Energy absorption by 'sparse' systems: beyond linear response theory

The analysis of the response to driving in the case of weakly chaotic or weakly interacting systems should go beyond linear response theory. Due to the 'sparsity' of the perturbation matrix, a resistor-network picture of transitions between energy levels is essential. The Kubo formula is modified, replacing the 'algebraic' average over the squared matrix elements by a 'resistor-network' average. Consequently, the response becomes semi-linear rather than linear. Some novel results have been obtained in the context of two prototype problems: the heating rate of particles in billiards with vibrating walls; and the Ohmic Joule conductance of mesoscopic rings driven by electromotive force. The results obtained are contrasted with the 'Wall formula' and the 'Drude formula'.