Diffusion in the standard map Consider the standard map (also known as Chirikov map):
$$
    p_{n+1} = p_n + K \sin(\theta_n) \\
\theta_{n+1} = \theta_n + p_{n+1} 
$$
I know that the diffusion coefficient according to the Einstein relation is defined through the relation:
$$
\sigma^2 = 2Dt
$$
being $\sigma$ the variance of the position in the case of the $1$-D random walker. Can you figure out why the diffusion coefficient for the standard map is given by:
$$
D = \lim_{n \rightarrow \infty} \frac{<(p_n - p_0)^2>}{n} \quad ?
$$
I mean, the $n$ makes sense, but what about the variance, why this is given just in terms of $p$, what happens with the $x$.
 A: The following is a passage from Diffusion in the standard map (pdf) by Itzhack Dana and Shmuel Fishman:

A major difficulty in the analysis of chaotic behavior of Hamiltonian systems is the proximity of chaotic and regular orbits on various scales [2]. Thus, for $K \approx 1$, the phase plane of (1) is an intricate mixture of regular and chaotic orbits [3]. For $K \gg 1$ chaotic orbits fill almost the entire phase plane, but "islets of stability", in particular accelerator modes [2, 7], are known to exist for $K$ arbitrarily large. When a chaotic orbit approaches such an islet, it may wander near it in a "regular" fashion. The area of each islet generally decreases when $K$ increases.
Distinctive features of a stochastic or random motion, which allow a statistical description of it, are the rapid decay of correlations and diffusion. The decay of correlations in the chaotic region is related to the local instability, measure Lyapunov characteristic exponent [1, 2, 8]. For $K \gg 1$, the decay of correlations is actually exponential, with the decay exponent proportional to the Lyapunov exponent [8]. The existence of a definite characteristic time for the decay of correlations and the resulting statistical independence generally imply diffusion in the unbounded direction of the action $I$ for $K > K_\mathrm{c}$ [...]. Therefore a definite diffusion coefficient $D$ can be associated with the chaotic motion
$$ D = \lim_{n\to\infty} \frac{\big\langle(I_n-I_0)^2\big\rangle}{n}, $$
where the average is taken over some ensemble of initial positions $(I_0,\theta_0)$ within the chaotic region. For large $K$, where the Lyapunov exponent is large, diffusion was    verified unambiguously [2].

