Anomalous transport and observable average in the standard map

Abstract: The distribution of finite time observable averages and transport in low dimensional Hamiltonian systems is studied. Finite time observable average distributions are computed, from which an exponent $\alpha$ characteristic of how the maximum of the distributions scales with time is extracted. To link this exponent to transport properties, the characteristic exponent $\mu(q)$ of the time evolution of the different moments of order $q$ related to transport are computed. As a testbed for our study the standard map is used. The stochasticity parameter $K$ is chosen so that either phase space is mixed with a chaotic sea and islands of stability or with only a chaotic sea. Our observations lead to a proposition of a law relating the slope in $q=0$ of the function $\mu(q)$ with the exponent $\alpha$.