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 a displaceable wall (piston). The motion is completely chaotic but with a 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 which 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 semilinear response characteristics of the system, hence leading to a prediction regarding the energy absorption rate of cold atoms in optical billiards with vibrating walls.