Burgers' Flows as Markovian Diffusion Processes

TL;DR

This paper explores how the Burgers' equation can be interpreted as a Markovian diffusion process, connecting fluid dynamics with probabilistic models and quantum theory, extending understanding of passive matter transport.

Contribution

It provides a complete characterization of diffusive transport governed by Burgers velocity fields, linking deterministic flows with stochastic diffusion processes.

Findings

Burgers' flows can be interpreted as Markovian diffusion processes.

The analysis extends to incompressible and compressible media.

Connections are made to quantum probabilistic dynamics.

Abstract

We analyze the unforced and deterministically forced Burgers equation in the framework of the (diffusive) interpolating dynamics that solves the so-called Schr\"{o}dinger boundary data problem for the random matter transport. This entails an exploration of the consistency conditions that allow to interpret dispersion of passive contaminants in the Burgers flow as a Markovian diffusion process. In general, the usage of a continuity equation $\partial_t\rho =-\nabla (\vec{v}\rho)$, where $\vec{v}=\vec{v}(\vec{x},t)$ stands for the Burgers field and $\rho $ is the density of transported matter, is at variance with the explicit diffusion scenario. Under these circumstances, we give a complete characterisation of the diffusive transport that is governed by Burgers velocity fields. The result extends both to the approximate description of the transport driven by an incompressible fluid and to motions in an infinitely compressible medium. Also, in conjunction with the Born statistical postulate in quantum theory, it pertains to the probabilistic (diffusive) counterpart of the Schr\"{o}dinger picture quantum dynamics.