The standard map as an example of crossing from Boltzmann-Gibbs statistics to Tsallis statistics.

As well known, Boltzmann-Gibbs statistics is the correct way of
thermostatistically approaching ergodic systems. On the other hand,
nontrivial ergodicity breakdown and strong correlations typically
drag the system into out-of-equilibrium states where Boltzmann-Gibbs
statistics fails. For a wide class of such systems, it has been shown
in recent years that the correct approach is to use Tsallis statistics
instead. Here we show how the dynamics of the paradigmatic conservative
(area-preserving) standard map, exhibits, in an exceptionally clear
manner, the crossover between both statistics. Our results unambiguously
illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis
statistics.
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The standard map is defined as
$ p_{i+1} = p_i - K \sin x_i,\,\, x_{i+1}=x_i+p_{i+1} $,
where $p$ and $x$ are taken as modulo $2\pi$.
This map has very rich properties depending on the map parameter $K$.
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The phase space of the standard map presents regions of positive Lyapunov
exponents coexisting with regions of zero Lyapunov exponents.
The positive Lyapunov regions present mixing and thus the system is ergodic
in those regions.
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For sufficiently low values of the control parameter $K$,
the phase space is almost entirely dominated by zero Lyapunov behavior
and the distributions (obtained through time averaging,
along the lines of central limit theorems) are $q$-Gaussians.
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As the value of $K$ increases the measure of the zero Lyapunov regions
decreases, and we see a continuous crossing between $q$-Gaussians distributions
(Tsallis statistics) and Gaussian ones (Boltzmann-Gibbs statistics).
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Remarkably enough, the distributions originated from initial conditions
taken inside the region of islands, instead of over the entire phase space,
yield one and the same value $q=1.935$, independently on whether
we consider one or many of these regions, and independently from $K$.
Initial conditions taken within the chaotic sea always yield Gaussians.
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Since various important physical systems from particle confinement
in magnetic traps to autoionization of molecular Rydberg states,
through particle dynamics in accelerators and comet dynamics,
can be reduced to the standard map, our results are expected to enlighten
and enable an improved interpretation of diverse experimental
and observational results.