"Weak Quantum Chaos" and its resistor network modeling

Weakly chaotic or weakly interacting systems have a wide regime where the common random matrix theory modeling does not apply. As an example we consider cold atoms in a nearly integrable optical billiard with displaceable wall ("piston"). The motion is completely chaotic but with small Lyapunov exponent. The Hamiltonian matrix does not look like one taken from a Gaussian ensemble, but rather it is very sparse and textured. This can be characterized by parameters $s$ and $g$ that reflect the percentage of large elements, and their connectivity, respectively. For $g$ we use a resistor network calculation that has a direct relation to the semi-linear response characteristics of the system, hence leading to a novel prediction regarding the rate of heating of cold atoms in optical billiards with vibrating walls.