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Burgers dynamics for Poisson point process initial conditions

Patrick Valageas

TL;DR

This work studies 1D Burgers dynamics in the inviscid limit starting from Poisson-point initial conditions for the velocity potential with intensity $oldsymbol{oldsymbol{\lambda(oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{oldsymbol{ rac{α}{}}}}}}}}}}}$, controlled by the exponent $oldsymbol{α>3/2}$. The authors exploit a geometric first-contact parabola representation to derive exact one- and two-point Eulerian distributions, void and shock statistics, and velocity-density power spectra, revealing self-similar dynamics across times and a heavy-tailed, $oldsymbol{α}$-dependent probability structure. In the limit $oldsymbol{α\to∞}$, the tails become Gaussian, and the results converge to Kida's Gaussian initial-condition regime with vanishing large-scale power; this connects rare-event dominated Burgers dynamics to classical diffusion-like behavior. The work highlights how rare initial peaks govern the nonlinear formation of shocks and voids, providing a precise solvable benchmark for universality classes in nonlinear transport and structure formation, with clear pathways to higher dimensions and stochastic forcing extensions.

Abstract

We investigate the statistical properties of one-dimensional Burgers dynamics evolving from stochastic initial conditions defined by a Poisson point process for the velocity potential, with a power-law intensity. Thanks to the geometrical interpretation of the solution in the inviscid limit, in terms of first-contact parabolas, we obtain explicit results for the multiplicity functions of shocks and voids, and for velocity and density one- and two-point correlation functions and power spectra. These initial conditions gives rise to self-similar dynamics with probability distributions that display power-law tails. In the limit where the exponent $α$ of the Poisson process that defines the initial conditions goes to infinity, the power-law tails steepen to Gaussian falloffs and we recover the spatial distributions obtained in the classical study by Kida (1979) of Gaussian initial conditions with vanishing large-scale power.

Burgers dynamics for Poisson point process initial conditions

TL;DR

This work studies 1D Burgers dynamics in the inviscid limit starting from Poisson-point initial conditions for the velocity potential with intensity , controlled by the exponent . The authors exploit a geometric first-contact parabola representation to derive exact one- and two-point Eulerian distributions, void and shock statistics, and velocity-density power spectra, revealing self-similar dynamics across times and a heavy-tailed, -dependent probability structure. In the limit , the tails become Gaussian, and the results converge to Kida's Gaussian initial-condition regime with vanishing large-scale power; this connects rare-event dominated Burgers dynamics to classical diffusion-like behavior. The work highlights how rare initial peaks govern the nonlinear formation of shocks and voids, providing a precise solvable benchmark for universality classes in nonlinear transport and structure formation, with clear pathways to higher dimensions and stochastic forcing extensions.

Abstract

We investigate the statistical properties of one-dimensional Burgers dynamics evolving from stochastic initial conditions defined by a Poisson point process for the velocity potential, with a power-law intensity. Thanks to the geometrical interpretation of the solution in the inviscid limit, in terms of first-contact parabolas, we obtain explicit results for the multiplicity functions of shocks and voids, and for velocity and density one- and two-point correlation functions and power spectra. These initial conditions gives rise to self-similar dynamics with probability distributions that display power-law tails. In the limit where the exponent of the Poisson process that defines the initial conditions goes to infinity, the power-law tails steepen to Gaussian falloffs and we recover the spatial distributions obtained in the classical study by Kida (1979) of Gaussian initial conditions with vanishing large-scale power.

Paper Structure

This paper contains 25 sections, 115 equations, 11 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. Limit $\alpha \to \infty$
  8. Numerical realizations
  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 10 more sections

Figures (11)

  • Figure 1: A realization of the system for the cases $\alpha=2.3$ (upper row) and $\alpha=5$ (lower row) at time $t=1$. Left column: the initial velocity potential $\psi_0(x)$ (red dashed curve) and the final velocity potential $\psi(x,t)$ (blue solid curve). Middle left column: velocity field $v(x,t)$. Middle right column: Lagrangian map $x(q,t)$. Right column: mass and location of the shocks.
  • Figure 2: Left panels: one-point probability distribution $P_0(\tilde{q})=P_0(\tilde{v})$ of the Lagrangian coordinate $\tilde{q}$, or of the velocity $\tilde{v}$, from Eq.(\ref{['eq:P_0-q']}). We use the rescaled coordinate $\tilde{q}$ as in (\ref{['eq:tilde-def']}), to illustrate the convergence to Eq.(\ref{['eq:P0-q-alpha-inf']}) in the limit $\alpha\to\infty$. We display our results on linear scales (left panel) and logarithmic scales (middle panel), for the cases $\alpha=2.3, 2.5, 2.8, 3.1, 3.5, 4, 5, \infty$. The horizontal dashed lines in the left panel are the values $P_0(\tilde{q}=0)$ of Eq.(\ref{['eq:P0_q=0']}). The slope at large $\tilde{q}$ in the middle panel becomes steeper for larger $\alpha$. The dashed lines in the middle panel are the asymptotic power-laws (\ref{['eq:P0-large-q']}). Right panel: variance $\langle \tilde{q}^2 \rangle = \langle \tilde{v}^2 \rangle$ of the Lagrangian displacement and of the velocity. The horizontal dashed line is the limit $\alpha\to\infty$, from Eq.(\ref{['eq:P0-q-alpha-inf']}).
  • Figure 3: Void probability $P_{\rm void}(\tilde{x})$ from Eq.(\ref{['eq:Pvoid']}), for the cases $\alpha=2.3, 2.5, 2.8, 3.1, 3.5, 4, 5, \infty$, as in Fig. \ref{['fig:P0_q']}. Again we use the rescaled coordinate $\tilde{x}$ to illustrate the convergence to the limit $\alpha\to\infty$. The slope at large $\tilde{x}$ in the right panel increases with $\alpha$. The dashed lines in the right panel are the asymptotic power-laws (\ref{['eq:Pvoid-x-0-large-x']}), while the dotted line is the asymptotic result (\ref{['eq:Pvoid-alpha-infty-asymp']}) for the case $\alpha=\infty$.
  • Figure 4: Left panel: cumulative void multiplicity function $n_{\rm void}(> \tilde{x})$ from Eq.(\ref{['eq:n-void-R-alpha']}). Middle panel: void multiplicity function $n_{\rm void}(\tilde{x})$ from Eq.(\ref{['eq:n-void-R-alpha']}). Right panel: rescaled number density of voids $\tilde{N}_{\rm void}$ as a function of $\alpha$. In the left and middle panels, the dashed lines are the power laws associated with Eq.(\ref{['eq:nvoid-power-law']}), whereas the dotted lines are the asymptotic regimes associated with Eq.(\ref{['eq:Pvoid-alpha-infty-asymp']}). In the right panel the horizontal dotted line is the value (\ref{['eq:Nvoid-asymp']}) in the limit $\alpha\to\infty$.
  • Figure 5: Velocity correlation $B_{\tilde{v}}(\tilde{x})$ for the cases $\alpha=2.8, 3.1, 3.5, 4, 5, \infty$. In the right panel the dashed lines are the asymptotic power laws (\ref{['eq:Bv-asymp']}) while the dotted line is the asymptotic result (\ref{['eq:Bv-asymp-alpha-infty']}) for $\alpha=\infty$.
  • ...and 6 more figures