Burgers dynamics for Poisson point process initial conditions

TL;DR

This paper analyzes the statistical properties of one-dimensional Burgers dynamics starting from Poisson point process initial conditions, revealing explicit shock and void distributions, correlation functions, and self-similar power-law behaviors.

Contribution

It provides explicit analytical results for shock and void statistics, correlation functions, and power spectra for Burgers dynamics with Poisson initial conditions, connecting to classical Gaussian cases.

Findings

Explicit multiplicity functions for shocks and voids

Power-law tails in velocity and density distributions

Recovery of Gaussian distributions as Poisson intensity increases

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 $\alpha$ 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.