Energy absorption by ‘sparse’ systems: beyond linear response theory

Abstract

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. Respectively, the obtained results are contrasted with the "Wall formula" and the "Drude formula".

Quantum thermalization: anomalous slow relaxation due to percolation-like dynamics

“…Due to the wide distribution of transition strengths, the obtained network is glassy. This glassiness is reminiscent of the sparsity that arises in integrable systems due to selection rules [28].…”

“…This striking breakdown of QCC is due to the percolation-like nature of energy spreading. As appropriate for a percolation process, D(x) should be estimated from the conductivity of the 'resistor network' that is formed by the quantum transitions [28]. Such evaluation gives the proper weight to low-resistance, well-connected links, as opposed to the over-estimated democratic weighing of equation (7).…”

“…However, such dense networks do not always exist. The quantum network of transitions is generally sparse [28], resulting in a percolation-like process of energy spreading, that is dominated by bottlenecks and preferred pathways. As a result, the Kubo formula grossly overestimates the thermalization rate and QCC is lost even when the underlying classical dynamics is highly chaotic.…”

Thermalization of Bipartite Bose–Hubbard Models

“…As outlined in ref , the stochastic-like spreading dynamics of P t ( x ) in the chaotic regime is captured by the master equation with rates given by a Fermi golden rule (FGR) prescription and restricted to the band | E x ,ν x – E x ′,ν′ x ′ | < 1/τ where the bandwidth 1/τ corresponds to the width of the power-spectrum of the perturbation, estimated by its variance The kinetic eq can be coarse grained to give a Fokker–Planck diffusion equation in x space , where g ̃( x ) is the density of states within the allowed energy shell. Proper evaluation of the diffusion coefficient D ( x ) requires a resistor-network calculation (see refs and for details).…”

Nonequilibrium steady state and induced currents of a mesoscopically glassy system: Interplay of resistor-network theory and Sinai physics

“…Having a log-wide distribution of time scales is typical for hopping in a random energy landscape, where the rates depend exponentially on the barrier heights. It also arises in driven quasi-integrable systems, where due to approximate selection rules there is a "sparse" fraction of large coupling elements, while the majority become very small [26].…”