  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.

Quantum Physics;Mesoscale and Nanoscale Physics;Nuclear Theory