Can It Be Used for Quantum Computers or Ultimate Theories? Recent Developments in the Theory and Experiment of Quantum-like Phenomena (HQFA, HQFT)

This article describes quantum-like phenomena within classical systems, known as Hydrodynamic Quantum Field Analogs (Theory). It is an extremely interesting phenomenon that has been rapidly developing. The developments in both theory and experiment over the past ten years have been summarized below using OpenAI o1 pro + Deep Search.

下記の英訳版です。（Below is the English translation version:）

https://qiita.com/ilabotakeda/items/209c4c1b1d3a09a45a7d

# Hydrodynamic Quantum Field Analogs: Developments in the Past Decade

## Introduction
In recent years, a research field called **Hydrodynamic Quantum Field Analogs** has attracted growing attention. This field seeks to recreate phenomena observed in quantum mechanics or quantum field theory using classical hydrodynamic systems. The discovery in 2005 of “**walking droplets**” by Yves Couder and Emmanuel Fort, in which droplets self-propel on the surface of a vibrating fluid bath by resonating with waves they themselves generate, marks the starting point of this area [1†L28–L36]. In this system, the droplet (particle) behaves as an excitation of the fluid surface (wave), concretely illustrating in a classical setting the concept of “a particle as a field excitation” familiar from quantum field theory [1†L28–L36]. The walking-droplet system has also been likened to the macroscopic realization of the pilot-wave picture proposed in the 1920s by Louis de Broglie, sparking significant interest in the bridge it offers between the probabilistic behavior of quantum mechanics and the deterministic trajectories of classical mechanics [1†L32–L39].

In this article, I summarize **the theoretical developments of the past decade**, focusing on hydrodynamic systems that exhibit quantum-like phenomena, often referred to as **quantum analogs**. Special attention is given to analogs of quantum vacuum, nonequilibrium fluid dynamics aspects, relationships with the Casimir effect, and the dynamics of solitons and vortices. I also describe experimental verifications and potential applications related to superfluidity and Bose–Einstein condensates (BEC), as well as surface-wave phenomena. Moreover, I review contributions by leading researchers—especially John W. M. Bush—and discuss the relevant theoretical models, including mathematical formulations. Finally, I shed light on the interface of theory and experiment and outline future perspectives.

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## 1. Analogs of the Quantum Vacuum
In quantum field theory, the **quantum vacuum** is the lowest-energy state, yet it exhibits zero-point fluctuations and virtual particle–antiparticle creation, producing observable effects such as the Casimir effect and Hawking radiation. Efforts have been made to reproduce analogs of these quantum-vacuum phenomena in hydrodynamic systems.

One approach is to regard **classical fluid-surface waves** as analogs of the quantum vacuum. For example, in experiments that realize **water-wave analogs of the Casimir effect**, two parallel plates are placed in a bath of random surface waves driven in an oscillating fluid container. A reduction in the amplitude of waves between the plates can lead to an attractive force that pulls them together [5†L175–L183]. This directly visualizes the three key elements of the quantum Casimir effect—vacuum fluctuations, mode suppression between plates, and the resulting attraction—and matches qualitatively with the concept of the electromagnetic zero-point fluctuations in the quantum vacuum [5†L175–L183]. Just as certain wave modes are cut off between the plates, leading to unbalanced radiation pressure, the classical system demonstrates how vacuum-like wave fluctuations can induce an effective force [5†L175–L183] [5†L169–L177] [4†L5–L13].

**Hawking radiation analogs** also stem from quantum vacuum effects. Following William Unruh’s proposal, “acoustic black holes” or “white holes” have been studied using sound waves or shallow-water waves, and **acoustic Hawking radiation** has been tested. For instance, in rubidium BECs (a superfluid), constructing a horizon (a discontinuity in the speed of sound) led to the observation of spontaneous Hawking radiation in the form of phonon pairs [23†L27–L35]. Jeff Steinhauer’s 2016 experiment at nanokelvin temperatures identified thermal phonon emission from this horizon, suggesting the spontaneous creation of pairs out of a quantum-vacuum-like condensate [23†L27–L35]. Similarly, in fluid flow past an obstacle, the transformation of upstream long waves into short waves across a horizon has been investigated as an analog of the thermal properties of Hawking radiation in a white-hole configuration [25†L100–L109]. Classical fluid waves thus provide a different perspective on phenomena that originate in quantum vacuum fluctuations.

In the walking-droplet system, a vibrating fluid bath serves as a kind of **“background field”**, mediating forces that act on the droplet. Hence, the combination of bath + surface wave can be regarded as a **classical model of the quantum vacuum (or pilot wave)**. Indeed, Bush and colleagues have proposed a **Hydrodynamic Quantum Field Theory (HQFT)** using this system [16†L369–L377]. In that model, the “quantum particle” has an internal high-frequency oscillation (of order \(2\omega_c\), related to its “mass” \(m_0\)) that perturbs the surrounding scalar field, creating a pilot wave that guides the particle [17†L23–L31][17†L25–L33]. Under appropriate assumptions, the pilot wave structure for a free particle traveling at constant velocity features de Broglie and Compton wavelengths, and the average momentum satisfies \(p = h/\lambda_{\mathrm{dB}}\) (where \(\lambda_{\mathrm{dB}}\) is the de Broglie wavelength) [17†L31–L40]. This suggests that pilot-wave interactions with a vacuum-like background might reproduce quantum-like motion—an attempt to reconstruct quantum theory from insights gained in the walking-droplet system.

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## 2. The Role of Nonequilibrium Fluid Dynamics
Many hydrodynamic quantum-analog systems operate in fundamentally **nonequilibrium states**. The system is continuously driven, and quantum-like behaviors arise when energy input balances dissipation. That external driving is necessary to reproduce intrinsic zero-point fluctuations or oscillations characteristic of quantum systems in a classical context.

In the **walking-droplet** system, the fluid bath is driven by vertical oscillations \(\bigl(a(t) = A_0 \cos \omega t\bigr)\), seeding standing waves. This is a **parametric (Faraday) instability** phenomenon in which the fluid surface spontaneously develops pattern-forming waves above a critical driving acceleration \(\gamma_F\). In typical walking-droplet experiments, the system is operated just below this **Faraday threshold** \((\gamma \lesssim \gamma_F)\). Under threshold, waves ordinarily damp out, but the droplet’s periodic impacts keep re-exciting waves before they decay—leading to a special nonequilibrium state with **“memory”** of past collisions. This **“path-memory”** effect means the droplet’s motion depends on its entire wave history, showing non-Markovian behavior [1†L39–L47]. Because the system is **temporally nonlocal** (the past influences the present), the droplet can sense distant obstacles, reminiscent of the nonlocality seen in quantum wavefunctions [1†L39–L47].

Other phenomena also rely on nonequilibrium forcing. In **water-wave Casimir analog experiments**, random waves are driven in a shaken water tank; in Hawking radiation analogs, fluid flow is established to create a horizon. In every case, a **static equilibrium** would not yield these effects; it is the interplay of driving and dissipation that allows quantumlike behavior to emerge. This underscores that hydrodynamic quantum analogs are not mere static models but rather demonstrate how **orderly quantum-like phenomena** can arise in a **nonequilibrium driven–dissipative system**. From the viewpoint of nonequilibrium statistical mechanics and chaos theory, these systems are compelling testbeds for emergent phenomena.

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## 3. Classical Analogs of the Casimir Effect
The **Casimir effect**—an attraction (or repulsion) between conducting plates due to zero-point energy of the quantum vacuum—has long been deemed a hallmark of quantum physics. Nonetheless, hydrodynamic systems have reproduced classical analogs of it. In the **water-wave Casimir** experiment by Denardo and colleagues (2009), random gravity–capillary waves were excited in a shaken container; two vertical plates inserted parallel to each other experienced an attractive force when wave amplitude was suppressed between them [5†L175–L183]. This is conceptually akin to the “dead-water” phenomenon in marine contexts, in which two large ships in close proximity can be drawn together by reduced wave activity between them [5†L160–L168]. Denardo et al. observed direct analogs of the wave generation, wave suppression, and plate attraction inherent to the Casimir picture [5†L175–L183]. They also discussed how an appropriately chosen cutoff in the wave spectrum can lead to repulsive forces, paralleling the possibility of repulsive Casimir forces in quantum field scenarios [5†L169–L177]. Such classical demonstrations provide intuitive insight into the mechanism behind the quantum Casimir force.

In walking-droplet systems, an **effective Casimir-like potential** sometimes arises between multiple droplets. Each droplet emits waves that interfere, and if interference suppresses wave intensity between droplets, they can be pulled together by a **wave-mediated attraction**. This phenomenon is typically dynamic rather than purely static, involving time-dependent coupling. Nonetheless, it is **wave-mediated** and reminiscent of Casimir interactions. Additional effects (e.g., air flow, meniscus forces) can complicate the picture, but at least the notion of **particles interacting via a shared wave field** has been experimentally demonstrated. This hydrodynamic system thus illustrates how an intermediary field can produce an effective interparticle force—an analog of the vacuum-mediated Casimir effect.

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## 4. Solitons and Vortex Dynamics
In quantum field theory, **solitons** and **vortices** arise as localized structures resulting from nonlinearities and topology. They are of great importance in both condensed matter (e.g., superconductors) and high-energy physics (e.g., cosmic strings). Classical fluids have long been known to exhibit vortex rings and solitary waves, inspiring, for example, Lord Kelvin’s 19th-century atomic vortex-ring model [8†L230–L238]. Modern work on hydrodynamic quantum analogs naturally explores soliton-like and vortex-like excitations.

In the walking-droplet system, one might think it differs from solitons since a droplet is effectively the “particle” and the wave is the “field.” Yet droplet and wave are inseparably coupled, moving together in a **solitary-wave-like** manner. Indeed, the droplet continually taps its wave’s crest for propulsion, and the wave is continually regenerated by the droplet’s impacts. This is reminiscent of a **particle–wave bound state**, akin to a soliton. Unlike standard solitary waves, however, the walking-droplet soliton requires **external vibration** to sustain it. Even so, theoretical models show that a stable, traveling droplet–wave solution does exist, often taking a **Bessel-function form** for the wave profile that decays radially [12†L43–L47][12†L19–L27]. This solution reveals how the droplet’s bounce frequency synchronizes with the wave field’s phase to maintain stable self-propulsion [15†L39–L47].

Regarding **vortices**, classical fluid vortices and quantum vortices in superfluids or BECs provide parallel lines of study. In superfluid helium or atomic condensates, **quantized vortices** are topological defects where the phase of the condensate changes by \(2\pi\). Classical fluids do not quantize circulation, but vortex rings exhibit fascinating dynamics and can even be knotted. Recently, experimental achievements include **tying knots** in vortex rings, testing Kelvin’s old hypothesis about knotted vortices. These phenomena are more topological than specifically quantum, yet they open avenues for comparing classical vortex structures with their quantum counterparts (quantum vortices, cosmic strings, etc.).

Interestingly, small **toroidal vortices** can form beneath droplets as they bounce. Research by Chu, Tsai, and others shows that a circular vortex ring is generated in the fluid each time a droplet strikes, with geometry depending on droplet size and impact conditions [7†L37–L43]. When two droplets bounce near each other, these vortices can become asymmetric, imparting lateral forces that cause the droplets to pair up or orbit each other [7†L37–L43]. This phenomenon can be regarded as a **wave (or vortex)–mediated interparticle coupling**, reminiscent of quantum field interactions. Bush and collaborators have studied how droplets can “entangle” or coordinate with each other through their shared wave field, fueling speculation about potential **entanglement analogies** [20†L67–L75].

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## 5. Applications to Superfluids and Bose–Einstein Condensates
Hydrodynamic analogies to quantum field theory also apply to **quantum fluids** themselves. Superfluid helium and atomic Bose–Einstein condensates (BECs) are macroscopic quantum states, but at large scales they behave like fluids described by the Gross–Pitaevskii equation (a nonlinear Schrödinger equation). This dual nature—quantum on small scales, but fluid-like on large scales—enables **quantum analog** experiments to reproduce phenomena from other fields.

The **Hawking radiation analog** in a BEC is a prime example. Because a BEC supports sound waves, Jeff Steinhauer at the Technion created a horizon (a boundary with discontinuous sound speed) in a cigar-shaped condensate to test acoustic Hawking radiation [23†L27–L35]. In 2014, he realized a “black hole laser” with two horizons where phonons would bounce back and forth, amplifying to yield spontaneous emission. Then in 2016, he observed **spontaneous Hawking radiation** from a single horizon, detecting thermal correlations in phonon pairs [23†L27–L35]. This gave indirect though compelling evidence for quantum Hawking-like emission, shining new light on quantum field theoretical predictions.

Superfluid helium has also been studied for potential analogies. Helium II can be described by a **two-fluid model** with unique modes like second sound, absent in normal fluids. Researchers have investigated whether inflationary cosmology or cosmic defect creation can be mimicked in superfluid helium. In quantum turbulence (tangled networks of quantized vortex lines), the energy dissipation mechanisms differ from those in classical turbulence, yet large-scale simulations and low-temperature experiments reveal a Kolmogorov-like \(k^{-5/3}\) cascade. These parallels between quantum and classical turbulence highlight deeper correspondences and differences.

Moreover, **dark solitons** and **quantized vortices** in BECs serve as topological excitations, paralleling classical shallow-water solitons and vortex rings. The collision or passage of dark solitons in a BEC can mirror classical soliton collisions, while vortex rings or vortex lattices show striking resemblances to classical flows. Artificial gauge fields induced in BECs by rotation or optical means simulate **Landau levels** and even quantum-Hall-like states. Though these are genuinely quantum states, the conceptual framework—“using one physical system to model phenomena in another”—aligns with the broader notion of **quantum analog experiments**.

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## 6. Connections with Surface-Wave Phenomena
Most **hydrodynamic quantum analogs** are strongly tied to **surface-wave phenomena**. The walking-droplet system notably relies on **Faraday waves**, standing waves excited in a fluid under vertical vibration, discovered by Michael Faraday in 1831. Mathematically, in the shallow-water approximation, Faraday waves arise from Floquet or Mathieu-type instabilities governed by the dispersion relation \(\omega^2 = \bigl(gk + \tfrac{\sigma}{\rho}k^3\bigr)\tanh(kH)\), where \(g\) is gravitational acceleration, \(\sigma\) is surface tension, \(\rho\) is density, and \(H\) is fluid depth.

In the walking-droplet context, the **phase** of the droplet’s bounce with respect to the container’s vibration is crucial. Near half the forcing frequency, strong Faraday waves are generated, effectively creating a **resonant walking** condition that maximizes wave amplitude and droplet propulsion [15†L33–L41][15†L43–L50]. If the droplet’s bouncing phase drifts away from this resonance, wave amplitude decreases, and the droplet’s motion can become slow or chaotic [15†L47–L54]. The droplet’s stable walking depends critically on this **impact phase**—a key parameter in Moláček & Bush’s (2013) analytical model [15†L43–L50].

A major question concerns the extent to which wave properties such as **diffraction and interference** manifest in the droplet system. In early experiments, Couder et al. (2006) observed that droplets passing through **two slits** in a barrier produce interference-like patterns in their final angular distribution, reminiscent of single-particle interference in optics or quantum mechanics [19†L49–L57][19†L61–L65]. Intriguingly, droplets sometimes sense the presence of a slit they do not physically traverse (a “which-slit” puzzle) [19†L61–L65]. Critics argue that reproducing stable interference fringes requires **high memory**, meaning the forcing acceleration \(\gamma\) must be extremely close to the Faraday threshold. Later attempts (Andersen et al.) found no clear interference fringes; they instead reported strong correlation between incoming and outgoing droplet trajectories [19†L67–L74]. Rode et al. directly measured the wave field behind the slits and found minimal difference if one slit was blocked [19†L73–L76]. This suggests the conditions for producing robust interference patterns are extremely delicate in a classical fluid environment. Still, the droplet system stands as a tangible demonstration of the **particle–wave duality** concept, spurring theoretical work on pilot-wave dynamics and nonlocal quantum potentials [19†L19–L27]. Further refined experiments with real-time wave-field imaging may clarify the interplay between wave interference and droplet trajectories.

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## 7. Experimental Observations and Verifications
**Experimental studies of the walking-droplet system** have driven this field’s progress, revealing a variety of quantum-analog phenomena:

- **Analog of Tunneling:** Couder’s group (with E. Fort, A. Eddi, etc.) placed submerged barriers (“underwater walls”) in shallow fluid regions and observed whether droplets could cross them. Classically, if the droplet’s kinetic energy is insufficient to surmount the barrier, it should reflect. But in a high-memory state, droplet–wave resonance can provide extra momentum, enabling **occasional barrier crossing** [20†L19–L27][20†L31–L39]. In Eddi’s experiments, about 14 out of 110 droplet impacts (≈13%) crossed the barrier, seemingly at random [20†L19–L27]. This strongly evokes quantum tunneling. Subsequent theory (Hubert et al., 2015) used simplified models of droplet–wave interactions to reproduce a **non-monotonic tunneling probability** vs. forcing amplitude \(\gamma\). Too little forcing yields a weak wave, while too much forcing creates intense standing waves behind the barrier, blocking crossing, so an intermediate value of \(\gamma\) maximizes tunneling [20†L49–L57]. Tadrist et al. recently found that the tunneling vs. reflection outcome is determined by tiny velocity variations from collisions preceding the barrier approach, implying that in principle the dynamics are deterministic but highly sensitive, appearing probabilistic [20†L67–L75]. This suggests that, unlike genuine quantum indeterminacy, classical chaos can mimic **random-like tunneling** in the droplet system.

- **Quantized Orbits (Landau Level Analogs):** When an external central-like force is applied to a droplet, discrete stable orbits can emerge. Fort et al. [11†L1985–L1993][11†L1998–L2006] rotated the container at angular velocity \(\Omega\), imparting a Coriolis force (analogous to a Lorentz force) on the droplet. At high memory, the droplet’s orbital radius took on discrete values, quantized approximately at \(R_n \sim n\, \lambda_F/2\), with \(n\) a positive integer and \(\lambda_F\) the Faraday wavelength [11†L1998–L2006]. Hence only certain radii were stable, reminiscent of **Landau levels** for electrons in a magnetic field [11†L2001–L2005]. Eddi et al. [11†L2003–L2009] extended this to pairs of droplets, noting that orbit stability and separation depend on whether the pair rotates with or against the container’s rotation—somewhat analogous to a **Zeeman-like splitting** [11†L2003–L2009].

- **Quantum-Well / Stationary Wave States:** A droplet confined in a region (e.g., a circular corral) can exhibit discrete **stationary wave patterns**. Harris & Bush observed that a droplet in a circular boundary can remain localized on certain radii or angular positions, matching wave “antinodes” in the standing pattern. This phenomenon parallels quantum corral experiments (e.g., Friedel oscillations in electron states bounded by adatoms on metal surfaces) [20†L1–L4]. While friction prevents a truly time-independent probability distribution, long-time averages show enhanced droplet density at wave antinodes. Thus the **analog of particle-in-a-box** or quantum bound states emerges.  

All these experiments not only visualize interesting phenomena but **test theoretical models**. For instance, in the orbital quantization experiment, measured orbit radii were compared to Oza et al.’s linear stability analysis, showing quantitative agreement [11†L2011–L2019]. Similarly, tunneling probabilities were matched with droplet–wave simulations [20†L49–L57]. This synergy of experiment and theory has propelled the field forward, clarifying both the utility and limits of hydrodynamic quantum analogies.

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## 8. Key Researchers and Related Literature
Below is a summary of **key researchers** and notable achievements:

- **Yves Couder** (École Normale Supérieure, Île-de-France):  
  Discovered the walking-droplet phenomenon in 2005 [1†L28–L36] and led pioneering experimental efforts with Emmanuel Fort. Their work demonstrated droplet “interference” in a double-slit geometry [19†L49–L57], tunneling-like behavior [20†L31–L39], and orbital quantization in a rotating system [11†L1998–L2006], establishing the basis of fluid-based quantum analogs.

- **John W. M. Bush** (MIT):  
  A leading fluid dynamicist who spearheaded theoretical work on walking droplets. Starting with Couder’s findings, Bush (and collaborators, particularly Moláček) developed detailed models for the droplet’s vertical bouncing (2013, *J. Fluid Mech.*) [15†L39–L47] and the coupling of droplet motion with surface waves (“pilot-wave theory”). In 2015, Bush published a comprehensive review “Pilot-Wave Hydrodynamics” (*Annu. Rev. Fluid Mech.*) [4†L17–L25], followed by a more extensive review “Hydrodynamic quantum analogs” (*Rep. Prog. Phys.* 2021) [1†L28–L36]. Bush’s group (including Anand U. Oza, Daniel M. Harris, etc.) further investigated the system using stroboscopic models [12†L49–L58], orbital stability in rotating frames [11†L2011–L2019], and other phenomena.

- **Emmanuel Fort** (ESPCI Paris):  
  Key figure alongside Couder in France. Fort is a lead or coauthor on many of the seminal experimental papers and also contributed to theoretical modeling (e.g., the “path-memory” model [7†L43–L50] and rotating-frame numerical simulations [11†L1998–L2006]). His group, including Alban Perrard and Arnaud Labousse, has worked extensively on droplet–wave coupling, chaos, and statistical analysis [20†L37–L45].

- **Other Researchers**:  
  Earlier studies of classical wave-based quantum analogs, such as experiments on Aharonov–Bohm–like effects with water waves, were conducted by Michael Berrey, Markus R. Dennin, etc. [9†L7–L10]. On the “analogue gravity” front, Silke Weinfurtner (Univ. of Nottingham) and Daniele Faccio (Univ. of Glasgow) have performed fluid-gravity experiments involving Hawking-radiation analogs, while Jeff Steinhauer (Technion) made significant advances in BEC-based Hawking experiments. Theorists such as P. J. Sáenz, M. Durey, and R. Merlin have explored the statistical and quantum-interpretational aspects of walking droplets.

Representative references include Couder et al., *Nature* (2005) [1†L28–L36] (discovery of walking droplets), Couder & Fort, *Phys. Rev. Lett.* (2006) [19†L49–L57] (single-particle interference), Fort et al., *PNAS* (2010) [11†L1998–L2006] (orbital quantization), Eddi et al., *Phys. Rev. Lett.* (2009) [20†L31–L39] (tunneling analog), Bush et al., *Annu. Rev. Fluid Mech.* (2015) [4†L17–L25] (review), Bush & Oza, *Rep. Prog. Phys.* (2021) [1†L28–L36] (comprehensive review), and Durey & Bush, *Front. Phys.* (2020) [16†L369–L377] (HQFT theory).

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## 9. Theoretical Models and Mathematical Background
To quantitatively describe hydrodynamic quantum analogs, various **theoretical models** have been proposed. For walking-droplet systems, they can be broadly classified into **(A) continuum models based on fluid dynamics** and **(B) simplified or “toy” models**. We outline the key frameworks and their mathematical basis below.

### (A) Continuum Models (Direct Solution of Fluid Equations)
The most fundamental approach solves the underlying **PDE system** (Navier–Stokes equations with a free surface) for the fluid’s surface displacement \(h(\mathbf{r},t)\) and velocity potential \(\phi(\mathbf{r},t)\), along with the droplet’s Newtonian mechanics. This is computationally expensive, so practical models use approximations such as shallow-water or axisymmetric assumptions.  

Moláček & Bush (2013) first modeled the droplet’s vertical bounce via a damped spring-like system, fitted from measurements [14†L39–L47]. The wave created by each droplet impact is approximated by a radially symmetric **Bessel function**:
\[
h(\mathbf{r}, t) 
= \sum_{t_n < t} H_0\, J_0\bigl(k_F|\mathbf{r}-\mathbf{x}(t_n)|\bigr)\, e^{-\frac{t-t_n}{T_M}},
\]
where \(t_n\) is each collision time, \(\mathbf{x}(t_n)\) is the droplet position, \(k_F=2\pi/\lambda_F\) is the Faraday wavenumber, \(T_M\) is the memory time (governing wave decay), and \(H_0\) is the initial amplitude from a single impact [12†L13–L21][12†L19–L27]. The exponential decay accounts for **wave memory**, so older wave contributions persist for a duration \(T_M\).  

The droplet’s horizontal motion obeys Newton’s second law,
\[
m \ddot{\mathbf{x}}(t) + D\, \dot{\mathbf{x}}(t) 
= -\nabla U\bigl(\mathbf{x}(t), t\bigr),
\]
where \(D\) is an effective drag, and the **effective potential** \(U\) generally satisfies \(U \propto F\, h(\mathbf{x},t)\) [12†L49–L58]. The droplet moves “downhill” on the local slope of the surface, i.e., 
\[
-\nabla U(\mathbf{x}(t),t) 
= -F\, \nabla h(\mathbf{x}(t),t).
\]
Here, \(F\) is a coupling constant. The system thus becomes a **delayed differential equation** involving all past collisions. Numerical solutions reproduce phenomena like orbital quantization and tunneling [20†L49–L57]. A **stroboscopic model** sometimes simplifies calculations by updating droplet position and velocity once per vibration cycle (a discrete map), effectively capturing the same memory dynamics in a less computationally demanding way [12†L49–L58].  

### (B) Simplified (“Toy”) Models
Even the continuum approach remains complex, so further simplified models have been proposed. Labousse & Perrard (2014), for instance, treated the droplet + wave as coupled **Rayleigh oscillators**, highlighting forced self-oscillation [20†L37–L45]. In this approach, droplet coordinate \(x(t)\) is coupled to an internal wave phase or amplitude, forming a nonlinear forced system. This can yield random switching between stable states (mimicking tunneling). Hubert et al. (2015) used this model to interpret the statistical distribution of tunneling events [20†L37–L45]. Variants with a central Coulomb-like force were also explored, aiming to investigate “hydrodynamic hydrogen-like orbits.”

### (C) Generalized Pilot-Wave Theory
Dagan & Bush (2017), Durey & Bush (2020), and others have extended walking-droplet insights into a proposed **HQFT** that might apply to real quantum particles [16†L369–L377]. In these models, the particle has an internal oscillation \(f(t)=\sin(2\omega_c t)\) (the Compton frequency) that excites a pilot wave \(\Psi(\mathbf{r},t)\). In one dimension, for instance,
\[
\Psi(x,t) 
= \int^t \!ds\,\, K\bigl(x - x_p(s)\bigr)\, \cos\!\bigl(\omega_{\mathrm{dB}}(t-s)\bigr)\, e^{-\alpha\,(t-s)},
\]
where the kernel \(K(\cdot)\) describes wave propagation from the particle position \(x_p(s)\), \(\omega_{\mathrm{dB}}\) is the frequency corresponding to the de Broglie wavelength, and \(\alpha\) is a decay constant [17†L49–L57][17†L13–L21]. \(\Psi\) obeys a wave equation akin to the d’Alembert equation with boundary conditions, while the particle obeys
\[
m \ddot{x}_p(t) 
= -\kappa\, \partial_x \Psi\bigl(x,t\bigr)\Bigr|_{x=x_p(t)},
\]
where \(\kappa\) is a coupling constant [16†L373–L381][16†L383–L389]. Above a certain threshold, the particle–wave system “locks” into a self-propelled solution, analogous to a walking droplet’s resonance [16†L369–L377]. Analyses suggest the resulting motion recovers de Broglie and Bohm-type quantum relations in a particular limit [17†L31–L39][17†L37–L45]. Though still primarily conceptual, this approach aims to show how pilot-wave-like mechanics could underlie quantum phenomena, inspired by hydrodynamic analogs.

Across all these models, the main idea is **“the particle is guided by the wave it generates.”** This resonates with de Broglie’s 1920s pilot-wave idea and Bohm’s 1950s quantum potential theory. Classical hydrodynamics has given tangible form to these concepts, prompting re-evaluation of standard quantum interpretations.

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## 10. Conclusions and Outlook
This review has surveyed theoretical developments of **hydrodynamic quantum field analogs** over the last decade. Since the discovery of the walking-droplet phenomenon, numerous quantumlike behaviors—wave–particle duality, discrete orbits, tunneling, etc.—have been confirmed experimentally and analyzed theoretically. The analogy with quantum vacuum is demonstrated by classical water-wave versions of the Casimir effect and by nonequilibrium-driven analogs of Hawking radiation. The essential role of nonequilibrium in sustaining these phenomena has also become clear. Investigations of solitons and vortices show parallels between classical fluid structures and quantum topological defects. Furthermore, superfluid or BEC studies extend these analogies, revealing deeper correspondences between “genuinely quantum” fluids and classical systems.

Notably, walking droplets have reopened discussions about the **foundations of quantum mechanics**—particularly the wavefunction’s nonlocality and the possibility of a hidden, deterministic substructure. While early skeptics dismissed such parallels as superficial, contemporary work points to structural commonalities in which **probabilistic or indeterminate outcomes** can emerge from a **classical system with driving, dissipation, and memory** [8†L239–L247][8†L251–L259]. John Bush’s HQFT posits a “transparent medium” (the quantum vacuum) through which pilot waves guide particles, reminiscent of de Broglie–Bohm theory [16†L369–L377]. If validated, such models might offer an alternative, or at least a complement, to standard quantum theory [1†L33–L41].

There are, however, clear **limits and open questions**. Walking droplets are millimeter-scale objects, far removed from Planck scales. Whether this analogy can be extended to subatomic scales or whether new physics emerges remains unclear. The necessity of **energetic driving** also differs fundamentally from typical quantum phenomena, which often appear in low-energy ground states. Meanwhile, replicating stable interference in two-slit-like configurations has proven experimentally delicate, highlighting differences between robust quantum coherence and the memory-limited nature of classical fluids. Future directions may include:

- **More refined experiments**: High-speed imaging and advanced measurement to directly visualize droplet wave fields in real time. Multi-droplet “entanglement analogs,” tests of the speed at which wave information propagates, and other lines of inquiry could refine the analogy.

- **Extensions to other systems**: Searching for pilot-wave-like phenomena in electrohydrodynamics (charged droplets and electromagnetic fields), elastic membranes, or granular media, to assess whether these analogs are universal or fluid-specific.

- **Direct comparison with quantum systems**: Parallel investigations using BECs, optical systems, or superconductors to see if the same equations (nonlinear Schrödinger-type, etc.) describe both classical analogs and genuine quantum states. This might shed light on whether there is a continuous bridge between classical wave–particle phenomena and quantum reality.

Overall, hydrodynamic quantum field analogs have become a **frontier** for exploring the “essence of quantum phenomena” from an alternative vantage point. The achievements of the past decade demonstrate both the power and subtlety of classical analogies for quantum behavior. Whether the droplet system will ultimately “solve” the quantum mystery remains uncertain, but it has undoubtedly provided fresh insights and new impetus for research into the interplay of waves, particles, and information.

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