Universality of Algebraic Decays in Hamiltonian Systems

Hamiltonian systems with a mixed phase space typically exhibit an algebraic decay of correlations and of Poincaré recurrences, with numerical experiments over finite times showing system-dependent power-law exponents. We conjecture the existence of a universal asymptotic decay based on results for a Markov tree model with random scaling factors for the transition probabilities. Numerical simulations for different Hamiltonian systems support this conjecture and permit the determination of the universal exponent.

Figure 1

(a) Survival probability $P_I(t)$ for 100 realizations of random scaling factors (thin lines) and their logarithmic average (thick line). Inset: binary tree with root $\overline{s}$ and absorbing site $D\overline{s}$. (b) Rescaled survival probabilities $P_I(t)t^{z_I}$ with $z_I$ from Eq. (9), showing the same asymptotic power-law exponents. (c) Variance ${\sigma }^2(t)$ of $\ln [P_I(t)t^{z_I}]$ showing boundedness.

Figure 2

Power-law exponent $z_I$ vs $⟨A⟩$ according to analytic estimate Eq. (9) (line) and from fitting the asymptotic decay of the numerically computed exact evolution, Eq. (2), for a single realization of scaling factors (circles). Here the distribution for the scaling factor $A_s$ ($B_s$) is chosen to be uniform on an interval centered around $⟨A⟩$ ($⟨B⟩=0.3$) with width 0.12 (0.1) and is independent from $B_s$ ($A_s$).

Figure 3

(a) Survival probabilities $P(t)$ for 100 area preserving maps (thin lines), defined in the text, and their logarithmic average (thick line). Inset: phase space of one of these maps, showing the main island and the border line to the escape region. (b) Rescaled survival probabilities $P(t)t^z$ with the fitted exponent $z=1.57$ from (a). (c) Variance ${\sigma }^2(t)$ of $\ln [P(t)t^z]$ showing boundedness.