Breakdown of Linear Response in Uniformly Hyperbolic Systems with Hierarchical Structure

TL;DR

This paper shows that linear response theory can fail in uniformly hyperbolic systems with hierarchical structures, due to multiscale activation of transport channels, leading to divergence of mobility as force approaches zero.

Contribution

It introduces a new deterministic mechanism for nonlinear transport response caused by hierarchical structures in hyperbolic systems, challenging previous assumptions.

Findings

Linear response breaks down in hierarchical hyperbolic systems.

Effective mobility diverges as applied force approaches zero.

Hierarchical activation thresholds create fractal-like transport behavior.

Abstract

Linear response theory asserts that sufficiently small external biases produce currents proportional to the applied force and forms the theoretical foundation of nonequilibrium transport. Here we demonstrate that linear response can break down even in uniformly hyperbolic deterministic systems when hierarchical asymmetry is present. Using a minimal class of uniformly expanding chaotic maps with hierarchical multiscale structure, we show that progressively finer transport channels become dynamically active as the applied bias decreases. The resulting force current relation is monotone and exhibits a hierarchical, fractal-like organization of activation thresholds. As a consequence, the effective mobility diverges as F to 0, demonstrating breakdown of linear response despite strong chaos and uniform hyperbolicity. The effect arises from deterministic multiscale activation rather than intermittency, stochastic noise, or singular invariant measures. These results identify hierarchy as an independent deterministic mechanism for nonperturbative transport response and demonstrate that uniform hyperbolicity alone does not guarantee the validity of linear response.