[Paper Review] Noise induced dissipation in Lebesgue-measure preserving maps on $d-$dimensional torus

Albert Fannjiang, Lech Wołowski|ArXiv.org|Sep 18, 2002

Mathematical Dynamics and Fractals14 references4 citations

TL;DR

This paper investigates noise-induced dissipation in Lebesgue-measure preserving maps on the d-dimensional torus, showing that nonergodic maps exhibit O(1/ε) dissipation time while ergodic toral automorphisms—such as cat maps—have O(ln(1/ε)) dissipation time, with the constant tied to minimal dimensionally averaged entropy across irreducible blocks. The analysis reduces the problem to an arithmetic-optimization framework using convexity, Diophantine approximation, and arithmetic progression theorems.

ABSTRACT

We consider dissipative systems resulting from the Gaussian and $alpha$-stable noise perturbations of measure-preserving maps on the $d$ dimensional torus. We study the dissipation time scale and its physical implications as the noise level $\vep$ vanishes. We show that nonergodic maps give rise to an $O(1/\vep)$ dissipation time whereas ergodic toral automorphisms, including cat maps and their $d$-dimensional generalizations, have an $O(\ln{(1/\vep)})$ dissipation time with a constant related to the minimal, {\em dimensionally averaged entropy} among the automorphism's irreducible blocks. Our approach reduces the calculation of the dissipation time to a nonlinear, arithmetic optimization problem which is solved asymptotically by means of some fundamental theorems in theories of convexity, Diophantine approximation and arithmetic progression. We show that the same asymptotic can be reproduced by degenerate noises as well as mere coarse-graining. We also discuss the implication of the dissipation time in kinematic dynamo.

Motivation & Objective

• To understand how noise induces irreversibility and dissipation in conservative, measure-preserving dynamical systems on the d-dimensional torus.
• To characterize the dissipation time scale as a function of noise level ε and the ergodic properties of the underlying map.
• To establish a rigorous asymptotic link between dissipation time and dynamical invariants such as entropy and irreducible block structure in toral automorphisms.
• To show that the same dissipation time scaling arises not only from Gaussian or α-stable noise but also from degenerate noise and coarse-graining procedures.

Proposed method

• Formalize the noisy dynamics via a composition operator Tε,α = Gε,α ∘ UF, where UF is the Koopman operator of the base map F and Gε,α is the α-stable noise kernel in Fourier space.
• Define the dissipation time ndiss as the smallest n such that ‖Tⁿε,α‖ < 1/e, capturing the onset of significant L²-norm decay.
• Reduce the problem of estimating ndiss to a nonlinear arithmetic optimization problem involving Diophantine approximation and lattice point distribution.
• Apply fundamental theorems from convex analysis and arithmetic progression theory to asymptotically solve the optimization problem.
• Use spectral decomposition and block-diagonalization of toral automorphisms to isolate irreducible components and compute minimal entropy contributions.
• Establish equivalence between dissipation time scaling and dynamical invariants such as entropy and irreducibility of characteristic polynomials.

Experimental results

Research questions

• RQ1 How does the dissipation time scale with noise level ε for non-ergodic measure-preserving maps on the d-torus?
• RQ2 What determines the dissipation time for ergodic toral automorphisms, particularly in terms of their spectral and arithmetic structure?
• RQ3 Can the dissipation time scaling be recovered using degenerate noise or coarse-graining instead of full stochastic perturbations?
• RQ4 How is the dissipation time related to the minimal dimensionally averaged entropy among irreducible blocks of a toral automorphism?

Key findings

• Nonergodic maps on the d-torus exhibit an O(1/ε) dissipation time scale as ε → 0.
• Ergodic toral automorphisms, including cat maps and their d-dimensional generalizations, have an O(ln(1/ε)) dissipation time scale.
• The constant in the O(ln(1/ε)) scaling is determined by the minimal, dimensionally averaged entropy among the irreducible blocks of the automorphism.
• The same asymptotic dissipation time scaling is reproduced under degenerate noise and coarse-graining, indicating robustness of the result.
• The dissipation time is linked to the irreducibility of the characteristic polynomial of the automorphism via Diophantine approximation and arithmetic progression theorems.
• The analysis confirms that the dissipation time is a robust measure of dynamical instability under stochastic perturbations, even in conservative systems.