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Extending the Droplet-Wave Statistical Correspondence in Walking Droplet Dynamics

Skyler Mao, David Darrow

TL;DR

The paper rigorously extends the PDF-MWF correspondence for walking droplets beyond restrictive single-impact models to arbitrary droplet-bath interactions and multi-droplet configurations by employing evolution-system theory on a Banach space. It proves that if the droplet configuration admits a stationary distribution $\rho$, the long-time mean wave field satisfies $\overline{\eta}(x) = \int \eta_B(x,\boldsymbol{\alpha})\rho(\boldsymbol{\alpha})\,d\boldsymbol{\alpha}$ (Theorem in Appendix), thereby generalizing prior results and removing strict ergodicity requirements. Numerically, the authors validate the generalized relation in a 1-D conformal-map model, demonstrating agreement between measured and predicted mean fields across geometries and memory parameters, and they show that the mean field can reveal correlations between droplets in multi-droplet setups. They also outline applications to multi-droplet statistics, arbitrary interaction models, and non-resonant bouncing, with implications for understanding ponderomotive forces and extending the framework to other time-periodic PDEs. Overall, the work strengthens the quantum-analogous interpretation of walking droplets and provides a versatile tool for diagnosing and predicting complex pilot-wave dynamics.

Abstract

Walking droplets -- millimetric oil droplets that self-propel across the surface of a vibrating fluid bath -- exhibit striking emergent statistics that remain only partially understood. In particular, in a variety of experiments, a robust correspondence has been observed between the droplet's statistical distribution and the time-average of the wave field that guides it. M. Durey, P. A. Milewski, and J. W. M. Bush, Chaos 28, 096108 (2018) rigorously established such a correspondence for single-droplet systems with a single, instantaneous droplet-bath impact during each vibration period, but numerical and experimental evidence suggests that the correspondence should hold far more broadly. Laboratory droplet systems, for instance, often exhibit complex bouncing modes that do not adhere to these hypotheses. We attempt to complete this program in the present work, rigorously extending this statistical correspondence to account for arbitrary droplet-bath impact models, multi-droplet interactions, and non-resonant bouncing. We investigate this correspondence numerically in systems of one and two droplets in 1-D geometries, and we highlight how the time-averaged wave field can distinguish between correlated and uncorrelated pairs of droplets.

Extending the Droplet-Wave Statistical Correspondence in Walking Droplet Dynamics

TL;DR

The paper rigorously extends the PDF-MWF correspondence for walking droplets beyond restrictive single-impact models to arbitrary droplet-bath interactions and multi-droplet configurations by employing evolution-system theory on a Banach space. It proves that if the droplet configuration admits a stationary distribution , the long-time mean wave field satisfies (Theorem in Appendix), thereby generalizing prior results and removing strict ergodicity requirements. Numerically, the authors validate the generalized relation in a 1-D conformal-map model, demonstrating agreement between measured and predicted mean fields across geometries and memory parameters, and they show that the mean field can reveal correlations between droplets in multi-droplet setups. They also outline applications to multi-droplet statistics, arbitrary interaction models, and non-resonant bouncing, with implications for understanding ponderomotive forces and extending the framework to other time-periodic PDEs. Overall, the work strengthens the quantum-analogous interpretation of walking droplets and provides a versatile tool for diagnosing and predicting complex pilot-wave dynamics.

Abstract

Walking droplets -- millimetric oil droplets that self-propel across the surface of a vibrating fluid bath -- exhibit striking emergent statistics that remain only partially understood. In particular, in a variety of experiments, a robust correspondence has been observed between the droplet's statistical distribution and the time-average of the wave field that guides it. M. Durey, P. A. Milewski, and J. W. M. Bush, Chaos 28, 096108 (2018) rigorously established such a correspondence for single-droplet systems with a single, instantaneous droplet-bath impact during each vibration period, but numerical and experimental evidence suggests that the correspondence should hold far more broadly. Laboratory droplet systems, for instance, often exhibit complex bouncing modes that do not adhere to these hypotheses. We attempt to complete this program in the present work, rigorously extending this statistical correspondence to account for arbitrary droplet-bath impact models, multi-droplet interactions, and non-resonant bouncing. We investigate this correspondence numerically in systems of one and two droplets in 1-D geometries, and we highlight how the time-averaged wave field can distinguish between correlated and uncorrelated pairs of droplets.

Paper Structure

This paper contains 11 sections, 6 theorems, 50 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on Walking Droplet Modeling
  3. Numerical PDF-MWF Correspondence
  4. Applications of PDF-MWF Correspondence
  5. Multi-Droplet Statistics
  6. Arbitrary Droplet-Bath Impact Models
  7. Non-Resonant Walking Droplets
  8. Conclusion
  9. Statement and Proof of Claim \ref{['claim:main_heuristic']}
  10. Classical Floquet Theory for the Finite-Dimensional Case
  11. Statistical Correspondence in the PDE Case

Key Result

Lemma 1

Suppose $L(t)$ and $b(t)$ are $T$-periodic and continuous. By Floquet's theorem floquet, the solutions to $\dot{y}=L(t)y$ can be organized into a fundamental matrix $Y(t)\in\mathbb C^{d\times d}$ that takes the form $Y(t) = Q(t)e^{Bt}$, where $B\in\mathbb C^{d\times d}$ is constant and $Q(t)\in\math for any non-negative integer $n$ and any time offset $t_0\in[0,T]$; here, the quantities $C(t_0)=Q(

Figures (5)

  • Figure 1: (a) Snapshot of the instantaneous wave field of a walking droplet in an elliptical bath. Shading corresponds to the slope of the wave field, rather than its height, as the former is more easily detected from an overhead camera Saenz2018. (b) The time-averaged slope of the wave field, which is closely related to the time-averaged wave height (i.e., the mean wave field). (c) Probability density function of the droplet, showing a strong visual similarity with the time-averaged wave slope. A correspondence between the probability density and mean wave field has been widely observed, but only proven under strict assumptions on the droplet-wave interaction. Images modified with permission from P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, "Statistical projection effects in a hydrodynamic pilot-wave system," Nature Physics 14, 315–319 (2018). Copyright 2018 Springer Nature.
  • Figure 2: (a) A snapshot of the droplet model, showing the bath topography (black, partially shown), the wave field (blue), and the droplet (red). (b) A droplet (red) starts at $-0.5~\mathrm{cm}$ and traverses the $3~\mathrm{cm}$ domain with memory parameter $\Gamma/\Gamma_F=0.85$. Snapshots of the underlying wavefield (blue) are shown at each Faraday period. (c) The long-term mean wave field (black) is calculated by averaging the value of the wave field at the start of each Faraday period.
  • Figure 3: Mean wave fields (MWFs) and droplet position PDFs for six simulations, stopped after $2\times 10^4$ Faraday periods $T_F$. In each, a single droplet moves around within a rectangular cavity of width (a-c)$3~\mathrm{cm}$ and (d-f)$4~\mathrm{cm}$, with various memory parameters $\Gamma/\Gamma_F$. For each MWF, we compare the measured MWF against the prediction of Claim \ref{['claim:main_heuristic']}, noting a close match in each case. We show each PDF calculated at times $1.2\times 10^4\,T_F$, $1.6\times10^4\,T_F$, and $2.0\times 10^4\,T_F$ to demonstrate convergence.
  • Figure 4: (a) A droplet (red) starts at $-0.5~\mathrm{cm}$ and traverses the $3~\mathrm{cm}$ domain with memory parameter $\Gamma/\Gamma_F=0.99$. The trajectory of the droplet is chaotic, in contrast to the numerical experiments of Figures \ref{['fig:exdrop']} and \ref{['fig:mwfcomp']}. (b) The PDF of the droplet's position at times $0.8\times 10^5\,T_F$, $1.0\times 10^5\,T_F$, and $1.2\times 10^5\,T_F$, demonstrating statistical convergence. (c) The measured MWF at time $1.2\times 10^5\,T_F$ aligns closely with the predicted MWF of Claim \ref{['claim:main_heuristic']}.
  • Figure 5: Two droplets move in each of the asymmetrical two-cavity systems, which we label 'uncorrelated' and 'correlated', respectively. The uncorrelated domain consists of two cavities separated by a wide central barrier, preventing the droplets from communicating with one another; the correlated domain replaces the barrier with another cavity, such that the droplets interact through their shared wave field. One droplet starts at $-1.20~\mathrm{cm}$ (red) and another starts at $1.00~\mathrm{cm}$ (black). (a,d) The uncorrelated and correlated systems, respectively, during the first 200 Faraday periods. (b,e) PDFs of the two systems after $1.8\times 10^4$ Faraday periods. (c,f) The comparison between the measured MWFs, the predicted MWFs, and the sum of the two MWFs that arise when one of the two droplets is removed from the system.

Theorems & Definitions (21)

  • Claim 1
  • Lemma 1
  • proof
  • Theorem 1
  • Remark 1
  • proof
  • Definition 1
  • Example 1
  • Definition 2
  • Remark 2
  • ...and 11 more