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Burgers dynamics for Poisson point process initial conditions of the Weibull class

Patrick Valageas

TL;DR

This work analyzes the deterministic 1D Burgers equation in the inviscid limit with stochastic, Poisson-generated initial velocity potentials drawn from a Weibull-class power-law intensity $\lambda(\psi_0)=a\psi_0^{\alpha}$ for $\alpha>-1$. Using the geometric framework of first-contact parabolas and Legendre transforms, the authors derive exact, closed-form statistics for Eulerian and Lagrangian observables, including one- and two-point distributions, void and shock multiplicities, and their associated spectra; they also prove a factorization of the $n$-point distributions into two-point conditional probabilities and establish self-similar evolution with a scale $L(t) \propto t^{(\alpha+1)/(2\alpha+3)}$. The results exhibit stretched-exponential tails whose exponents depend on $\alpha$, connecting to Gaussian/Fréchet limits in appropriate regimes, and provide a unified, tractable description of Burgers turbulence for this broad initial-condition class. These analytical expressions illuminate how discreteness in the initial Poisson configuration seeds shocks and voids, shaping large-scale correlations and the density and velocity spectra across Eulerian and Lagrangian frameworks.

Abstract

We derive the statistical properties of one-dimensional Burgers dynamics with stochastic initial conditions for the velocity potential defined by a Poisson point process whose intensity follows a power law with exponent $α> -1$. Working in the inviscid limit and exploiting the geometrical construction of solutions in terms of first-contact parabolas, we derive explicit analytical expressions for a broad set of statistical quantities. These include the one- and two-point probability distributions of the velocity, the multiplicity functions of voids and shocks, and the velocity and density correlation functions together with their associated power spectra. We also show that the full hierarchy of $n$-point distributions factorizes into a sequence of two-point conditional probabilities. This class of initial conditions leads to self-similar evolution and produces probability distributions characterized by stretched-exponential tails, with tail exponents spanning the full range from unity to infinity. The associated characteristic length scale grows as a power law of time, with an exponent lying between zero and one half.

Burgers dynamics for Poisson point process initial conditions of the Weibull class

TL;DR

This work analyzes the deterministic 1D Burgers equation in the inviscid limit with stochastic, Poisson-generated initial velocity potentials drawn from a Weibull-class power-law intensity for . Using the geometric framework of first-contact parabolas and Legendre transforms, the authors derive exact, closed-form statistics for Eulerian and Lagrangian observables, including one- and two-point distributions, void and shock multiplicities, and their associated spectra; they also prove a factorization of the -point distributions into two-point conditional probabilities and establish self-similar evolution with a scale . The results exhibit stretched-exponential tails whose exponents depend on , connecting to Gaussian/Fréchet limits in appropriate regimes, and provide a unified, tractable description of Burgers turbulence for this broad initial-condition class. These analytical expressions illuminate how discreteness in the initial Poisson configuration seeds shocks and voids, shaping large-scale correlations and the density and velocity spectra across Eulerian and Lagrangian frameworks.

Abstract

We derive the statistical properties of one-dimensional Burgers dynamics with stochastic initial conditions for the velocity potential defined by a Poisson point process whose intensity follows a power law with exponent . Working in the inviscid limit and exploiting the geometrical construction of solutions in terms of first-contact parabolas, we derive explicit analytical expressions for a broad set of statistical quantities. These include the one- and two-point probability distributions of the velocity, the multiplicity functions of voids and shocks, and the velocity and density correlation functions together with their associated power spectra. We also show that the full hierarchy of -point distributions factorizes into a sequence of two-point conditional probabilities. This class of initial conditions leads to self-similar evolution and produces probability distributions characterized by stretched-exponential tails, with tail exponents spanning the full range from unity to infinity. The associated characteristic length scale grows as a power law of time, with an exponent lying between zero and one half.

Paper Structure

This paper contains 23 sections, 77 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Equations of motion and Initial conditions
  3. Equations of motion
  4. Geometrical interpretation and Legendre transform
  5. Initial condition
  6. Self-similar dynamics
  7. Numerical realizations
  8. Limit $\alpha \to \infty$
  9. One-point Eulerian distributions
  10. Two-point Eulerian distributions
  11. Two first-contact parabolas
  12. Void probabilities
  13. Probability of an empty interval
  14. Multiplicity function of voids and distance between shocks
  15. Two-point velocity correlation and energy power spectrum
  16. ...and 8 more sections

Figures (9)

  • Figure 1: A realization of the system for the cases $\alpha=-1/2$ (top row) and $\alpha=2$ (bottom row) at time $t=1$. Left column: the initial velocity potential $\psi_0(x)$ (red dashed curve) and the evolved velocity potential $\psi(x,t)$ (blue solid curve). Middle column: velocity field $v(x,t)$. Right column: mass and location of the shocks.
  • Figure 2: One-point probability distribution $P_0(q)=P_0(v)$ of the Lagrangian coordinate $q$, or of the velocity $v$, from Eq.(\ref{['eq:P_0-q']}). We display our results on a linear scale (left panel) and a logarithmic scale (right panel), for the cases $\alpha= -0.5, 0$, and $2$. The dotted lines in the right panel are the asymptotic results (\ref{['eq:P0-large-q']}).
  • Figure 3: Void probability $P_{\rm void}(x)$ from Eq.(\ref{['eq:Pvoid']}), for the cases $\alpha=-1/2, 0$, and $2$, as in Fig. \ref{['fig:P0_q']}. The dotted lines in the right panel are the asymptotic stretched exponentials (\ref{['eq:Pvoid-x-0-large-x']}).
  • Figure 4: Left panel: cumulative void multiplicity function $n_{\rm void}(>x)$ from Eq.(\ref{['eq:n-void-R-alpha']}). Middle panel: void multiplicity function $n_{\rm void}(x)$ from Eq.(\ref{['eq:n-void-R-alpha']}). Right panel: number density of voids $N_{\rm void}$ as a function of $\alpha$ (solid blue line). In the left and middle panels, the dotted lines are the stretched exponentials associated with Eq.(\ref{['eq:nvoid-power-law']}). In the right panel the dotted and dashed lines are the asymptotic regimes (\ref{['eq:Nvoid-asymp']}).
  • Figure 5: Velocity correlation $B_v(x)$ for the cases $\alpha= -0.5, 0$, and $2$. In the right panel the dotted lines are the asymptotic stretched exponentials (\ref{['eq:Bv-asymp']}).
  • ...and 4 more figures