Weak Ergodicity Breaking and Aging of Chaotic Transport in Hamiltonian Systems

Momentum diffusion is a widespread phenomenon in generic Hamiltonian systems. We show for the prototypical standard map that this implies weak ergodicity breaking for the superdiffusive transport in coordinate direction with an averaging-dependent quadratic and cubic increase of the mean-squared displacement (MSD), respectively. This is explained via integrated Brownian motion, for which we derive aging time dependent expressions for the ensemble-averaged MSD, the distribution of time-averaged MSDs, and the ergodicity breaking parameter. Generalizations to other systems showing momentum diffusion are pointed out.

Figure 1

Ensemble-averaged MSD for different aging times $t_a$ numerically obtained from $N=5\times {10}^5$ trajectories generated by the standard map for $k=1.5$. The trajectories were started close to the hyperbolic fixed point at $(x,p)=(0,0)$. Also shown in the figure is the theoretical result of Eq. (5), which was derived from integrated Brownian motion.

Figure 2

Time-averaged MSD ($t_a=0$) of four different trajectories of length $T=2\times {10}^{10}$ numerically generated by the standard map for $k=1.5$. Again, all trajectories were started close to the hyperbolic fixed point at $(x,p)=(0,0)$. The theoretical curve corresponds to Eq. (7) and describes the ensemble average of time-averaged MSDs for integrated Brownian motion.

Figure 3

Scatter distribution of rescaled time-averaged MSDs ($\tau =100$) for three different aging times $t_a$ numerically determined from $N={10}^5$ trajectories of length $T={10}^5$ generated by the standard map for $k=1.5$. The numerical data agree very well with the theoretical curves, which were obtained from a numerical Laplace inversion of Eq. (10).

Figure 4

Numerically obtained ergodicity breaking parameter ($\tau =100$) for different values of $\lambda =t_a/T$ compared with EB from Eq. (12). $N={10}^5$ trajectories of length $T={10}^5$ generated by the standard map for $k=1.5$ were used for the numerical determination.