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We investigate the crossing of an energy barrier by a self-propelled particle described by a Rayleigh friction term. We reveal the existence of a sharp transition in the external force field whereby the amplitude dramatically increases. This corresponds to a saddle point transition in the velocity flow phase space, as would be expected for any type of repulsive force field. We use this approach to rationalize the results obtained by Eddi et al. [Phys. Rev. Lett. 102, 240401 (2009)] who studied the interaction between a drop propelled by its accompanying wave field and a submarine obstacle. This wave particle entity can overcome potential barrier, suggesting the existence of a “macroscopic tunneling effect.” We show that the effect of self-propulsion is sufficiently strong to generate crossing of the high-energy barrier. By assuming a random distribution of initial angles, we define a probability distribution to cross the potential barrier that matches with the data of Eddi et al. This probability is similar to the one encountered in statistical physics for Hamiltonian systems, i.e., a Boltzmann exponential law.

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... The system has extended the range of classical physics to include many features previously thought to be exclusively quantum [7][8][9]. Hydrodynamic quantum analogs achieved with this system now include tunneling [10][11][12], Landau levels [13,14], Zeeman splitting [15], and Friedel oscillations [16,17]. Quantized orbits arise for droplets walking in either a rotating frame [13,14,18] or a simple harmonic potential [19][20][21]. ...

... The particle oscillations and coupled particle-wave system are thus consistent with relativity [67]. The coupled Klein-Gordon equation (12) and guidance equation (13) are discretized using finite differences. An explicit finite-difference method was derived to solve the Klein-Gordon wave equation, and a Runge-Kutta scheme is employed to advance in time the guidance equation (13), which is nonlinear due to the dependence of γ on the particle speed [68]. ...

... In Section 3.1, we explore the effect of particle vibration and translation on the pilot-wave field using kinematic simulations; specifically, we prescribe a constant particle speed, so the trajectory equation (13) need not be solved. This constraint is relaxed in Section 3.2, where we investigate the fully dynamic coupling between the particle and its pilot-wave expressed in (12) and (13), as results in the particle's selfpropulsion. ...

We revisit de Broglie’s double-solution pilot-wave theory in light of insights gained from the hydrodynamic pilot-wave system discovered by Couder and Fort [1]. de Broglie proposed that quantum particles are characterized by an internal oscillation at the Compton frequency, at which rest mass energy is ex-changed with field energy. He further proposed that the resulting pilot-wave field satisfies the Klein-Gordon equation. While he developed a guidance equation for the particle, he did not specify how the particle generates the wave. Informed by the hydrodynamic pilot-wave system, we explore a variant of de Broglie’s mechanics in which the form of the Compton-scale dynamic interaction between particle and pilot wave is specified. The particle is modeled as a localized periodic disturbance of the Klein-Gordon field at twice the Comp-ton frequency. We simulate the evolution of the particle position by assuming that the particle is propelled by the local gradient of its pilot wave field. Resonance is achieved between the particle and its pilot wave, leading to self-excited motion of the particle. The particle locks into quasi-steady motion characterized by a mean momentum ̄p=ħk, where k is the wavenumber of the surrounding matter waves. Speed modulations along the particle path arise with the de Broglie wavelength and frequency ck. The emergent dynamics is strongly reminiscent of that arising in the hydrodynamic pilot-wave system, on the basis of which we anticipate the emergence of quantum statistics in various settings. Our results suggest the potential value of a new hydrodynamically-inspired pilot-wave theory for the motion of quantum particles.

... In this article, we address the question of unpredictability in an experimental configuration proposed as a macroscopic analog of quantum tunneling [19][20][21]: a walking droplet is launched toward a submerged boundary that it may cross or not (see Fig. 1). The first experimental demonstration of walker tunneling was done by Eddi et al. [19]. ...

... This experiment has been analysed theoretically by Hubert et al. [21] and Nachbin [20]. Hubert et al. [21] showed that one possible reason for the "unpredictable tunneling" of walkers was the lack of control of initial parameters such as the droplet velocity (which was controlled at ±5% in the original experiment) and the incident angle. ...

... This experiment has been analysed theoretically by Hubert et al. [21] and Nachbin [20]. Hubert et al. [21] showed that one possible reason for the "unpredictable tunneling" of walkers was the lack of control of initial parameters such as the droplet velocity (which was controlled at ±5% in the original experiment) and the incident angle. Nachbin [20] examined experimentally a simplified one-dimensional confined version of walker tunneling and showed that the wave construction in the second cavity may allow the walker to cross the barrier. ...

A walker is a macroscopic coupling of a droplet and a capillary wave field that exhibits several quantumlike properties. In 2009, Eddi et al. [Phys. Rev. Lett. 102, 240401 (2009)] showed that walkers may cross a submerged barrier in an unpredictable manner and named this behavior “unpredictable walker tunneling.” In quantum mechanics, tunneling is one of the simplest arrangements where similar unpredictability occurs. In this paper, we investigate how unpredictability can be unveiled for walkers through an experimental study of walker tunneling with precision. We refine both time and position measurements to take into account the fast bouncing dynamics of the system. Tunneling is shown to be unpredictable until a distance of 2.6 mm from the barrier center, where we observe the separation of reflected and transmitted trajectories in the position-velocity phase-space. The unpredictability is unlikely to be attributable to either uncertainty in the initial conditions or to the noise in the experiment. It is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. We compare this macroscopic system to a tunneling quantum particle that is subjected to repeated measurements of its position and momentum. We show that, despite the different theoretical treatments of these two disparate systems, similar patterns emerge in the position-velocity phase space.

... 1 Modifying pulsations of the medium 1.1 Inhomogeneous medium in walking droplets experiments By using a container with different fluid depths in walking droplets experiments, authors have studied tunnel-like effects [16,17] (see also [18,19] for theoretical investigations) and non-specular reflection of walking droplets [20]. A barrier with a different thickness in the vibrating cell changes properties of the system at the location of the barrier, in particular the threshold for Faraday instability. ...

... for the corresponding effective Schrödinger equation (19) with ω m (r) = a Ωm x c 2 m . Definition (6) of the modulating wave ψ leads to the (real-valued) transverse wave: ...

... A solution of the effective Schrödinger equation (19) with this effective potential and for a standing modulating wave ψ(x, t) = F (x) e − i ω t is such that (cf. e.g. ...

In this paper we suggest a macroscopic toy system in which a potential-like energy is generated by a nonuniform pulsation of the medium (i.e. pulsation of transverse standing oscillations that the elastic medium of the system tends to support at each point). This system is inspired by walking droplets experiments with submerged barriers. We ﬁrst show that a Poincaré-Lorentz covariant formalization of the system causes inconsistency and contradiction. The contradiction is solved by using a general covariant formulation and by assuming a relation between the metric associated with the elastic medium and the pulsation of the medium. (Calculations are performed in a Newtonian-like metric, constant in time). We ﬁnd (i) an eﬀective Schrödinger equation with external potential, (ii) an eﬀective de Broglie-Bohm guidance formula and (iii) an energy of the ‘particle’ which has a direct counterpart in general relativity as well as in quantum mechanics. We analyze the wave and the ‘particle’ in an eﬀective free fall and with a harmonic potential. This potentiallike energy is an eﬀective gravitational potential, rooted in the pulsation of the medium at each point. The latter, also conceivable as a natural clock, makes easy to understand why proper time varies from place to place.

... This unique feedback between the droplet and the wavefield dynamics is at the core of a stream of research mainly motivated by analogies with quantum systems [27][28][29][30][31][32][33][34][35][36][37][38][39] . In addition, recent numerical, theoretical and experimental studies [40][41][42][43][44][45][46][47] have shown that the memory of the walker leads to run-and-tumble-like chaotic dynamics [48][49][50] , similar to Marangoni-driven drops 51 , or particles in in-silico superfluids 52 . ...

... The experimental data also collapses adequately onto the master curve. It is interesting to compare the velocity potential used here, namely ΦðvÞ ¼ ϕ 0 jvj À v 0 À Á 2 =2, with the one used in previous investigations performed at lower memories 36,56 which is stiffer with the presence of v 4 terms. This suggest that the constrain on the self-propulsion speed v 0 is softer at high memory than at short memory. ...

Information storage is a key element of autonomous, out-of-equilibrium dynamics, especially for biological and synthetic active matter. In synthetic active matter however, the implementation of internal memory in self-propelled systems is often absent, limiting our understanding of memory-driven dynamics. Recently, a system comprised of a droplet generating its guiding wavefield appeared as a prime candidate for such investigations. Indeed, the wavefield, propelling the droplet, encodes information about the droplet trajectory and the amount of information can be controlled by a single scalar experimental parameter. In this work, we show numerically and experimentally that the accumulation of information in the wavefield induces the loss of time correlations, where the dynamics can then be described by a memory-less process. We rationalize the resulting statistical behavior by defining an effective temperature for the particle dynamics where the wavefield acts as a thermostat of large dimensions, and by evidencing a minimization principle of the generated wavefield. Memory and information storage play an important role in biological systems, however challenging to implement in synthetic active matter. The authors show that the wave field, propelling the particle, acts as a memory repository, and an excess of memory leads to a memory-less particle dynamics.

... This unique feedback between the droplet motion and the wave field dynamics is at the core of a rich and fruitful stream of research mainly motivated by the tantalizing analogy with quantum systems (27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38). Complementary to this motivation, recent numerical, theoretical and experimental studies (39)(40)(41)(42)(43)(44)(45)(46) have shown that the memory of the walker lead to chaotic dynamics characterized by anomalous diffusion similar to the run and tumble observed in bacteria (47)(48)(49), with Marangonidriven drops (50) or particles in in silico superfluid (51). ...

... The experimental data collapses also adequately onto the master curve. It is interesting to compare the velocity potential used here, namely Φ( v) = φ0 (| v| − v0) 2 /2, with the one used in previous investigations performed at lower memories (36,54) which is stiffer with the presence of v 4 terms. This suggest that the constrain on the self-propulsion speed v0 is softer at high memory than at short memory. ...

Information storage, for short "memory", is a key element of autonomous, out-of-equilibrium dynamics, in particular in biological entities. In synthetic active matter, however, the implementation of internal memory in agents is often limited or even absent. As a consequence, most of the investigations in the field of active matter had no choice but to ignore the influence of memory on the dynamics of these systems. We take here the opportunity to explore this question by leveraging one of the very few experimental physical system in which memory can be described in terms of a single and most importantly tunable scalar quantity. Here we consider a particle propelled at a fluid interface by self-generated stationary waves. The amount of souvenirs stored in the wave-memory field can be tuned, allowing for a throughout investigation of the properties of this memory-driven dynamics. We show numerically and experimentally that the accumulation of information in the wave field induces the loss of long-range time correlations. The dynamics can then be described by a memory-less process. We rationalize the resulting statistical behavior by defining an effective temperature for the particle dynamics and by evidencing a minimization principle for the wave field.

... Hubert et al. [192] developed an analytical model describing the dynamics of non-Hamiltonian particles crossing an energy barrier via their self-propulsion mechanism in the presence of a Rayleigh friction created from the waves emitted by the droplet. Under the influence of the force field, ...

... Schematic diagram of the self-propelled droplet moving with initial velocity V toward the potential barrier E p corresponding to a force field F f , b trajectories for two different values of the force field where the shaded region indicates the application of the force field, Hubert et al. [192] • The third state is known as global structure, M e = 150, in which the droplet is propelled and guided in a fully coherent structure, corresponding to a welldefined organization of the pivotal surface wave. The mechanism of this third state was explained by Perrard et al. [187]. ...

This article considers additional phenomena that complement the earlier topics addressed by Ibrahim [(Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press, Cambridge, 2005), (ASME J Fluids Eng 137(9):090801, 2015)]. The first phenomenon is the localized Faraday waves known as oscillons, which were observed in granular materials and liquid layers subjected to parametric excitation. Extreme waves, known as rogue, generated in the Faraday surface ripples, are related to the increase in the horizontal mobility of oscillating solitons (oscillons), and their horizontal motion is random over a limited range of excitation acceleration amplitude. Parametric excitation of water in a Hele–Shaw cell and the associated localized standing surface waves of large amplitude will be discussed. The surface wave pattern exhibited a certain similarity with the three-dimensional axisymmetric oscillon. Faraday waves in superfluid Fermi–Bose mixtures and their wave function will be addressed in terms of position and time as described by the Schrödinger equation with time-dependent parabolic potential. The phenomenon of walking fluid droplets on Faraday waves constitutes the majority portion of this article. Different regimes of droplet motion in terms of droplet physical properties, the fluid bath excitation acceleration amplitude and frequency will be discussed. The droplet trajectory diffraction, when passes through a slit, shares the same random features of electron diffraction. The duality of the droplet-wave field together with the path-memory-driven nonlocality and other related topics will be assessed. This article is complemented with the fascinating phenomenon of the stone and bombs skipping/ricochet over water surface.

... Therefore, the effect of the submarine obstacles are not taken into account and needs to be added by hand in the simulation. We use the ansatz theorized in [24]: we account for the submarine obstacle by using an external potential U (r, θ). The r direction accounts for the confinement due to the annulus while the θ direction contains the periodic potential. ...

... Note that the memory has been kept constant in each simulation with Me = 20. One can see that the dynamics observed mimics the experimental one, validating our hypothesis: a change of the wavefield amplitude indeed allows the walker to be reflected by the external potential, similarly to the studies of Eddi et al [9] and later explained theoretically by Hubert et al [24]. As the amplitude of the wavefield gets smaller at the Bragg's condition, the active mechanism propelling the droplet gets weaker. ...

A walking droplet possesses fascinating properties due to its peculiar wave/particle interaction. The self-propelling motion of such a droplet is driven by the Faraday instability triggered around the droplet at each impact. We studied in this article how such a droplet behaves in an annular cavity where a periodic pattern is placed underneath the liquid-air interface, altering the Faraday instability. We show that, while the annulus ensures a circular motion of the droplet, the periodic pattern affects the global droplet motion. Similarly to electromagnetic waves in photonic crystals, the average droplet speed nearly vanishes when the pattern has a characteristic length close to half the Faraday wavelength. This effect opens ways to design guides, reflectors, lattices and metamaterials for such macroscopic particles.

... In this hydrodynamic quantum analogy (HQA), droplets interact in resonance with a quasi-monochromatic wavefield they generate and exhibit a self-propelling mechanism. This analog has extended the range of classical physics to include many features previously thought to be exclusively quantum, including tunneling [13][14][15][16], Landau levels [17][18][19], quantum harmonic oscillator [20,21], the quantum corral [22][23][24][25][26], the quantum mirage [25], and Friedel oscillations [27]. Remarkably, Couder was able to demonstrate in his hydrodynamic analogy a mechanism for single-particle diffraction [28]. ...

This chapter explores a deterministic hydrodynamically-inspired ensemble interpretation for free relativistic particles, following the original pilot wave theory conceptualized by de Broglie in 1924 and recent advances in hydrodynamic quantum analogs. We couple a one-dimensional periodically forced Klein-Gordon wave equation and a relativistic particle equation of motion, and simulate an ensemble of multiple uncorrelated particle trajectories. The simulations reveal a chaotic particle dynamic behavior, highly sensitive to the initial random condition. Although particles in the simulated ensemble seem to fill out the entire spatiotemporal domain, we find coherent spatiotemporal structures in which particles are less likely to cross. These structures are characterized by de Broglie's wavelength and the relativistic modulation frequency kc. Markedly, the probability density function of the particle ensemble correlates to the square of the absolute wave field, solved here analytically, suggesting a classical deterministic interpretation of de Broglie's matter waves and Born's rule.

... In this hydrodynamic quantum analogy (HQA), droplets interact in resonance with a quasi-monochromatic wavefield they generate and exhibit a selfpropelling mechanism. This analog has extended the range of classical physics to include many features previously thought to be exclusively quantum, including tunneling [12][13][14][15], Landau levels [16][17][18], the quantum harmonic oscillator [19,20], the quantum corral [21][22][23][24][25], the quantum mirage [24], and Friedel oscillations [26]. ...

We present a classical hydrodynamic analog of free relativistic quantum particles inspired by de Broglie’s pilot wave theory and recent developments in hydrodynamic quantum analogs. The proposed model couples a periodically forced Klein–Gordon equation with a nonrelativistic particle dynamics equation. The coupled equations may represent both quantum particles and classical particles driven by the gradients of locally excited Faraday waves. Exact stationary solutions of the coupled system reveal a highly nonlinear mechanism responsible for the self-propulsion of free particles, leading to the onset of unsteady motion. Although the model is essentially nonrelativistic, a stabilizing mechanism for any particle traveling close to the wave signaling speed emerges through the coupling with the wavefield. Consequently, inline particle oscillations comparable to de Broglie’s wavelength are realized through this fully-classical model, suggesting a new classical interpretation for the motion of relativistic quantum particles.

... In this hydrodynamic quantum analogy (HQA), droplets interact in resonance with a quasi-monochromatic wavefield they generate and exhibit a self-propelling mechanism. This analog has extended the range of classical physics to include many features previously thought to be exclusively quantum, including tunneling [12][13][14][15], Landau levels [16][17][18], quantum harmonic oscillator [19,20], the quantum corral [21][22][23][24][25], the quantum mirage [24], and Friedel oscillations [26]. ...

We present a classical hydrodynamic analog of free relativistic quantum particles inspired by de Broglie's pilot wave theory and recent developments in hydrodynamic quantum analogs. The proposed model couples a periodically forced Klein-Gordon equation with a nonrelativistic particle dynamic equation. The coupled equations may represent both quantum particles and classical particles driven by the gradients of locally excited Faraday waves. Exact stationary solutions of the coupled system reveal a highly nonlinear mechanism responsible for the self-propulsion of free particles, leading to the onset of unsteady motion. Although the model is essentially nonrelativistic, a stabilizing mechanism for any particle traveling close to the wave signaling speed emerges through the coupling with the wavefield. Consequently, inline particle oscillations comparable to de Broglie's wavelength formula are realized through this fully-classical model, suggesting a new classical interpretation for the motion of relativistic quantum particles.

... The non-Markovian feature of the droplet dynamics gives rise to behavior that might be mistakenly inferred to be spatially non-local if the influence of the wave field is not adequately resolved 24 . Of particular interest, here is the hydrodynamic analog of unpredictable quantum tunneling 37 , as has been demonstrated both experimentally 38,39 and numerically 40,41 . ...

Superradiance occurs in quantum optics when the emission rate of photons from multiple atoms is enhanced by inter-atom interactions. When the distance between two atoms is comparable to the emission wavelength, the atoms become entangled and their emission rate varies sinusoidally with their separation distance due to quantum interference. We here explore a theoretical model of pilot-wave hydrodynamics, wherein droplets self-propel on the surface of a vibrating bath. When a droplet is confined to a pair of hydrodynamic cavities between which it may transition unpredictably, in certain instances the system constitutes a two-level system with well-defined ground and excited states. When two such two-level systems are coupled through an intervening cavity, the probability of transition between states may be enhanced or diminished owing to the wave-mediated influence of its neighbour. Moreover, the tunneling probability varies sinusoidally with the coupling-cavity length. We thus establish a classical analog of quantum superradiance. Pilot-wave hydrodynamics has been employed to design classical analogues of various quantum phenomena. Here, coupling between two liquid droplets is reported to show qualitative similarities to superradiance of a coupled 2-atom system.

... The droplet thus navigates a potential landscape of its own making. This non-Markovian feature of the droplet dynamics gives rise to behavior that might be mistakenly inferred to be spatially non-local if the influence of the wave-field is not adequately resolved [19] Of particular interest here is the hydrodynamic analog of unpredictable quantum tunneling [26], as has been demonstrated both experimentally [27,28] and numerically [29,30]. ...

Superradiance and subradiance occur in quantum optics when the emission rate of photons from multiple atoms is enhanced and diminished, respectively, owing to interaction between neighboring atoms. We here demonstrate a classical analog thereof in a theoretical model of droplets walking on a vibrating bath. Two droplets are confined to identical two-level systems, a pair of wells between which the drops may tunnel, joined by an intervening coupling cavity. The resulting classical superradiance is rationalized in terms of the system's non-Markovian, pilot-wave dynamics.

... The authors demonstrate that the reflection or transmission of a walker over a submerged barrier is unpredictable; moreover, the crossing probability decreases exponentially with increasing barrier width, as in the case of quantum tunneling [267,268]. Walker tunneling was revisited theoretically by Hubert et al [7], who used the Rayleigh oscillator model of Labousse and Perrard [232] (section 4.6) to capture the observed crossing statistics. ...

The walking droplet system discovered by Yves Couder and Emmanuel Fort presents an example of a vibrating particle self-propelling through a resonant interaction with its own wave field. It provides a means of visualizing a particle as an excitation of a field, a common notion in quantum field theory. Moreover, it represents the first macroscopic realization of a form of dynamics proposed for quantum particles by Louis de Broglie in the 1920s. The fact that this hydrodynamic pilot-wave system exhibits many features typically associated with the microscopic, quantum realm raises a number of intriguing questions. At a minimum, it extends the range of classical systems to include quantum-like statistics in a number of settings. A more optimistic stance is that it suggests the manner in which quantum mechanics might be completed through a theoretical description of particle trajectories. We here review the experimental studies of the walker system, and the hierarchy of theoretical models developed to rationalize its behavior. Particular attention is given to enumerating the dynamical mechanisms responsible for the emergence of robust, structured statistical behavior. Another focus is demonstrating how the temporal nonlocality of the droplet dynamics, as results from the persistence of its pilot wave field, may give rise to behavior that appears to be spatially nonlocal. Finally, we describe recent explorations of a generalized theoretical framework that provides a mathematical bridge between the hydrodynamic pilot-wave system and various realist models of quantum dynamics.

... Here the wave persistence defines a memory time during which the positional information is stored and can be read or erased, similarly to a Turing machine [23]. Besides wave-particle-inspired dynamics [19,20,24,25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], walking droplets exhibit cascades of bifurcation to chaos in Coriolis and Coulomb force field [43] as well as intermittency in harmonic potential [36,[43][44][45][46][47][48]. Nonsteady propulsions have been reported in asynchronous bouncing modes [49,50] and speed limit cycle and chaotic behavior for the free particle [51] have been investigated for synchronous bouncing modes. ...

We present a wave-memory-driven system that exhibits intermittent switching between two propulsion modes in free space. The model is based on a pointlike particle emitting periodically cylindrical standing waves. Submitted to a force related to the local wave-field gradient, the particle is propelled, while the wave field stores positional information on the particle trajectory. For long memory, the linear motion is unstable and we observe erratic switches between two propulsive modes: linear motion and diffusive motion. We show that the bimodal propulsion and the stochastic aspect of the dynamics at long time are generated by a Shil'nikov chaos. The memory of the system controls the fraction of time spent in each phase. The resulting bimodal dynamics shows analogies with intermittent search strategies usually observed in living systems of much higher complexity.

... With the walking droplet system, a number of hydrodynamic quantum analogs (HQA) have been discovered in the laboratory and investigated both experimentally and theoretically. Examples include tunneling across barriers [14][15][16] and refraction from single and double slits. [17][18][19] A number of static and dynamic bound states of multiple droplets have been discovered, including crystal lattices, 20 orbiting pairs, 21,22 ratcheting pairs, 23 and promenading pairs. ...

Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269–292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is “path memory” [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech. 675, 433–463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting on the droplet depends explicitly on its past. The second is the resonance condition between droplet and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient is chaos, which in several systems is characterized by unpredictable switching between unstable periodic orbits. This focus issue is devoted to recent studies of and relating to pilot-wave hydrodynamics, a field that attempts to answer the following simple but provocative question: Might deterministic chaotic pilot-wave dynamics underlie quantum statistics?

... One such candidate could be a subsurface barrier with which a hydrodynamic analog of quantum tunneling has been demonstrated. [32][33][34] If the height and the width of such a barrier are suitably tuned, then in principle it should be possible to have a subsurface barrier reflect or transmit a single walker with a 50% probabil- ity. However, the reflection of a walker from a subsurface barrier is known to be sensitive to the incident angle and it might be difficult to overcome this subtlety in prac- tice. ...

We present a numerical study of two-droplet pair correlations for in-phase droplets walking on a vibrating bath. Two such walkers are launched toward a common point of intersection. As they approach, their carrier waves may overlap and the droplets have a non-zero probability of forming a two-droplet bound state. The likelihood of such pairing is quantified by measuring the probability of finding the droplets in a bound state at late times. Three generic types of two-droplet correlations are observed: promenading, orbiting, and chasing pair of walkers. For certain parameters, the droplets may become correlated for certain initial path differences and remain uncorrelated for others, while in other cases, the droplets may never produce droplet pairs. These observations pave the way for further studies of strongly correlated many-droplet behaviors in the hydrodynamical quantum analogs of bouncing and walking droplets.

... One such candidate could be a subsurface barrier with which a hydrodynamic analog of quantum tunneling has been demonstrated. [32][33][34] If the height and the width of such a barrier are suitably tuned, then in principle it should be possible to have a subsurface barrier reflect or transmit a single walker with a 50% probability. However, the reflection of a walker from a subsurface barrier is known to be sensitive to the incident angle and it might be difficult to overcome this subtlety in practice. ...

We present a numerical study of two-droplet pair correlations for in-phase droplets walking on a vibrating bath. Two such walkers are launched towards a common origin. As they approach, their carrier waves may overlap and the droplets have a non-zero probability of forming a two-droplet bound state. The likelihood of such pairing is quantified by measuring the probability of finding the droplets in a bound state at late times. Three generic types of two-droplet correlations are observed: promenading, orbiting and chasing pair of walkers. For certain parameters, the droplets may become correlated for certain initial path differences and remain uncorrelated for others, while in other cases the droplets may never produce droplet pairs. These observations pave the way for further studies of strongly correlated many-droplet behaviors in the hydrodynamical quantum analogs of bouncing and walking droplets.

A classical wave–particle entity (WPE) can be realized experimentally as a droplet walking on the free surface of a vertically vibrating liquid bath, with the droplet’s horizontal walking motion guided by its self-generated wave field. These self-propelled WPEs have been shown to exhibit analogs of several quantum and optical phenomena. Using an idealized theoretical model that takes the form of a Lorenz-like system, we theoretically and numerically explore the dynamics of such a one-dimensional WPE in a sinusoidal potential. We find steady states of the system that correspond to a stationary WPE as well as a rich array of unsteady motions, such as back-and-forth oscillating walkers, runaway oscillating walkers, and various types of irregular walkers. In the parameter space formed by the dimensionless parameters of the applied sinusoidal potential, we observe patterns of alternating unsteady behaviors suggesting interference effects. Additionally, in certain regions of the parameter space, we also identify multistability in the particle’s long-term behavior that depends on the initial conditions. We make analogies between the identified behaviors in the WPE system and Bragg’s reflection of light as well as electron motion in crystals.

A classical wave-particle entity can be realized experimentally as a droplet walking on the free surface of a vertically vibrating liquid bath, with the droplet's horizontal walking motion guided by its self-generated wave field. These self-propelled wave-particle entities have been shown to exhibit analogs of several quantum and optical phenomena. Using an idealized theoretical model that takes the form of a Lorenz-like system, we theoretically and numerically explore the dynamics of such a one-dimensional wave-particle entity in a sinusoidal potential. We find steady states of the system that correspond to a stationary wave-particle entity as well as a rich array of unsteady motions such as back-and-forth oscillating walkers, runaway oscillating walkers and various types of irregular walkers. In the parameter space formed by the dimensionless parameters of the applied sinusoidal potential, we observe patterns of alternating unsteady behaviors suggesting interference effects. Additionally, in certain regions of the parameter space, we also identify multistability in the particle's long-term behavior that depends on the initial conditions. We make analogies between the identified behaviors in the wave-particle entity system and Bragg's reflection of light as well as electron motion in crystals.

We investigate the dynamics of a deterministic self-propelled particle endowed with coherent memory. We evidence experimentally and numerically that it exhibits several stable free states. The system is composed of a self-propelled drop bouncing on a vibrated liquid driven by the waves it emits at each bounce. This object possesses a propulsion memory resulting from the coherent interference of the waves accumulated along its path. We investigate here the transitory regime of the buildup of the dynamics which leads to velocity modulations. Experiments and numerical simulations enable us to explore unchartered areas of the phase space and reveal the existence of a self-sustained oscillatory regime. Finally, we show the coexistence of several free states. This feature emerges both from the spatiotemporal nonlocality of this path memory dynamics as well as the wave nature of the driving mechanism.

Bouncing fluid droplets can walk on the surface of a vibrating bath forming a wave-particle association. Walking droplets have many quantum-like features. Research efforts are continuously exploring quantum analogues and respective limitations. Here, we demonstrate that two oscillating particles (millimetric droplets) confined to separate potential wells exhibit correlated dynamical features, even when separated by a large distance. A key feature is the underlying wave mediated dynamics. The particles’ phase space dynamics is given by the system as a whole and cannot be described independently. Numerical phase space histograms display statistical coherence; the particles’ intricate distributions in phase space are statistically indistinguishable. However, removing one particle changes the phase space picture completely, which is reminiscent of entanglement. The model here presented also relates to nonlinearly coupled oscillators where synchronization can break out spontaneously. The present oscillator-coupling is dynamic and can change intensity through the underlying wave field as opposed to, for example, the Kuramoto model where the coupling is pre-defined. There are some regimes where we observe phase-locking or, more generally, regimes where the oscillators are statistically indistinguishable in phase-space, where numerical histograms display their (mutual) most likely amplitude and phase.

Active Brownian particles, also referred to as microswimmers and nanoswimmers, are biological or manmade microscopic and nanoscopic particles that can self-propel. Because of their activity, their behavior can only be explained and understood within the framework of nonequilibrium physics. In the biological realm, many cells perform active Brownian motion, for example, when moving away from toxins or towards nutrients. Inspired by these motile microorganisms, researchers have been developing artificial active particles that feature similar swimming behaviors based on different mechanisms; these manmade micro- and nanomachines hold a great potential as autonomous agents for healthcare, sustainability, and security applications. With a focus on the basic physical features of the interactions of active Brownian particles with a crowded and complex environment, this comprehensive review will put the reader at the very forefront of the field of active Brownian motion, providing a guided tour through its basic principles, the development of artificial self-propelling micro- and nanoparticles, and their application to the study of nonequilibrium phenomena, as well as the open challenges that the field is currently facing.

We study in detail the hydrodynamic theories describing the transition to
collective motion in polar active matter, exemplified by the Vicsek and active
Ising models. Using a simple phenomenological theory, we show the existence of
an infinity of propagative solutions, describing both phase and microphase
separation, that we fully characterize. We also show that the same results hold
specifically in the hydrodynamic equations derived in the literature for the
active Ising model and for a simplified version of the Vicsek model. We then
study numerically the linear stability of these solutions. We show that stable
ones constitute only a small fraction of them, which however includes all
existing types. We further argue that in practice, a coarsening mechanism leads
towards phase-separated solutions. Finally, we construct the phase diagrams of
the hydrodynamic equations proposed to qualitatively describe the Vicsek and
active Ising models and connect our results to the phenomenology of the
corresponding microscopic models.

It has recently been demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic realm. The walker, consisting of a droplet plus its guiding wavefield, is a spatially extended object. We here examine the dependence of the walker mass and momentum on its velocity. Doing so indicates that, when the walker's time scale of acceleration is long relative to the wave decay time, its dynamics may be described in terms of the mechanics of a particle with a speed-dependent mass and a nonlinear drag force that drives it towards a fixed speed. Drawing an analogy with relativistic mechanics, we define a hydrodynamic boost factor for the walkers. This perspective provides a new rationale for the anomalous orbital radii reported in recent studies.

Bouncing walking droplets possess fascinating properties due to their
peculiar wave/particule interaction. In order to study such walkers in a 1d
system, we considered the case of one or more droplets in an annular cavity. We
show that, in this geometry, walking droplets form a string of synchronized
bouncing droplets that share a common coherent wave propelling the group at a
speed faster than single walkers. The formation of this coherent wave and the
collective behavior of droplets is captured by a model.

Yves Couder, Emmanuel Fort, and coworkers recently discovered that a millimetric droplet sustained on the surface of a vibrating fluid bath may self-propel through a resonant interaction with its own wave field. This article reviews experimental evidence indicating that the walking droplets exhibit certain features previously thought to be exclusive to the microscopic, quantum realm. It then reviews theoretical descriptions of this hydrodynamic pilot-wave system that yield insight into the origins of its quantumlike behavior. Quantization arises from the dynamic constraint imposed on the droplet by its pilot-wave field, and multimodal statistics appear to be a feature of chaotic pilot-wave dynamics. I attempt to assess the potential and limitations of this hydrodynamic system as a quantum analog. This fluid system is compared to quantum pilot-wave theories, shown to be markedly different from Bohmian mechanics and more closely related to de Broglie’s original conception of quantum dynamics, his double-solution theory, and its relatively recent extensions through researchers in stochastic electrodynamics.

The transmission of information can couple two entities of very different nature, one of them serving as a memory for the other. Here we study the situation in which information is stored in a wave field and serves as a memory that pilots the dynamics of a particle. Such a system can be implemented by a bouncing drop generating surface waves sustained by a parametric forcing. The motion of the resulting "walker" when confined in a harmonic potential well is generally disordered. Here we show that these trajectories correspond to chaotic regimes characterized by intermittent transitions between a discrete set of states. At any given time, the system is in one of these states characterized by a double quantization of size and angular momentum. A low dimensional intermittency determines their respective probabilities. They thus form an eigenstate basis of decomposition for what would be observed as a superposition of states if all measurements were intrusive.

A bouncing droplet on a vibrated bath can couple to the waves it generates, so that it becomes a propagative walker. Its propulsion at constant velocity means that a balance exists between the permanent input of energy provided by the vibration and the dissipation. Here we seek a simple theoretical description of the resulting non-Hamiltonian dynamics with a walker immersed in a harmonic potential well. We demonstrate that the interaction with the recently emitted waves can be modeled by a Rayleigh-type friction. The Rayleigh oscillator has well defined attractors. The convergence toward them and their stability is investigated through an energetic approach and a linear stability analysis. These theoretical results provide a description of the dynamics in excellent agreement with the experimental data. It is thus a basic framework for further investigations of wave-particle interactions when memory effects are included.

We present the results of a combined experimental and theoretical investigation of droplets walking on a vertically vibrating fluid bath. Several walking states are reported, including pure resonant walkers that bounce with precisely half the driving frequency, limping states, wherein a short contact occurs between two longer ones, and irregular chaotic walking. It is possible for several states to arise for the same parameter combination, including high- and low-energy resonant walking states. The extent of the walking regime is shown to be crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measure the tangential coefficient of restitution of drops impacting a quiescent bath. We then analyse the spatio-temporal evolution of the standing waves created by the drop impact and obtain approximations to their form in the small-drop and long-time limits. By combining theoretical descriptions of the horizontal and vertical drop dynamics and the associated wave field, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes. The critical requirement for walking is that the drop achieves resonance with its guiding wave field. We also rationalize the observed dependence of the walking speed on system parameters: while the walking speed is generally an increasing function of the driving acceleration, exceptions arise due to possible switching between different vertical bouncing modes. Special focus is given to elucidating the critical role of impact phase on the walking dynamics. The model predictions are shown to compare favourably with previous and new experimental data. Our results form the basis of the first rational hydrodynamic pilot-wave theory.

From the formation of animal flocks to the emergence of coordinated motion in bacterial swarms, populations of motile organisms at all scales display coherent collective motion. This consistent behaviour strongly contrasts with the difference in communication abilities between the individuals. On the basis of this universal feature, it has been proposed that alignment rules at the individual level could solely account for the emergence of unidirectional motion at the group level. This hypothesis has been supported by agent-based simulations. However, more complex collective behaviours have been systematically found in experiments, including the formation of vortices, fluctuating swarms, clustering and swirling. All these (living and man-made) model systems (bacteria, biofilaments and molecular motors, shaken grains and reactive colloids) predominantly rely on actual collisions to generate collective motion. As a result, the potential local alignment rules are entangled with more complex, and often unknown, interactions. The large-scale behaviour of the populations therefore strongly depends on these uncontrolled microscopic couplings, which are extremely challenging to measure and describe theoretically. Here we report that dilute populations of millions of colloidal rolling particles self-organize to achieve coherent motion in a unique direction, with very few density and velocity fluctuations. Quantitatively identifying the microscopic interactions between the rollers allows a theoretical description of this polar-liquid state. Comparison of the theory with experiment suggests that hydrodynamic interactions promote the emergence of collective motion either in the form of a single macroscopic 'flock', at low densities, or in that of a homogenous polar phase, at higher densities. Furthermore, hydrodynamics protects the polar-liquid state from the giant density fluctuations that were hitherto considered the hallmark of populations of self-propelled particles. Our experiments demonstrate that genuine physical interactions at the individual level are sufficient to set homogeneous active populations into stable directed motion.

This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments. This approach offers a unified framework for the mechanical and statistical properties of living matter: biofilaments and molecular motors in vitro or in vivo, collections of motile microorganisms, animal flocks, and chemical or mechanical imitations. A major goal of this review is to integrate several approaches proposed in the literature, from semimicroscopic to phenomenological. In particular, first considered are “dry” systems, defined as those where momentum is not conserved due to friction with a substrate or an embedding porous medium. The differences and similarities between two types of orientationally ordered states, the nematic and the polar, are clarified. Next, the active hydrodynamics of suspensions or “wet” systems is discussed and the relation with and difference from the dry case, as well as various large-scale instabilities of these nonequilibrium states of matter, are highlighted. Further highlighted are various large-scale instabilities of these nonequilibrium states of matter. Various semimicroscopic derivations of the continuum theory are discussed and connected, highlighting the unifying and generic nature of the continuum model. Throughout the review, the experimental relevance of these theories for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material is discussed. Promising extensions toward greater realism in specific contexts from cell biology to animal behavior are suggested, and remarks are given on some exotic active-matter analogs. Last, the outlook for a quantitative understanding of active matter, through the interplay of detailed theory with controlled experiments on simplified systems, with living or artificial constituents, is summarized.

On a vertically vibrating fluid interface, a droplet can remain bouncing indefinitely. When approaching the Faraday instability onset, the droplet couples to the wave it generates and starts propagating horizontally. The resulting wave–particle association, called a walker, was shown previously to have remarkable dynamical properties, reminiscent of quantum behaviours. In the present article, the nature of a walker's wave field is investigated experimentally, numerically and theoretically. It is shown to result from the superposition of waves emitted by the droplet collisions with the interface. A single impact is studied experimentally and in a fluid mechanics theoretical approach. It is shown that each shock emits a radial travelling wave, leaving behind a localized mode of slowly decaying Faraday standing waves. As it moves, the walker keeps generating waves and the global structure of the wave field results from the linear superposition of the waves generated along the recent trajectory. For rectilinear trajectories, this results in a Fresnel interference pattern of the global wave field. Since the droplet moves due to its interaction with the distorted interface, this means that it is guided by a pilot wave that contains a path memory. Through this wave-mediated memory, the past as well as the environment determines the walker's present motion.

Active Brownian particles (ABP) have served as phenomenological models of
self-propelled motion in biology. We study the effective diffusion coefficient of two one-dimensional ABP models
(simplified depot model and Rayleigh-Helmholtz model) differing in their
nonlinear friction functions. Depending on the choice of the friction function the diffusion coefficient
does or does not attain a minimum as a function of noise intensity. We furthermore discuss the case of an additional bias
breaking the left-right
symmetry of the system. We show that this bias induces a drift and that it generally reduces the
diffusion coefficient. For a finite range of values of the bias, both models can exhibit a maximum in
the diffusion coefficient vs. noise intensity.

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.

We experimentally study a monolayer of vibrated disks with a built-in polar asymmetry which enables them to move quasibalistically on a large persistence length. Alignment occurs during collisions as a result of self-propulsion and hard core repulsion. Varying the amplitude of the vibration, we observe the onset of large-scale collective motion and the existence of giant number fluctuations with a scaling exponent in agreement with the predicted theoretical value.

A droplet bouncing on a vibrated bath becomes a "walker" moving at constant velocity on the interface when it couples to the surface wave it generates. Here the motion of a walker is investigated when it collides with barriers of various thicknesses. Surprisingly, it undergoes a form of tunneling: the reflection or transmission of a given incident walker is unpredictable. However, the crossing probability decreases exponentially with increasing barrier width. This shows that this wave-particle association has a nonlocality sufficient to generate a quantumlike tunneling at a macroscopic scale.

Small drops can bounce indefinitely on a bath of the same liquid if the container is oscillated vertically at a sufficiently high acceleration. Here we show that bouncing droplets can be made to 'walk' at constant horizontal velocity on the liquid surface by increasing this acceleration. This transition yields a new type of localized state with particle-wave duality: surface capillary waves emanate from a bouncing drop, which self-propels by interaction with its own wave and becomes a walker. When two walkers come close, they interact through their waves and this 'collision' may cause the two walkers to orbit around each other.

We study sustained oscillations in two-dimensional oscillator systems driven by Rayleigh-type negative friction. In particular we investigate the influence of mismatch of the two frequencies. Further we study the influence of external noise and nonlinearity of the conservative forces. Our consideration is restricted to the case that the driving is rather weak and that the forces show only weak deviations from radial symmetry. For this case we provide results for the attractors and the bifurcations of the system. We show that for rational relations of the frequencies the system develops several rotational excitations with right/left symmetry, corresponding to limit cycles in the four-dimensional phase space. The corresponding noisy distributions have the form of hoops or tires in the four-dimensional space. For irrational frequency relations, as well as for increasing strength of driving or noise the periodic excitations are replaced by chaotic oscillations.

We study a model of Brownian particles which are pumped with energy by means of a non-linear friction function, for which different types are discussed. A suitable expression for a non-linear, velocity-dependent friction function is derived by considering an internal energy depot of the Brownian particles. In this case, the friction function describes the pumping of energy in the range of small velocities, while in the range of large velocities the known limit of dissipative friction is reached. In order to investigate the influence of additional energy supply, we discuss the velocity distribution function for different cases. Analytical solutions of the corresponding Fokker-Planck equation in 2d are presented and compared with computer simulations. Different to the case of passive Brownian motion, we find several new features of the dynamics, such as the formation of limit cycles in the four-dimensional phase-space, a large mean squared displacement which increases quadratically with the energy supply, or non-equilibrium velocity distributions with crater-like form. Further, we point to some generalizations and possible applications of the model. Comment: 10 pages, 12 figures

We consider Brownian particles with the ability to take up energy from the environment, to store it in an internal depot, and to convert internal energy into kinetic energy of motion. Provided a supercritical supply of energy, these particles are able to move in a ``high velocity'' or active mode, which allows them to move also against the gradient of an external potential. We investigate the critical energetic conditions of this self-driven motion for the case of a linear potential and a ratchet potential. In the latter case, we are able to find two different critical conversion rates for the internal energy, which describe the onset of a directed net current into the two different directions. The results of computer simulations are confirmed by analytical expressions for the critical parameters and the average velocity of the net current. Further, we investigate the influence of the asymmetry of the ratchet potential on the net current and estimate a critical value for the asymmetry in order to obtain a positive or negative net current. Comment: accepted for publication in European Journal of Physics B (1999), for related work see http://summa.physik.hu-berlin.de/~frank/active.html

Since the subject of traffic dynamics has captured the interest of physicists, many astonishing effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by so-called ``phantom traffic jams'', although they all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction of the traffic volume cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize in lanes, while similar systems are ``freezing by heating''? Why do self-organizing systems tend to reach an optimal state? Why do panicking pedestrians produce dangerous deadlocks? All these questions have been answered by applying and extending methods from statistical physics and non-linear dynamics to self-driven many-particle systems. This review article on traffic introduces (i) empirically data, facts, and observations, (ii) the main approaches to pedestrian, highway, and city traffic, (iii) microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts like a general modelling framework for self-driven many-particle systems, including spin systems. Subjects such as the optimization of traffic flows and relations to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are discussed as well. Comment: A shortened version of this article will appear in Reviews of Modern Physics, an extended one as a book. The 63 figures were omitted because of storage capacity. For related work see http://www.helbing.org/

Eddi et al. [Phys. Rev Lett. 102, 240401 (2009)] presented experimental results demonstrating the unpredictable tunneling of a classical wave-particle association as may arise when a droplet walking across the surface of a vibrating fluid bath approaches a submerged barrier. We here present a theoretical model that captures the influence of bottom topography on this wave-particle association and so enables us to investigate its interaction with barriers. The coupled wave-droplet dynamics results in unpredictable tunneling events. As reported in the experiments by Eddi et al. and as is the case in quantum tunneling [Gamow, Nature (London) 122, 805 (1928)], the predicted tunneling probability decreases exponentially with increasing barrier width. In the parameter regimes examined, tunneling between two cavities suggests an underlying stationary ergodic process for the droplet's position.

Couder et al. ( Nature , vol. 437 (7056), 2005, p. 208) discovered that droplets walking on a vibrating bath possess certain features previously thought to be exclusive to quantum systems. These millimetric droplets synchronize with their Faraday wavefield, creating a macroscopic pilot-wave system. In this paper we exploit the fact that the waves generated are nearly monochromatic and propose a hydrodynamic model capable of quantitatively capturing the interaction between bouncing drops and a variable topography. We show that our reduced model is able to reproduce some important experiments involving the drop–topography interaction, such as non-specular reflection and single-slit diffraction.

We present the results of a numerical investigation of the emergence of chaos in the orbital dynamics of droplets walking on a vertically vibrating fluid bath and acted upon by one of the three different external forces, specifically, Coriolis, Coulomb, or linear spring forces. As the vibrational forcing of the bath is increased progressively, circular orbits destabilize into wobbling orbits and eventually chaotic trajectories. We demonstrate that the route to chaos depends on the form of the external force. When acted upon by Coriolis or Coulomb forces, the droplet's orbital motion becomes chaotic through a period-doubling cascade. In the presence of a central harmonic potential, the transition to chaos follows a path reminiscent of the Ruelle-Takens-Newhouse scenario.

Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale exists for their behaviour in the presence of boundaries, as arises in a number of key quantum analogue systems. We here present the results of a combined experimental and theoretical study of the interaction of walking droplets with a submerged planar barrier. Droplets exhibit non-specular reflection, with a small range of reflection angles that is only weakly dependent on the system parameters, including the angle of incidence. The observed behaviour is captured by simulations based on a theoretical model that treats the boundaries as regions of reduced wave speed, and rationalized in terms of momentum considerations.

Biological motion and human traffic require energy supply from external sources. We develop here a model for the dynamics of driven entities which includes hydrodynamic interactions in order to adapt the model to the dynamics of swarms moving in dense fluids. Our entities have the ability to use the energy contained in an internal energy depot or an external energy inflow for the acceleration of motion. As a prototype of such entities we study Brownian particles having the ability to take up energy from their environment, to store it in an internal energy depot and to convert internal energy into kinetic energy. The motion of the particles is described by Langevin equations which include a dissipative force term resulting from the driving and equations for the dynamics of the depot. The hydrodynamic interactions are modeled by an Oseen-type tensorial force. It is shown that hydrodynamic interactions lead to the synchronization of the directions of motion leading to several new collective modes of the dynamics, including spontaneous rotations of the swarm.

John William Strutt, third Baron Rayleigh (1842-1919), was an English physicist best known as the co-discoverer of the element argon, for which he received the Nobel Prize in Physics in 1904. Rayleigh graduated from Trinity College, Cambridge, in 1865 and after conducting private research was appointed Cavendish Professor of Experimental Physics in 1879, a post which he held until 1884. These highly influential volumes, first published between 1877 and 1878, contain Rayleigh's classic account of acoustic theory. Bringing together contemporary research and his own experiments, Rayleigh clearly describes the origins and transmission of sound waves through different media. This textbook was considered the standard work on the subject for many years and provided the foundations of modern acoustic theory. Volume 1 discusses the origin and transmission of sound waves in harmonic vibrations, the vibrations of bars, stretched strings, plates and membranes, through mathematical models and experimental discussions.

We study the dynamics of an active Brownian particle with a nonlinear friction function located in a spatial cubic potential. For strong but finite damping, the escape rate of the particle over the spatial potential barrier shows a nonmonotonic dependence on the noise intensity. We relate this behavior to the fact that the active particle escapes from a limit cycle rather than from a fixed point and that a certain amount of noise can stabilize the sojourn of the particle on this limit cycle.

How bacteria regulate, assemble and rotate flagella to swim in liquid media is reasonably well understood. Much less is known about how some bacteria use flagella to move over the tops of solid surfaces in a form of movement called swarming. The focus of bacteriology is changing from planktonic to surface environments, and so interest in swarming motility is on the rise. Here, I review the requirements that define swarming motility in diverse bacterial model systems, including an increase in the number of flagella per cell, the secretion of a surfactant to reduce surface tension and allow spreading, and movement in multicellular groups rather than as individuals.

- J W M Bush

J. W. M. Bush, Ann. Rev. Fluid Mech. 47, 269 (2015).