Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exponential mixing in a hydrodynamic pilot--wave theory with singular potentials

Hung D. Nguyen, Anand U. Oza

Abstract

We conduct an analysis of a stochastic hydrodynamic pilot-wave theory, which is a Langevin equation with a memory kernel that describes the dynamics of a walking droplet (or "walker") subjected to a repulsive singular potential and random perturbations through additive Gaussian noise. Under suitable assumptions on the singularities, we show that the walker dynamics is exponentially attracted toward the unique invariant probability measure. The proof relies on a combination of the Lyapunov technique and an asymptotic coupling specifically tailored to our setting. We also present examples of invariant measures, as obtained from numerical simulations of the walker in two-dimensional Coulomb potentials. Our results extend previous work on the ergodicity of stochastic pilot-wave dynamics established for smooth confining potentials.

Exponential mixing in a hydrodynamic pilot--wave theory with singular potentials

Abstract

We conduct an analysis of a stochastic hydrodynamic pilot-wave theory, which is a Langevin equation with a memory kernel that describes the dynamics of a walking droplet (or "walker") subjected to a repulsive singular potential and random perturbations through additive Gaussian noise. Under suitable assumptions on the singularities, we show that the walker dynamics is exponentially attracted toward the unique invariant probability measure. The proof relies on a combination of the Lyapunov technique and an asymptotic coupling specifically tailored to our setting. We also present examples of invariant measures, as obtained from numerical simulations of the walker in two-dimensional Coulomb potentials. Our results extend previous work on the ergodicity of stochastic pilot-wave dynamics established for smooth confining potentials.

Paper Structure

This paper contains 15 sections, 10 theorems, 222 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Physical background
  3. Overview of the main results.
  4. Organization of the paper
  5. Main results
  6. Functional setting
  7. Main assumptions and well-posedness
  8. Geometric ergodicity
  9. Moment estimates
  10. Geometric ergodicity
  11. Asymptotic strong Feller property
  12. Irreducibility
  13. Proof of the main results
  14. Numerical simulations of the invariant measure
  15. Auxiliary results

Key Result

Theorem 1.1

Under suitable assumptions on the functions $U$, $G$, $H$ and $K$, and suitable initial data $(x_0,v_0)$, let $(x(t),v(t))$ be the solution of eqn:droplet:original. Then, there exists a unique probability measure $\pi$ on $\mathcal{R}^d$ such that for every suitable observable $f\in C( \mathcal{R}^d for some positive constants $C$ and $c$ independent of $t$.

Figures (1)

  • Figure 1: Numerical simulations of Eq. \ref{['eqn:droplet:original']} for $m = 1$, $\sigma=1$, $U(x) = |x |^2/2$ and $G(x) = -\alpha\log(|x|)$, for $\alpha=1$ (panels (a) through (d)), $\alpha=3$ (panels (e) through (h)) and $\alpha=5$ (panels (i) through (l)). In the leftmost panels, the gray curves show the two-dimensional walker position $x(t)=(x_1(t),x_2(t))$ over the time interval $t_{\text{max}} -1000 < t < t_{\text{max}}$. The blue curve highlights a subset of this trajectory over the interval $t_{\text{max}} - 50 < t < t_{\text{max}}$. The corresponding time series for the radius $r=|x(t)|$ and speed $|v(t)|$ are shown in the middle panels. The rightmost panels show the walker's radial position probability density function $p(r)$. The variables are divided by $2\pi$, which corresponds to the approximate periodicity of the pilot-wave force function $H(x)=\text{J}_1(x)x/|x|$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 2.2
  • Remark 2.5
  • Remark 2.7
  • Definition 2.9
  • Remark 2.11
  • Theorem 2.12
  • Corollary 2.13
  • Remark 2.14
  • Lemma 3.1
  • ...and 16 more