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Linear response for macroscopic observables in high-dimensional systems.

Caroline L Wormell1, Georg A Gottwald1

  • 1School of Mathematics and Statistics, The University of Sydney, Sydney, New South Wales 2006, Australia.

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Linear response theory (LRT) explains chaotic system behavior, but its mechanism in complex systems is unclear. This study reveals how macroscopic observables in coupled systems exhibit LRT via self-generated noise, even when microscopic parts do not.

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Area of Science:

  • Statistical mechanics
  • Dynamical systems theory
  • Nonlinear dynamics

Background:

  • Linear response theory (LRT) predicts chaotic system behavior but doesn't apply universally.
  • The mechanism for macroscopic observables in complex dissipative chaotic systems exhibiting LRT, despite microscopic violations, is poorly understood.
  • High-dimensional coupled deterministic dynamical systems with mean-field coupling present a complex scenario for LRT.

Purpose of the Study:

  • To provide a comprehensive understanding of linear response in macroscopic observables of high-dimensional coupled deterministic dynamical systems.
  • To derive conditions under which LRT holds for these systems, both in finite dimensions and the thermodynamic limit.
  • To investigate the role of self-generated noise and emergent dynamics in the applicability of LRT.

Main Methods:

  • Derivation of stochastic reductions for macroscopic observables from microscopic system statistics.
  • Analysis of conditions for LRT in finite-dimensional systems and the thermodynamic limit.
  • Numerical evidence and analytical intuition to support theoretical findings.

Main Results:

  • Linear response in large, finite systems is induced by self-generated noise.
  • Demonstration of macroscopic LRT in the thermodynamic limit, even when microscopic subsystems violate LRT.
  • Identification of a converse case where macroscopic observables violate LRT despite microscopic subsystems adhering to it, linked to emergent dynamics.

Conclusions:

  • Self-generated noise is a key mechanism enabling linear response in macroscopic observables of large, coupled chaotic systems.
  • The applicability of LRT to macroscopic observables is not solely dependent on the microscopic constituents' adherence to it.
  • Emergent dynamics in macroscopic observables can lead to deviations from LRT, even in systems composed of individually linear-responsive parts.