Coarse-grained entropy decrease and phase-space focusing in Hamiltonian dynamics

We analyze the behavior of the coarse-grained entropy for classical probabilities in nonlinear Hamiltonians. We focus on the result that if the trajectory dynamics are integrable, the probability ensemble shows transient increases in the coherence, corresponding to an increase in localization of the ensemble and hence the phase-space density of the ensemble. We discuss the connection of these dynamics to the problem of cooling in atomic ensembles. We show how these dynamics can be understood in terms of the behavior of individual trajectories, allowing us to manipulate ensembles to create “cold” dense final ensembles. We illustrate these results with an analysis of the behavior of particular nonlinear integrable systems, including discussions of the spin-echo effect and the seeming violation of Liouville's theorem.

Figure 1

(Color online) Increase of CGE for delta-kick cooling.

Figure 2

(Color online) Initial condition for delta-kick cooling.

Figure 3

(Color online) Distribution after free expansion for delta-kick cooling. Notice the change in shape, as well as the decrease in peak height. The jagged character of the distribution comes from the finite number of ensemble members in each bin.

Figure 4

(Color online) Decrease of CGE followed by its increase for grad-H effect.

Figure 5

(Color online) Initial condition for grad-H effect.

Figure 6

(Color online) Previous state after being subject to a position-dependent kick that places it on the stable manifold of an unstable fixed point.

Figure 7

(Color online) Previous state after evolution in the pendulum potential, at $t=1.0$, when the distribution has contracted along the stable manifold.

Figure 8

(Color online) Same as above, at $t=2.5$. Note the change of scale on all three axes. The distribution is now sharply focused at the unstable fixed point.

Figure 9

(Color online) Same as above, at $t=3.25$. After the transient focusing, the distribution begins to stretch along the unstable manifold of the stable fixed point.

Figure 10

(Color online) Dephasing oscillations in the CGE for the ‘inverse dephasing’ effect with $\kappa =0.065$, $\Delta \kappa =0.01$. Notice the first prominent dip in the curve before the oscillating drift higher.

Figure 11

(Color online) Initial condition for the inverse dephasing effect, the microcanonical distribution for an energy shell, with a double-peak corresponding to the two unstable fixed points.

Figure 12

(Color online) At $T=190$ kicks, a transient stage verifying that because of the various frequencies participating, the distribution can actually be less focused on the way to greater focusing.

Figure 13

(Color online) At $T=320$ kicks, the peaks are clearly visible already, and can be seen to be not at the turning points.

Figure 14

(Color online) At $T=333$ kicks, at or near a maximum in the transient focusing from inverse dephasing.

Figure 15

(Color online) The distribution from Fig. 14 seen much later (at $T=2400$ kicks) showing that the dephasing has lead to an incoherent distribution spread across the accessible orbits.

Figure 16

(Color online) Time to peak in CGE versus perturbation for inverse dephasing effect.

Figure 17

(Color online) Maximum in change in CGE versus perturbation for inverse dephasing effect.