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Breakdown of Linear Response in Uniformly Hyperbolic Systems with Hierarchical Structure

Vinesh Vijayan, Priyadharshini B, Santhoshbalaji M, Mohanasundari M

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.

Breakdown of Linear Response in Uniformly Hyperbolic Systems with Hierarchical Structure

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.
Paper Structure (6 sections, 17 equations, 2 figures)

This paper contains 6 sections, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Uniform expansion and chaos
  4. Deterministic Chaos Without Linear Response
  5. Discussion
  6. Conclusion

Figures (2)

  • Figure 1: (a) Reduced hierarchical map $\tilde{T}(x)=T(x)\bmod 1$ showing a piecewise linear expanding structure with asymmetric discontinuities at all spatial scales. The inset demonstrates self-similar collapse under magnification, revealing the hierarchical organization of the map. (b) Multiscale response of trajectories $x_n$ as a function of the applied bias $F$, illustrating strong sensitivity across a dense set of force values.
  • Figure 2: (a) Devil's staircase current $J(F)$ exhibiting a monotone, fractal structure with hierarchical activation thresholds that accumulate as $F \to 0$. (b) (b) Log-log plot of mobility $\mu(F) = J(F)/F$ exhibiting unbounded growth as $F \to 0$, signaling breakdown of linear response. The dashed line indicates a reference slope $\sim F^{-1}$ shown for comparison only.