Statistical properties of the Burgers equation with Brownian initial velocity

TL;DR

This paper analyzes the statistical properties of the Burgers equation with Brownian initial velocity, deriving explicit distributions and correlations, and revealing exactness of certain cosmological models in a 1D setting.

Contribution

It provides closed-form distributions for velocity, density, and Lagrangian fields in the Burgers equation with Brownian initial conditions, highlighting exact cosmological model analogs.

Findings

Derived one-point velocity distribution in closed form

Established factorization and invariance properties of distributions

Confirmed the exactness of the stable-clustering and Press-Schechter models in 1D

Abstract

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its $n$-point correlations. In the same limit, we derive the $n-$point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.