[Paper Review] Diffusion in Energy Conserving Coupled Maps

Jean Bricmont, A. Kupiainen|arXiv (Cornell University)|Feb 18, 2011

Mathematical Dynamics and Fractals17 references4 citations

TL;DR

This paper establishes that energy-conserving coupled map lattices exhibit diffusive behavior in the scaling limit by modeling fast chaotic dynamics as a noise source acting on slow energy variables. Using a multi-scale Renormalization Group approach, it proves that the energy density satisfies a nonlinear diffusion equation with a conductivity function, under weak coupling and weak nonlinearity assumptions.

ABSTRACT

We consider a dynamical system consisting of subsystems indexed by a lattice. Each subsystem has one conserved degree of freedom ("energy") the rest being uniformly hyperbolic. The subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved. We prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit.

Motivation & Objective

• To establish the emergence of diffusive behavior in extended dynamical systems with conserved energy.
• To analyze the scaling limit of coupled map lattices where energy is conserved globally but not locally.
• To justify the nonlinear diffusion equation as the macroscopic limit of such systems using rigorous methods.
• To develop a framework where fast chaotic dynamics acts as a noise source for slow energy evolution.
• To prove that both nonlinearity and noise are irrelevant in the Renormalization Group sense, leading to diffusive scaling.

Proposed method

• Model the system as a coupled map lattice with local energy variables $E(x)$ and fast chaotic variables $\theta(x)$ on a lattice $\mathbb{Z}^d$.
• Define the uncoupled dynamics as $ (E(x), \theta(x)) \to (E(x), g(\theta(x), E(x))) $, preserving energy at each site.
• Introduce weak coupling between sites so that total energy remains conserved, and the energy current satisfies $ \dot{E}(t,x) = \nabla \cdot \mathbf{J}(t,x) $.
• Apply a multi-scale Renormalization Group (RG) method to analyze the slow energy dynamics under diffusive scaling of space and time.
• Use symbolic dynamics and telescoping expansions to represent the potential in the Gibbs measure, ensuring uniform Hölder continuity and localization.
• Prove that the SRB measure of the system arises as the pushforward of a Gibbs measure under a Hölder continuous conjugation, enabling control of the statistical properties.

Experimental results

Research questions

• RQ1 Can a deterministic, energy-conserving system with weakly coupled chaotic subsystems exhibit diffusive energy transport in the macroscopic limit?
• RQ2 Under what conditions does the energy density in such systems satisfy a nonlinear diffusion equation $ \partial_t E = \nabla \cdot (\kappa(E) \nabla E) $?
• RQ3 How do the fast chaotic dynamics and weak coupling interact to produce diffusive scaling?
• RQ4 Can the Renormalization Group method be used to show that nonlinearity and noise are irrelevant in the scaling limit?
• RQ5 What is the role of the Gibbs measure and SRB measure in characterizing the statistical behavior of the system in the limit?

Key findings

• The energy density in the coupled map lattice satisfies a nonlinear diffusion equation $ \partial_t E = \nabla \cdot (\kappa(E) \nabla E) $ in the diffusive scaling limit.
• The conductivity function $ \kappa(E) $ emerges from the statistical properties of the fast chaotic dynamics and is non-perturbative in nature.
• The Renormalization Group analysis shows that both the nonlinearity and the noise from the chaotic dynamics are irrelevant in the scaling limit.
• The SRB measure $ \nu $ of the full system is the pushforward of a Gibbs measure $ \mu $ under a Hölder continuous conjugation $ \Gamma $, ensuring statistical stability.
• The weak limit of the Gibbs measures $ \mu_R $ exists and defines a unique Gibbs measure $ \mu $ with summable, localized potential, satisfying the conditions for the existence of a physical measure.
• The system's long-time statistical behavior is governed by a Gibbs measure with exponentially decaying correlations, enabling rigorous control of the macroscopic limit.