[Paper Review] Dynamical random multiplicative cascade model in 1+1 dimensions

Juergen Schmiegel, H. C. Eggers+1 more|arXiv (Cornell University)|Jun 18, 2001

Complex Network Analysis Techniques1 references3 citations

TL;DR

This paper proposes a dynamical random multiplicative cascade model in 1+1 space-time dimensions that generalizes static multifractal models by incorporating continuous, causal stochastic evolution. By defining the energy dissipation field as a log-stable moving average over a stable white noise field with a time- and space-dependent window function, the model reproduces multifractal scaling in both equal-time and temporal two-point correlations, with scaling exponents matching those of the geometric model when large-scale corrections are applied.

ABSTRACT

Geometrical random multiplicative cascade processes are often used to model positive-valued multifractal fields such as for example the energy dissipation field of fully developed turbulence. A dynamical generalisation of these models is proposed, which describes the continuous and homogeneous stochastic evolution of the field in one space and one time dimension. Two-point correlation functions are calculated.

Motivation & Objective

• To develop a causal, continuous-time generalization of geometric random multiplicative cascade models (RMCMs) that can describe the stochastic evolution of positive-valued multifractal fields such as energy dissipation in turbulence.
• To ensure the model preserves the multifractal scaling properties of the original geometric RMCM while incorporating time evolution and causality.
• To derive and analyze equal-time and temporal two-point correlation functions to verify that the model exhibits the same multifractal scaling exponents as the static model.
• To demonstrate that the model's statistical properties, including moments and multiplier distributions, are consistent with experimental observations of turbulent energy dissipation.

Proposed method

• The energy dissipation field ε(x,t) is constructed as an exponential functional of a stable white noise field γ(x,t), integrated over a spatio-temporal window function f(x,t) that ensures causality (f=0 for t<0).
• The window function g(t) is defined to create a scale-invariant hierarchy of time scales, translating the discrete scale hierarchy of geometric RMCMs into a continuous temporal evolution.
• The model uses a stable Lévy noise field with index α ∈ (0,2], ensuring that the logarithmic field has stable increments and enabling exact calculation of moments via characteristic functions.
• Two-point correlation functions are derived by computing the overlap volume of integration regions for different space-time points, leading to expressions involving the spatio-temporal overlap volume V(l) or V(t).
• The multifractal scaling exponents τ(n) are defined via the stable distribution properties, with τ(n) ∝ σ^α / cos(πα/2) × V(l) × (n - n^α), linking the model's statistics to its underlying noise and geometry.
• A small extension of the window function beyond t = -T is introduced to cancel large-scale deviations in the two-point correlator, restoring exact power-law scaling over all scales.

Experimental results

Research questions

• RQ1 Can a continuous, causal stochastic model be constructed in 1+1 dimensions that generalizes the geometric random multiplicative cascade model while preserving its multifractal scaling properties?
• RQ2 How do equal-time and temporal two-point correlation functions behave in such a dynamical model, and do they exhibit the same multifractal scaling as in the static case?
• RQ3 What is the role of the spatio-temporal overlap volume in determining the scaling exponents of the correlation functions?
• RQ4 Can large-scale deviations in the two-point correlator be corrected to restore exact power-law scaling across all scales?

Key findings

• The equal-time two-point correlator is given by C_{n1,n2}(l) = (L/l)^{τ[n1,n2]} × exp(−τ[n1,n2](1−l/L)), showing multifractal scaling for η < l ≪ L.
• By extending the window function beyond t = -T by ΔT = (T−tη)η/(L−η), the large-scale deviation is canceled, yielding exact power-law scaling: C_{n1,n2}(l) = (L/l)^{τ[n1,n2]} for all η ≤ l ≤ L.
• The temporal two-point correlator scales as C_{n1,n2}(t) ≈ (T/t)^{τ[n1,n2]} for 0 ≪ t < T, with the same scaling exponents as in the equal-time case.
• The multifractal scaling exponents are defined as τ(n) = τ(2)(n−n^α)/(2−2^α), with τ(2) proportional to σ^α / cos(πα/2) × ηL(T−tη)/(L−η), linking the model to stable noise parameters.
• The model’s moments, multiplier distributions, and higher-order correlations are consistent with experimental data from turbulent energy dissipation fields.