# Leaking from the phase space of the Riemann-Liouville fractional standard map

## TL;DR

This paper investigates how orbits escape from the phase space of the Riemann-Liouville fractional standard map, revealing different behaviors in ergodic and non-ergodic regimes based on fractional order and nonlinearity.

## Contribution

It characterizes escape dynamics in the RL fractional standard map, connecting phase space ergodicity with escape probabilities and revealing scale invariance in ergodic cases.

## Key findings

- In ergodic phase space, escape times are scale invariant and proportional to (h/K)^2.
- Non-ergodic phase space exhibits non-universal, parameter-dependent escape behaviors.
- The RL-fSM generalizes the standard map by incorporating fractional derivatives, affecting escape dynamics.

## Abstract

In this work we characterize the escape of orbits from the phase space of the Riemann-Liouville (RL) fractional standard map (fSM). The RL-fSM, given in action-angle variables, is derived from the equation of motion of the kicked rotor when the second order derivative is substituted by a RL derivative of fractional order $\alpha$. Thus, the RL-fSM is parameterized by $K$ and $\alpha\in(1,2]$ which control the strength of nonlinearity and the fractional order of the RL derivative, respectively. Indeed, for $\alpha=2$ and given initial conditions, the RL-fSM reproduces Chirikov's standard map. By computing the survival probability $P_{\text{S}}(n)$ and the frequency of escape $P_{\text{E}}(n)$, for a hole of hight $h$ placed in the action axis, we observe two scenarios: When the phase space is ergodic, both scattering functions are scale invariant with the typical escape time $n_{\text{typ}}=\exp\langle \ln n \rangle \propto (h/K)^2$. In contrast, when the phase space is not ergodic, the scattering functions show a clear non-universal and parameter-dependent behavior.