moments (upper) and their power-law exponents ˛ (lower) for the DNLS model for D 0 (red), D 0:1 (green), D 0:2 (blue), D 0:3 (magenta), D 0:4 (cyan), and D 0:5 (black). Right column: the second moments (upper) and their power-law exponents ˛ (lower) for the NQKR model for D 0 (red), D 0:17 (green), D 0:25 (blue), D 0:33 (magenta), D 0:5 (cyan), and D 1:5 (black).

Shepelyansky performed the first pioneering study on subdiffusive spreading and destruction of dynamical localization for ˇ ¤ 0 in [69]. t/ lead to inconclusive results. Gligoric et al. log10 t/ instead (note here that the second moment m2 is equivalent to the rotor energy E). The results impressively obtain a regime of weak chaos with ˛ 1=3, and also strong chaos with ˛ 1=2. The original simulations of Shepelyansky [69] were performed in the crossover region between strong and weak chaos, leading to incorrect fitting results—which are however between the two weak and strong chaos limits, as expected.

7 Testing the Predictions In this chapter we will review numerical results which test the above predictions. We will in particular discuss the crossover from strong to weak chaos, the scaling of the density profiles, the impact of different powers of nonlinearity and different lattice dimensions, and the temperature dependence of heat conductivity. We will also extend the discussion to quasiperiodic Aubry-Andre localization, dynamical localization with kicked rotors, Wannier-Stark localization, and time-dependent ramping protocols of the nonlinearity strength which speed up the slow subdiffusive spreading process up to normal diffusion.

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