Burgers dynamics for Poisson point process initial conditions of the Weibull class
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 . This choice realizes a broad and analytically tractable family of strongly non-Gaussian initial states, relevant as limiting cases for problems ranging from turbulence phenomenology to simplified models of structure formation in cosmology. 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.
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BibTeX
@article{valageas2026burgers,
title = {{Burgers dynamics for Poisson point process initial conditions of the Weibull class}},
author = {Patrick Valageas},
journal = {Journal of Statistical Mechanics: Theory and Experiment},
year = {2026},
month = may,
doi = {10.1088/1742-5468/ae6071},
eprint = {2512.09813},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05759200v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Burgers dynamics for Poisson point process initial conditions of the Weibull class
AU - Patrick Valageas
PY - 2026
DA - 2026/05/15
JO - Journal of Statistical Mechanics: Theory and Experiment
DO - 10.1088/1742-5468/ae6071
UR - https://hal.science/hal-05759200v1
AB - 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 . This choice realizes a broad and analytically tractable family of strongly non-Gaussian initial states, relevant as limiting cases for problems ranging from turbulence phenomenology to simplified models of structure formation in cosmology. 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.
SN - 1742-5468
ER -