# Stochastic hydrodynamic velocity field and the representation of Langevin equations

## TL;DR

This paper introduces a stochastic velocity field approach to Langevin equations, simplifying the modeling of Brownian motion and hydrodynamic fluctuations, especially in complex scenarios like confined geometries.

## Contribution

It proposes a novel hydrodynamic/fluctuational framework using a stochastic velocity field derived from linear response theory, simplifying Langevin equations and incorporating higher-order correlations.

## Key findings

- Langevin equations become simpler with the stochastic velocity field approach.
- Hydrodynamic fluctuations can be modeled as Extended Poisson-Kac Processes.
- Higher-order correlations are crucial for short-time Brownian motion analysis.

## Abstract

The fluctuation-dissipation theorem, in the Kubo original formulation, is based on the decomposition of the thermal agitation forces into a dissipative contribution and a stochastically fluctuating term. This decomposition can be avoided by introducing a stochastic velocity field, with correlation properties deriving from linear response theory. Here, we adopt this field as the comprehensive hydrodynamic/fluctuational driver of the kinematic equations of motion. With this description, we show that the Langevin equations for a Brownian particle interacting with a solvent fluid become particularly simple and can be applied even in those cases in which the classical approach, based on the concept of a stochastic thermal force, displays intrinsic difficulties e.g., in the presence of the Basset force. We show that a convenient way for describing hydrodynamic/thermal fluctuations is by expressing them in the form of Extended Poisson-Kac Processes possessing prescribed correlation properties and a continuous velocity density function. We further highlight the importance of higher-order correlation functions in the description of the stochastic hydrodynamic velocity field with special reference to short-time properties of Brownian motion. We conclude by outlining some practical implications in connection with the statistical description of particle motion in confined geometries.