Maziar Heidari

Théophile Gaichies

Ludwik Leibler

Matthieu Labousse
email: matthieu.labousse@espci.psl.eu

Mechanical activation of reversible bonds by low amplitude high frequencies excitations

Introduction The control of ligand-receptor binding dynamics emerges as a central issue in the adhesives [START_REF] Heinzmann | Supramolecular polymer adhesives: advanced materials inspired by nature[END_REF], tires and plastics industry. In particular, it is an essential feature for determining the properties of all materials based on reversible networks (2,3) such as supramolecular architectures [START_REF] Tsuda | Spectroscopic visualization of sound-induced liquid vibrations using a supramolecular nanofibre[END_REF], dynamics polymer network [START_REF] Montarnal | Silica-like malleable materials from permanent organic networks[END_REF], and self-healing systems [START_REF] Cordier | Self-healing and thermoreversible rubber from supramolecular assembly[END_REF] and for controlling their fracture [START_REF] Ducrot | Toughening elastomers with sacrificial bonds and watching them break[END_REF][START_REF] Chen | Mechanochemistry unveils stress transfer during sacrificial bond fracture of tough multiple network elastomers[END_REF] and fatigue resistance [START_REF] Sanoja | Why is mechanical fatigue different from toughness in elastomers? The role of damage by polymer chain scission[END_REF]. Similarly in the biological system, the binding kinetics of receptorligand complex plays a crucial role in the cellular functioning. For example, the complexes embedded on the cell membrane are under thermal and active fluctuations with a wide range of spatiotemporal spectrum [START_REF] Turlier | Equilibrium physics breakdown reveals the active nature of red blood cell flickering[END_REF][START_REF] Hu | Binding constants of membrane-anchored receptors and ligands depend strongly on the nanoscale roughness of membranes[END_REF][START_REF] Seifert | Stochastic thermodynamics, fluctuation theorems and molecular machines Rep[END_REF]. A chemical reaction or ligand-receptor bond is governed by thermodynamics or kinetics [START_REF] Kramers | Brownian motion in a field of force and the diffusion model of chemical reactions[END_REF]. Under thermodynamical control, the rate of binding (𝑘 𝑜→𝑏 ) over unbinding (𝑘 𝑏→𝑜 ) is expected to satisfy the Kramers' relation [START_REF] Arrhenius | Über die Dissociationswärme und den Einfluss der Temperatur auf den Dissociationsgrad der Elektrolyte[END_REF] 𝑘 𝑏→𝑜 𝑘 𝑜→𝑏 = 𝑒 Δ𝐹/𝑘 𝐵 𝑇 where, Δ𝐹, is the free energy difference between the open (𝑜) state and the bound (𝑏) state, 𝑘 𝐵 , the Boltzmann constant and, 𝑇, the temperature. In contrast, under kinetic control, the Arrhenius rule [START_REF] Arrhenius | Über die Dissociationswärme und den Einfluss der Temperatur auf den Dissociationsgrad der Elektrolyte[END_REF] implies that reaction pathway possessing the smallest activation energy 𝐸 𝑎 is the fastest with a transition rate scaling as ∼ 𝑒 𝐸 𝑎 /𝑘 𝐵 𝑇 . Although fundamentally different, these two relations dictate that a reaction can only be favored by either changing the energy landscape or the temperature [START_REF] Rowan | Dynamic Covalent Chemistry[END_REF].

While the change of temperature is, by essence, unselective and can only be realized in a very narrow range, the decrease of an activation energy requires usually a catalytic strategy which is reaction-specific and which represent most of the time a scientific tour de force. Alternatively, the binding kinetics of a single receptor and its corresponding ligand may be influenced by external mechanical forces and different loading conditions. Constant force or pulling with constant velocity, have been thoroughly studied theoretically [START_REF] Bell | Models for the specific adhesion of cells to cells[END_REF][START_REF] Evans | Dynamic strength of molecular adhesion bonds[END_REF][START_REF] Hummer | Free energy reconstruction from nonequilibrium single-molecule pulling experiments[END_REF][START_REF] Seifert | Dynamic strength of adhesion molecules: Role of rebinding and self-consistent rates Europhys[END_REF][START_REF] Moreira | reversible bridging: When is kinetics relevant for adhesion?[END_REF][START_REF] Erdmann | Stability of Adhesion Clusters under Constant Force[END_REF][START_REF] Tang | Lifetime of ligand-receptor clusters under external force[END_REF][START_REF] Hummer | Kinetics from Nonequilibrium Single-Molecule Pulling Experiments[END_REF][START_REF] Dudko | Intrinsic Rates and Activation Free Energies from Single-Molecule Pulling Experiments[END_REF][START_REF] Janeš | First-Principle Coarse-Graining Framework for Scale-Free Bell-Like Association and Dissociation Rates in Thermal and Active Systems[END_REF][START_REF] Bullerjahn | Theory of rapid force spectroscopy[END_REF] and experimentally [START_REF] Strick | The Elasticity of a Single Supercoiled DNA Molecule[END_REF][START_REF] Merkel | Energy landscapes of receptorligand bonds explored with dynamic force spectroscopy[END_REF][START_REF] Dudko | Beyond the conventional description of dynamic force spectroscopy of adhesion bonds[END_REF][START_REF] Marshall | Direct observation of catch bonds involving cell-adhesion molecules[END_REF][START_REF] Fenz | Switching from Ultraweak to Strong Adhesion[END_REF]. Extensile mechanical states are known to promote the breaking of a bond. Also, oscillatory forcing of a ligand receptor association has been theoretically investigated [START_REF] Lin | Forced dissociation of a biomolecular complex under periodic and correlated random forcing[END_REF][START_REF] Lin | s Expression and the Generalized Garg Form for Forced Dissociation of a Biomolecular Complex[END_REF][START_REF] Chen | Simplified Analysis for the association of a constrained receptor to an oscillating ligand[END_REF] with the main anstaz following the spirit of a Bell's like theory [START_REF] Bell | Models for the specific adhesion of cells to cells[END_REF]. Experimentally, ultrasound waves were used to generate such strong oscillatory forces in the bulk to control the reaction and dissociate bonds. Spectacular examples in mechanochemistry, like mechanically-activated catalysis [START_REF] Piermattei | Activating catalysts with mechanical force[END_REF], and efficient antibiotic release pathways [START_REF] Huo | Mechanochemical bond scission for the activation of drugs[END_REF][START_REF] Zou | Activation of Antibiotic-Grafted Polymer Brushes by Ultrasound ACS Macro Lett[END_REF][START_REF] Yildiz | Sonopharmacology: controlling pharmacotherapy and diagnosis by ultrasound-induced polymer Mechanochemistry[END_REF][START_REF] Schaefer | The Role of Mass and Length in the Sonochemistry of Polymers[END_REF], were experimentally designed. Also, liquid-gel transition [START_REF] Cravotto | Molecular self-assembly and patterning induced by sound waves. The case of gelation[END_REF] in Rhodium(I)-Iridium(I) [START_REF] Paulusse | Reversible Switching of the Sol-Gel Transition with Ultrasound in Rhodium(I) and Iridium(I) Coordination Networks[END_REF], naphthylamide-based [START_REF] Pang | Switchable sol gel transition controlled by ultrasound and body temperature[END_REF] and diselenidecrosslinked [START_REF] Kharandiuk | Mechanoresponsive diselenide-crosslinked microgels with programmed ultrasound-triggered degradation and radical scavenging ability for protein protection[END_REF] networks were demonstrated. These networks form a class of ultrasound responsive polymer system [START_REF] Beyer | The Mechanical Activation of Covalent Bonds[END_REF][START_REF] Hickenboth | Biasing reaction pathways with mechanical force[END_REF][START_REF] Balkenende | Mechanochemistry with Metallosupramolecular Polymers[END_REF][START_REF] Davis | Force-induced activation of covalent bonds in mechanoresponsive polymeric materials[END_REF][START_REF] Caruso | Mechanically-Induced Chemical Changes in Polymeric Materials[END_REF][START_REF] Yan | Stimuli-responsive supramolecular polymeric materials[END_REF][START_REF] Jonesa | Gels with sense: supramolecular materials that respond to heat, light and sound[END_REF][START_REF] Manourasa | Field responsive materials: photo-, electro-,magnetic-and ultrasound-sensitive polymers[END_REF][START_REF] Izak-Nau | Polymer mechanochemistry-enabled pericyclic reactions[END_REF]. In these cases, the cavitation induced by the mechanical waves is thought to generate a shearing stress sufficient to break the bonds of the gel network. However, the microscopic mechanisms at stake remains unelucidated but should require strong local displacements. In contrast, we propose and predict, here, an alternative non-equilibrium mechanochemical strategy holding even for nanometric displacements which always favor the bond breaking, and whose core idea is sketched in Fig 1A and1B.

We consider two systems in which a ligand binds to receptors. In the first system, the ligand is a Brownian molecule, while in the second system, the ligand is attached to the end of a grafted polymer chain. We compare this equilibrium situation with that placed under a mechanical wave which sets the receptors into an oscillatory motion. This action is displacement-controlled. In this article, we demonstrate that this particular constrained motion favors the open state (Fig 1c). We show that, even for a very small motion amplitude, the open state is exponentially favored by friction above a characteristic forcing frequency whenever the system would be under extensile or compressive states. We call this out-of-equilibrium protocol mechanical pumping, a term inspired from its optical counterpart owing to some unexpected analogous mechanisms.

Theoretical predictions We introduce the mechanism at stake from a theoretical perspective before realizing a numerical proof of concept. As exemplified in Fig 1A,a surface with attached receptors is set into a displacement-controlled oscillatory motion of amplitude 𝐴 and angular frequency 𝜔.

For simplicity, we start our analysis by considering the case of a Brownian particle. When the Brownian particle detaches from the receptor, it does not feel the presence of the wall anymore and we expect that the open time should be almost independent on the wall motion, an assertion we shall confirm further. So ) [START_REF] Lin | s Expression and the Generalized Garg Form for Forced Dissociation of a Biomolecular Complex[END_REF]. In addition, in the referential comoving with the wall, the overdamped Brownian particle experiences a friction force along the direction of the wall motion, 𝐹 𝛾 (𝑡) = 𝛾𝐴𝜔cos (𝜔𝑡) with 𝛾 the friction coefficient. As a consequence, we infer the form for time dependent escaping rate 𝑘(𝑡) = 𝑘 0 exp (-𝐸 𝑎 (𝑡)/𝑘 𝐵 𝑇) by considering a time-dependent energy barrier 𝐸 𝑎 (𝑡) = 𝐸 𝑎,0 -𝐹 𝛾 (𝑡)𝜎 with, 𝜎, the effective spatial extension of the potential and, 𝐸 𝑎,0 , the binding energy at equilibrium. This adiabatic hypothesis is reminiscent of the Bell Ansatz ( 16) and has been shown to hold for constant and oscillatory forces [START_REF] Seifert | Dynamic strength of adhesion molecules: Role of rebinding and self-consistent rates Europhys[END_REF][START_REF] Tang | Lifetime of ligand-receptor clusters under external force[END_REF]. Note that the displacementcontrolled motion of the receptors induces a linear frequency dependence in the amplitude of 𝐹 𝛾 (𝑡), a situation which shall not be encountered with a force-controlled motion. After calculations and following [START_REF] Tang | Lifetime of ligand-receptor clusters under external force[END_REF][START_REF] Lin | s Expression and the Generalized Garg Form for Forced Dissociation of a Biomolecular Complex[END_REF], we get

𝑇 b→o Brownian = 𝑇 b→o Brownian (0) 𝐼 0 ( 𝛾𝐴𝜔𝜎 𝑘 𝐵 𝑇 ) , (1)
with 𝑇 b→o Brownian (0) the mean transition time when the traps are immobile (𝜔 = 0, 𝐴 = 0), and 𝐼 0 the modified Bessel function of first kind of order 0. Furthermore, 𝐼 0 behaves essentially as an exponentially growing function at the leading order so that balancing the denominator term in Eq.

with 𝑇 b→o Chain (0) the mean transition time when the traps are immobile. Both Equations 1 and 2 involve an 𝐼 0 function in the denominator. As 𝐼 0 diverges exponentially, we predict that the transition time from bound to open state is significantly reduced at high frequency. Therefore, when the receptors are mechanically pumped, the detachment process is effectively promoted while the attachment event remains unchanged.

Numerical scheme To assess the relevance of our theoretical predictions, we consider in Fig. 1C, a numerical model system composed of a ligand attached to the end of a grafted polymer (see methods section). It is facing a receptor field positioned at a non-permeable wall moving around a mean position ℎ 0 with an amplitude 𝐴 at a frequency 𝑓 = 𝜔/2𝜋. We use the generic coarsegrained Kremer-Grest polymer model [START_REF] Kremer | Dynamics of entangled linear polymer melts: A molecular dynamics simulation[END_REF][START_REF] Heidari | Ordering of Functional Groups by Confining Grafted Chains, Star Polymers, or Polymer-Stabilized Nanoparticles[END_REF] and unless explicitly stated otherwise, in what follows, lengths are expressed in monomer size, 𝑎, units, energy in 𝑘 𝐵 𝑇 units, timescales in monomer relaxation time, 𝜏 = √𝑚𝑎 2 /𝑘 𝐵 𝑇, units (see methods section), with 𝑚 the mass of particle and frequency in 𝜏 -1 unit.

Numerical Results

We present in Fig. 1D the typical numerical results by simulating the molecular dynamics of the chain systems, for a rest interwall distance ℎ 0 = 8 and oscillation amplitude 𝐴 = 2. We find that the probability of binding 𝑃 b sharply decreases when the oscillation of the wall exceeds a characteristic frequency 𝑓 * ≃ 10 -2 . For frequency 𝑓 < 𝑓 * the probability of binding reaches a plateau corresponding to its value at equilibrium. In sharp contrast, for frequency 𝑓 > 𝑓 * the probability of binding abruptly decreases to zero and we find a strong difference compared with the equilibrium binding probability. We show in Fig. S1 a top view of the cumulative path travelled by the ligand over a fixed and long period of time (2 × 10 5 𝜏) for increasing frequency.

When the wall is immobile, the binding is strong and the ligand spends most of its time attached.

In fact, it detaches very rarely and when it happens it either rebinds to the same receptor or explores laterally and rebind to another receptors. Rebinding process is controlled by diffusion and density of receptors when the wall oscillates. As the frequency increases, the detaching events are more probable and the ligand spend more time detached from the receptor field. As a consequence, its lateral diffusion is more pronounced and the ligand visits much more receptors. For example, at 𝑓 = 0.33 the ligand visits about 3 times more receptors than at equilibrium for an equivalent period of times.

We now examine the origin of the mechanical pumping and demonstrate that its efficiency lies in the combined effects of a gap opening and diffusion-controlled rebinding. The gap opening is evidenced in Fig. 2A by comparing the axial position probability distribution 𝑃 z of the ligand for increasing frequency. At low frequency, 𝑓 = 3.3 × 10 -3 < 𝑓 * for which the ligand is mostly bound, 𝑃 z is strongly peaked around 𝑧 = ℎ min and 𝑧 = ℎ max and there is a pronounced gap of probability between these modes. We explain this gap creation by observing that wall speed is faster in the region 𝑧 = ℎ 0 and slower in the region 𝑧 = ℎ min and 𝑧 = ℎ max . This non-uniform motion implies that the receptor spends more time around 𝑧 = ℎ min and 𝑧 = ℎ max which concentrates the position probability distribution 𝑃 z at these two positions. We refer to as pumping state (intermediate state, respectively) the bonding events corresponding to the vicinity of peak at We investigate this inversion of population by examining the mechanism controlling the detaching events. We start by measuring the mean escaping times, 𝑇 b→o chain , 𝑇 b→o Brownian , respectively, for the simulated chain and Brownian systems. We observe in Fig 3A that the mean escaping time decreases rapidly with the oscillation frequency above a well-identified crossover. Furthermore, for a fixed interwall distance, the mean escaping time decreases with the wall oscillation amplitude.

𝑧 = ℎ max (𝑧 = ℎ min ,
Finally, the mean escaping time increases with the dissipation coefficient, 𝛾. These three observations hold for both the chain system and the Brownian particles. We rationalize these combined effects by considering, first, the Brownian case. We observe in Fig. 3B that all data collapse onto a single master curve. Thus, the theoretical predictions (Eq. 1) and (Eq. 2) are in very good agreement with the numerical results for low and moderate angular frequency. We use 𝜎 = 0.4 𝑎 for the effective trap potential which is measured from numerical simulations (see Fig. S2).

The deviation at high angular frequency is expected as inertial terms which start playing a role in this frequency range, are neglected in the theoretical description.

Having rationalized the escaping times, we focus now on the open phases. When the ligand escapes from a receptor well it diffuses until it rebinds to a receptor. This receptor may be different from the departing one. We measure the open time defined as the mean time to switch from the open to the bound state, 𝑇 o→b chain and 𝑇 o→b Brownian , for the chain system and the Brownian particle, respectively. We observe in Fig. 3C that the open times vary very weakly with the wall oscillation frequency and are primarily prescribed by the interwall distance ℎ 0 (See also Fig. S6). As for the chain system, and given the error bars, the open times are approximatively independent on frequency. For ℎ 0 = 18, the chain must extend away from its natural end-to-end distance (𝑅 𝑒𝑒 = 15.7), an energetical cost of entropic nature which increases the open time. In contrast, for ℎ 0 = 8, the chain is compressed and the open time is significantly reduced. For the case of a Brownian particle, the errors are less pronounced and we observe for all parameter configurations, a plateau at low frequency followed by a slow decrease of the open times.

Finally, we observe in Fig. 3D an increase of the flux ratio 𝑘 b→o /𝑘 o→b with the receptor wall frequency for the chain and Brownian systems. At low frequency, the flux ratio converges towards the Kramers' two states model limit, as expected. At high frequency, the flux ratio strongly disagrees with Kramers' limit. Specifically, for strong binding, the flux ratio is strongly unbalanced towards unbinding events and exceeds by more than two orders of magnitude the

Conclusion and discussion

We introduce a protocol called mechanical pumping and demonstrated its effectiveness to prevent a ligand from binding to a receptor above a crossover frequency. By simulating both a Brownian particle and ligand attached to the end of a grafted polymer, we showed that the protocol does not depend on the particular details of the system. We show that the mechanism is driven by the relative friction between the monomers and the surrounding solvent with a mean escaping time depending exponentially on the monomer damping coefficient.

In this article, the Brownian case is used as a guidance while the polymeric system is more suitable for applications. However, for the latter case, several aspects remain to be discussed to assess its practical relevance. Firstly, the solvent is modelled as being implicit, which means that the molecular dynamics of the surrounding solvents is not simulated per se. However, the characteristic relaxation time of the solvent can be considered longer than the relevant range of oscillating period 1/𝑓. Thus, at short time scale, specifically during a couple of periods of oscillations, the pulled chain and the melt have a net relative motion and thus, friction.

Additionally, in both theory and simulations, we neglected any dynamical hydrodynamic effects of the solvent as effects, like e.g., acoustic streaming, are expected to arise at much longer time scale. Nevertheless, and while being beyond the scope of this article, the deterministic interactions between a soft or rigid wall and a floating object in Stokes regimes are usually known to generate a lifting, i.e. repulsive, force (56,57) over a large range of physical parameters. This additional effect is expected to enhance the mechanical pumping.

Secondly, in practice, the system can be implemented as sketched in Fig1A, wherein the receptor is attached onto a colloidal particle which can be set into motion by, e.g., and acoustic wave or an optical trapping. We may pursue one step ahead and design supramolecular networks, i.e., a gel, as pictured in Fig 4A&C made of ligands at the end of polymer chains, receptors and two types of colloidal particles. The polymer chains are grafted to the smallest particles. In contrast, the receptors are grafted onto the largest particles. As the speed of displacement upon radiation pressure scales as the particle size, this configuration ensures that the receptors are co-displaced with the largest particles, while the chain with floppy backbones are attached to particles which are weakly displaced. Interestingly, below the crossover frequency, the system is expected to behave as a soft solid and becomes liquid-like beyond. We note that this concept may be leveraged in mechanochemistry to induce liquid-gel transition with a mechanism differing from ultrasoundinduced cavitation [START_REF] Beyer | The Mechanical Activation of Covalent Bonds[END_REF][START_REF] Hickenboth | Biasing reaction pathways with mechanical force[END_REF][START_REF] Balkenende | Mechanochemistry with Metallosupramolecular Polymers[END_REF][START_REF] Davis | Force-induced activation of covalent bonds in mechanoresponsive polymeric materials[END_REF].

Thirdly and finally, we may wonder how realistic the crossover frequency is for practical applications and its dependency on the temperature. As the 𝑓 * depends on the inverse of the monomeric friction coefficient 𝛾, its evolution with temperature can be estimated by re-expressing the empirical Williams-Landel-Ferry (WLF) time-temperature relation [START_REF] Williams | The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids[END_REF] as

Log 10 ( 𝑓 * 𝑓 WLF * ) = 𝐶 1 (𝑇-𝑇 𝑔 ) 𝐶 2 +(𝑇-𝑇 𝑔 ) , (3)
where 𝐶 1 = 17.44 K -1 and 𝐶 2 = 51.6 K are two universal empirical coefficients. We denote 𝑇 𝑔 the glass temperature and 𝑓 WLF * (see supplementary information) an empirical constant which are both monomer-dependent. Eq.3 predicts the temperature dependence of the crossover frequency in a master curve as indicated in Fig. 4B. As we decrease the temperature towards the glass temperature, 𝑓 * strongly decreases. As a consequence, even if 𝑓 * may be large at room temperature, it will necessarily fall into an experimentally-accessible frequency range by decreasing the temperature above the glass transition. Following the WLF approach, and the crossover frequency dependence from our theory (see section Theoretical predictions), we conclude by providing some order of magnitude for 𝑓 * for some commercial polymer systems, specifically for the polybutadiene (cis-1,4-and trans-1,4-polybutadiene, PB), the Polydimethylsiloxane (PDMS) and the styrene-butadiene (S-SBR) and for adhesion energy typical of a hydrogen bond (see Fig 4D

  and supplementary information). For low temperature, but above the glass transition, 𝑓 * falls into the kHz-MHz regime which is accessible to piezoelectric devices, or Atomic Force Microscopy tips. At room temperature, the frequency range greatly exceeds ∼100 MHz. The latter frequency range is mostly beyond the reach of standard ultrasonic or piezoelectric devices.However, a flurry of ingenious optomechanical designs operating up to the 1GHz-100GHz mechanical regime[START_REF] Aspelmeyer | Cavity optomechanics[END_REF][START_REF] Gil-Santos | Light-mediated cascaded locking of multiple nanooptomechanical oscillators[END_REF] have emerged over the past decade. Combining mechanochemistry with optomechanics will allow experimental implementations of optomechanical chemical switches at room temperature and will open a new pathway toward the control of ligand-receptor, molecular, and supramolecular binding.

Fig. 1 .

 1 Fig. 1. Controlling a reaction by mechanical pumping. (A)-(D). Sketch of the two systems with

Fig. 2 Fig. 3 . 8 (Fig. 4

 2384 Fig. 2 Mechanism of Gap opening. (A) Ligand z-position probability density for 𝐴 = 2, ℎ 0 = 8 𝛾 = 10 and 𝑓 = 3.3 × 10 -3 (dark orange circles), 𝑓 = 2.× 10 -2 (yellow triangles), 𝑓 = 3 × 10 -1 (green square). (B) Ligand z-position probability density for increasing frequency for

  , we focus our analysis on the transitions from the bound to the open

	state. The mean transition time from bound to open state, 𝑇 b→o Brownian , specifically the mean
	escaping time, is given by 𝑇 b→o Brownian = -∫ 𝑡 𝑃 ̇𝑠(𝑡)𝑑𝑡 +∞ 0	. In this equation, 𝑃 𝑠 (𝑡) denotes the
	survival density probability and is related to the time dependent escaping rate, 𝑘(𝑡), by the relation
	𝑡 𝑃 𝑠 = exp(-∫ 𝑘(𝑡′)𝑑𝑡′ 0

  friction and the characteristic trapping force field. Guided by this physical picture, the critical force balance in the case of the second system, a ligand attached to the end of a grafted polymer, is expected to scale as 𝑧𝛾𝐴𝜔 chain * ≃ 𝐸 𝑎,0 / 𝜎 where 𝑧 is the characteristic number of monomer that frictions when the wall start moving upwards and 𝜔 chain * the corresponding crossover angular frequency. By recalling that a segment composed of 𝑧 monomers relaxes local stresses within a Rouse time 𝜏 𝑅 = 𝛾

					(𝑧𝑎) 2 3𝜋 2 𝑘 𝐵 𝑇	(53), we note that a segment of
	𝑧 1 𝜔	, or equivalently if the segment
	has 𝑧 * =	√3𝜋 𝑎	(	𝑘 𝐵 𝑇 𝛾𝜔	) 1/2
	𝑇 b→o Chain =	𝑇 b→o Chain (0) 𝑧(𝜔)𝛾𝐴𝜔𝜎 𝐼 0 ( 𝑘 𝐵 𝑇

1 with its nominator 𝑇 b→o Brownian (0) ∝ exp (𝐸 𝑎,0 /𝑘 𝐵 𝑇) predicts a crossover angular frequency, 𝜔 Brownian * given by 𝛾𝐴𝜔 Brownian * ≃ 𝐸 𝑎,0 / 𝜎. The latter relation is interpreted as a force balance between the driving * monomers can relax in timescale faster 1/𝜔 when 𝜏 𝑅 (𝑧 * ) = monomers. As a result, for the case of the ligand attached to the end of a grafted polymer, we predict that the master curve collapsing the mean transition time from bound to open state 𝑇 b→o chain /𝑇 b→o chain (0) must be a function solely of the product 𝑧𝛾𝐴𝜔 chain ∝ 𝐴(𝛾𝜔 chain ) 1/2 and yields ) ,

  respectively), and to open state the unbinding events in 𝑧 ∈ [0, ℎ min ]. At low frequency, (𝑓 = 3 × 10 -3 in Figs 2A and 2B) both the pumping and intermediate states are strongly populated while the open state is depleted. At intermediate frequency, 𝑓 = 0.02 , the open state starts populating. In contrast, the pumping state starts depopulating, while the intermediate state follows likewise but most moderately. At high frequency, 𝑓 = 0.3 > 𝑓

* , the pumping state is mostly unpopulated; the intermediate state exhibits a surviving signature and the open state is widely filled. The full evolution of the probability distribution profile with the pump frequency is shown in Fig. 2B. The pumping converts the two states model bound-open into a three states problem: one pumping state (bound), an intermediate state and an open state. We observe (i) an inversion of population from the pumping state to the intermediate state and (ii) that the open state is mostly populated from the intermediate state.

  In contrast at high frequency, the transition rate from the bound state to open state is strongly biased while the transition from open to bound states is mainly unaffected by the wall pumping. Their combined effect populates efficiently the open state. These two regimes are in perfect agreement with our theoretical predictions.

Kramers' limit. At equilibrium, the transition from the bound states to the open states is reactioncontrolled as predicted by Kramers' two state model (see supplementary information and Figure S3-S6).

Acknowledgments: ML would like to thank Joshua McGraw for insightful discussions about lifting forces between small objects and soft substrates. MH thanks Prof. Hartmut Löwen for his hospitality during the summer 2020 visit.

Funding: This work was funded by the ESPCI Paris and CNRS.

Data and materials availability: All data, code, and materials used in the analysis are available to any researcher for purposes of reproducing or extending the analysis upon reasonable request.


Konstantinos Papatryfonos

Louis Vervoort
email: lvervoort@hse.ru

André Nachbin
email: nachbin@impa.br

Matthieu Labousse
email: matthieu.labousse@espci.psl.eu

John W M Bush
email: bush@math.mit.edu

A platform for investigating Bell correlations in pilot-wave hydrodynamics *

Since its discovery in 2005, the hydrodynamic pilot-wave system has provided a concrete macroscopic realization of wave-particle duality and concomitant classical analogs of a growing list of quantum effects. The question naturally arises as to whether this system might support statistical states that violate Bell's inequality, and so yield a classical analog of quantum entanglement. We here introduce a new platform for addressing this question, a numerical model of coupled bipartite tunneling in the hydrodynamic pilot-wave system. We demonstrate that, under certain conditions, the Bell inequality is violated in a static Bell test owing to correlations induced by the wave-mediated coupling between the two subsystems. The establishment of non-factorizable states with two spatially separated classical particles introduces the possibility of novel forms of quantum-inspired classical computing.

Significance Millimeter-scale droplets self-propelling along the surface of a vibrating liquid bath represent a macroscopic realization of wave-particle duality, an oddity once thought to be exclusive to the microscopic realm. This classical pilot-wave system bears a strong resemblance to an early model of quantum dynamics proposed by Louis de Broglie, and has provided the basis for a surprising number of hydrodynamic quantum analogs. We here take the first step towards assessing the plausibility of achieving entanglement with this hydrodynamic system. Our numerical investigation of the dynamics of two distant droplets reveals violations of Bell's inequality that may be rationalized in terms of the wave-induced coupling between them. The resulting non-factorizable, spatially separated classical states may find application in quantum-inspired classical computing.

ered that a millimetric droplet may self-propel along the surface of a vibrating fluid bath through a resonant interaction with its own wave field. The resulting 'walker' consists of a droplet dressed in a quasi-monochromatic wave field, and represents a concrete, macroscopic example of wave-particle duality [START_REF] Fort | Path-memory induced quantization of classical orbits[END_REF]. Remarkably, this hydrodynamic pilot-wave system exhibits many features previously thought to be exclusive to the microscopic, quantum realm [START_REF] Bush | Pilot-wave hydrodynamics[END_REF][START_REF] Bush | Hydrodynamic quantum analogs[END_REF]. Notable examples include singleparticle diffraction and interference [START_REF] Couder | Single particle diffraction and interference at a macroscopic scale[END_REF][START_REF] Pucci | Walking droplets interacting with single and double slits[END_REF][START_REF] Ellegaard | Interaction of wavedriven particles with slit structures[END_REF], quantized orbits [START_REF] Fort | Path-memory induced quantization of classical orbits[END_REF][START_REF] Perrard | Self-organization into quantized eigenstates of a classical wave-driven particle[END_REF], unpredictable tunneling [START_REF] Eddi | Unpredictable tunneling of a classical wave-particle association[END_REF], Friedel oscillations [START_REF] Sáenz | A hydrodynamic analog of Friedel oscillations[END_REF], spin lattices [START_REF] Sáenz | Emergent order in hydrodynamic spin lattices[END_REF], and quantum-like statistics and statistical projection effects in corrals [START_REF] Harris | Wavelike statistics from pilot-wave dynamics in a circular corral[END_REF][START_REF] Sáenz | Spin lattices of walking droplets[END_REF]. In all instances, the emergent quantum behavior may be rationalized in terms of the droplet's non-Markovian pilotwave dynamics [START_REF] Bush | Hydrodynamic quantum analogs[END_REF]. Specifically, the instantaneous wave force imparted to the drop during impact depends on the droplet's history. Thus, the drop navigates a potential landscape of its own making [START_REF] Bush | Hydrodynamic quantum analogs[END_REF], and the hydrodynamic pilot-wave system is said to be endowed with 'memory' [START_REF] Eddi | Information stored in Faraday waves: the origin of a path memory[END_REF].

In several settings, long-range interactions in the walking-droplet system emerge dynamically through the influence of the pilot-wave field [START_REF] Bush | Hydrodynamic quantum analogs[END_REF]. For example, longrange lift forces are generated when a walking droplet interacts with a submerged pillar [START_REF] Harris | The interaction of a walking droplet and a submerged pillar: From scattering to the logarithmic spiral[END_REF] or well [START_REF] Sáenz | A hydrodynamic analog of Friedel oscillations[END_REF], and long-range correlations between distant walkers may be established through the influence of the intervening wave field [START_REF] Nachbin | Walking droplets correlated at a distance[END_REF][START_REF] Nachbin | The effect of isolation on two-particle correlations in pilot-wave hydrodynamics[END_REF]. Recently, Papatryfonos et al. [START_REF] Papatryfonos | Hydrodynamic superradiance in wave-mediated cooperative tunneling[END_REF] estab-lished a walking-droplet analog of superradiance, an effect originally attributed to quantum interference of two or more entangled atoms [START_REF] Devoe | Observation of superradiant and subradiant spontaneous emission of two trapped ions[END_REF][START_REF] Makarov | Metastable entangled states of atomic systems in macroscale: radiation dynamics and spectrum[END_REF][START_REF] Karnieli | Superradiance and subradiance due to quantum interference of entangled free electrons[END_REF], but subsequently rationalised in terms of classical electromagnetic wave interference [START_REF] Tanji-Suzuki | Chapter 4 -interaction between atomic ensembles and optical resonators: Classical description[END_REF]. The totality of the quantum-like features evident in the hydrodynamic pilot-wave system naturally raises the following question. Might this walking-droplet system provide a platform for demonstrating a classical analog of entanglement [START_REF] De La Peña | The emerging quantum[END_REF][START_REF] Vervoort | Are hidden-variable theories for pilot-wave systems possible?[END_REF]; specifically, might it permit violations of Bell's inequality?

Bell's Theorem was derived by John Bell in 1964 [START_REF] Bell | On the Einstein Podolsky Rosen paradox[END_REF] with a view to informing the Bohr-Einstein debate concerning the completeness of quantum theory [START_REF] Einstein | Can quantummechanical description of physical reality be considered complete?[END_REF]. Hidden variables are those variables that would be required for a complete description of quantum dynamics, including the position and momentum of a microscopic particle. Bell tests inform what class of hidden variable theories are viable candidates for a causally complete quantum theory. A Bell test can be performed on any probabilistic system consisting of two subsystems (A and B) on which one measures a dichotomic property X (with stochastic outcomes of +1 or -1) that depends on some 'analyzer setting' (α or β). The measurement X A made at the left measurement system depends on the analyzer setting α which may take values a or a ; likewise, the measurement (X B ) made at the right measurement system depends on β which may take values b or b . In the derivation of the Bell inequality (Eq. 1), it is assumed that the two subsystems undergo only local interactions; specifically, X A depends on α and not β; likewise, X B depends on β and not α. This assumption is referred to as 'Bell locality'. Another assumption made is that the hidden variables that prescribe X are independent of α and β, a condition referred to as 'measurement independence'. Bell's theorem [START_REF] Bell | On the Einstein Podolsky Rosen paradox[END_REF] implies that for any classical system for which Bell locality and measurement independence hold, the quantity

S (α = a, β = b, α = a , β = b ) = M (a, b) + M (a , b) + M (a, b ) -M (a , b ) must satisfy the inequality |S (a, b, a , b ) | ≤ 2 (1)
for any choice of measurement settings (a, a , b, b ).

Here,

M (α, β) is the average product, M (α, β) = X A ,X B X A X B P (X A , X B |α, β)
, where P (X A , X B |α, β) is the joint probability of measurements (X A , X B ) when the left and right analyzers are set to (α, β). We note that Eq. 1) is cast in the form of the CHSH inequality [START_REF] Clauser | Proposed experiment to test local hidden-variable theories[END_REF].

It has been well established that bipartite quantum systems can violate inequality 1 for a judicious choice of (a, a , b, b ), with a maximum value S = 2 √ 2 corresponding to the Tsirelson bound. Bell tests were first performed with static analyzer settings [START_REF] Aspect | Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: A new violation of Bell's inequalities[END_REF], so could not strictly rule out signalling between the two subsystems. Specifically, the left measurement X A could in principle be influenced by the right analyzer setting β (or, similarly, X B by α) through long-range interactions between the two subsystems. Such 'static' Bell tests have also been performed with pairs of massive entangled particles [START_REF] Rowe | Experimental violation of a Bell's inequality with efficient detection[END_REF]. Technological advances in quantum optics and electronics have enabled this 'locality' loophole to be closed via dynamic Bell tests [START_REF] Aspect | Experimental test of Bell's inequalities using time-varying analyzers[END_REF][START_REF] Weihs | Violation of bell's inequality under strict einstein locality conditions[END_REF][START_REF] Scheidl | Violation of local realism with freedom of choice[END_REF][START_REF] Hensen | Loopholefree bell inequality violation using electron spins separated by 1.3 kilometres[END_REF][START_REF] Brunner | Bell nonlocality[END_REF][START_REF] Tittel | Violation of bell inequalities by photons more than 10 km apart[END_REF][START_REF] Giustina | Significant-loophole-free test of bell's theorem with entangled photons[END_REF], in which the detector settings α and β are altered just prior to measurement, so that the two measurement events are spacelike separated. Considerable effort has been made to closing other loopholes, including the 'detection' loophole (that posits that the detection efficiency depends on (α, β)) and the measurement-independence loophole, through a series of Bell tests of increasing sophistication and precision [START_REF] Hensen | Loopholefree bell inequality violation using electron spins separated by 1.3 kilometres[END_REF][START_REF] Ansmann | Violation of bell's inequality in josephson phase qubits[END_REF][START_REF] Hofmann | Heralded entanglement between widely separated atoms[END_REF][START_REF] Giustina | Bell violation using entangled photons without the fair-sampling assumption[END_REF][START_REF] Rauch | Cosmic bell test using random measurement settings from high-redshift quasars[END_REF]. Nevertheless, Morgan [START_REF] Morgan | Bell inequalities for random fields[END_REF] and Vervoort [START_REF] Vervoort | Are hidden-variable theories for pilot-wave systems possible?[END_REF] have questioned whether the measurementindependence loophole can be closed in systems with a background medium, wherein the hidden field variables may be influenced by the analyzer settings. With a view to addressing this question, we here present a new platform for conducting Bell tests in the pilot-wave hydrodynamic system. Specifically, we devise and execute a static Bell test on the walking-droplet system, as a first step towards a dynamic test.

We consider a pair of walking droplets in the bipartite tunneling system introduced by Papatryfonos et al. [START_REF] Papatryfonos | Hydrodynamic superradiance in wave-mediated cooperative tunneling[END_REF] in their demonstration of a hydrodynamic analog of quantum superradiance (See Figure 1a). The two subsystems, labelled A and B, correspond to single, wave-generating particles confined to a pair of identical cavities separated by a barrier across which the particles may tunnel. Each particle generates waves and moves in response to them according to the mathematical model of walking droplets detailed in the Methods section. In each subsystem, the preferred cavity corresponds to the ground state |-> and the other to the excited state |+ >. The ground state may be either the inner or outer cavity, depending on the length of the outer cavities [START_REF] Papatryfonos | Hydrodynamic superradiance in wave-mediated cooperative tunneling[END_REF]. For the specific geometry considered here, the inner cavity of each subsystem is the ground state |->; the outer cavity, the excited state, |+ >.

The two subsystems are separated by a coupling cavity of variable length L c , and by barriers that are sufficiently high as to preclude the particles from tunneling into the coupling cavity. Waves are transmitted across the central cavity, and so provide the coupling between subsystems A and B. The efficiency of this coupling is prescribed by the geometry of the central cavity: by increasing its depth d c , the coupling may be increased, allowing the coupling cavity to serve as a nearly resonant transmission line [START_REF] Nachbin | Walking droplets correlated at a distance[END_REF]. Transitions between ground and excited states in the subsystems correspond to individual tunneling events, the rate of which depends on the depths of the left and right barriers, and the length of the coupling cavity, L c . As illustrated in Fig. 1a and1c, we identify the state (|+ > or |->) of each subsystem with the dichotomic property X in the optical Bell test, and the depths of the left and right barriers with the analyzer settings (respectively, α and β). The four possible combinations of (X A , X B ) are shown in Fig. 1c. We proceed by employing the numerical simulation method developed by Nachbin [START_REF] Nachbin | Walking droplets correlated at a distance[END_REF][START_REF] Nachbin | Tunneling with a hydrodynamic pilot-wave model[END_REF] to identify conditions under which Bell's inequality is violated. Pairs of entangled photons are produced at the source and sent in opposite directions to two measurement stations, conventionally named A and B. At each station, one of two measurement (polarizer) settings is selected randomly and the measurement is performed. The measurement outcomes, which can take on two possible values, +1 or -1, are noted. B) Schematic of our hydrodynamic Bell test. The system consists of a pair of drops (red and green) walking on the surface of a vibrating liquid bath (blue) that spans the solid substrate (grey). Each drop is confined to its subsystem, a pair of wells separated by barriers across which they may tunnel unpredictably at a rate influenced by the barrier depths α and β, as may assume values of a, a or b, b , respectively. C) The four possibilities for the drop configuration space, (XA, XB), in the hydrodynamic Bell test.

To collect statistics in a manner comparable to the optical Bell tests with entangled photons, we proceed as follows. Each run begins by placing the two particles at random positions within their own subsystem. Their trajectories are then calculated for 2000 Faraday periods. At that time, measurements are made in each subsystem. Specifically, we note the cavity in which the particle is located, and assign X the value of +1 if the particle is in the outer cavity or -1 if it is in the inner cavity. The methodology for collecting the data is detailed in the Methods section. As in our recent study of hydrodynamic superradiance [START_REF] Papatryfonos | Hydrodynamic superradiance in wave-mediated cooperative tunneling[END_REF], the barrier depths (α, β) prescribe the local tunneling probability, and may also influence that of the distant partner drop. We proceed by demonstrating that for judicious choice of pairs of measurement settings (α, β), Bell's inequality may be violated. Our main result is shown in Fig. 2, indicating a narrow parameter range in which the CHSH inequality is violated. In the remainder of this section we will analyze in detail how this violation comes about.

While the Bell inequality can be violated in our system with four different values of the measurement settings (a, a , b, b ), our exploration of the (a, a , b, b ) parameter-space indicates that a local maximum of S arises when b = a and b = a , the symmetric case in which one may write S(a, b, a , b ) = S(a, a, a , a ) = M (a, a)-M (a , a ) + 2M (a, a ). We deduced a maximum violation of S max = 2.49 ±0.04 when a = b = a * = 0.099 cm and a = b = a * = 0.1033 cm. In Fig. 2a, we plot S as a function of a for fixed a = a * , with the dashed line showing the limit S = 2 above which the CHSH inequality (Eq. 1) is violated. While the inequality is violated only for a narrow range of parameters settings, in this parameter regime, the violation is clear, and the statistical confidence of the violation is above 20 standard deviations. This behavior is reminiscent of the quantum case, where, without guidance from the theory, it is relatively difficult to find analyzer settings that allow for violation of the CHSH inequality, but for judiciously chosen settings, the inequality is violated substantially. Figures 2b and2c show a typical example of the convergence of the 'running average' with the number of runs which determines the relative error of our statistics. This approach indicates when our statistics have converged for each M (α, β) calculation, specifically when the relative error has fallen below the prescribed tolerance.

We proceed by detailing the manner in which Bell's inequality is violated in this classical pilot-wave system.

The maximum S value occurs for moderate barrier depths, for which the droplets may become strongly correlated through the background wave field. In Fig. 3a, we show typical trajectories for the three combinations of measurement settings (α, β) ∈ {(a * , a * ), (a * , a * ), (a * , a * )} that maximize S. For (a, a ) = (a * , a * ), S is maximized because M (a * , a * ) and M (a * , a * ) are large (see Fig. 3a top and middle panels), while M (a * , a * ) is relatively small (Fig. 3a lower panel). Fig. 3b corresponds to a shallow barrier, a = 0.0937 cm (the left-most value in Fig. 2a) and Fig. 3c to a relatively deep barrier, a = 0.11 cm (the right-most value in Fig. 2a). Figures 3b andc correspond to minima of S occurring when the a barrier is either too shallow (Fig. 3b) or too deep (Fig. 3c).

The degree of synchronization in the droplet tunneling depends on the extent to which the droplets are affected by the barrier depth in the distant station. When the barrier depth in one station is too small, the local particle is prevented from tunneling, regardless of the barrier depth in the other. The synchronization of states is thus reduced substantially. Conversely, when the barrier depth is too large, the particle generally tunnels across it, unaffected by the distant particle. Thus, the synchronization again remains relatively low. For intermediate barrier depths, each particle tunnels with a moderate probability that is strongly affected by the behavior of its distant partner. (We note that when only one drop is present, the variation of the measurement setting in the other subsystem does not influence its behavior. See Supplementary Information, Figure S1.) As in our previous study of superradiance [START_REF] Papatryfonos | Hydrodynamic superradiance in wave-mediated cooperative tunneling[END_REF], the wave-mediated correlation creates a collective behavior of the droplet pairs. In particular, when one of the droplets tunnels to its excited state, the probability of the second droplet doing likewise increases substantially. Thus, through its wave-mediated interaction with its partner, each droplet is affected by the barrier depth of the distant station.

Non-factorizable states arise in multi-and bipartite systems when the state of the whole cannot be simply defined in terms of the state of its subsystems [START_REF] Karimi | Classical entanglement?[END_REF][START_REF] Horodecki | Quantum entanglement[END_REF]. A canonical example is the singlet state of entangled photons. In our system, non-separability manifests itself through the fact that the joint probability of the two dichotomic states X A and X B , specifically P (X A , X B |α, β), is not equal to the product P (X A |α)P (X B |β). Classical non-factorizable states have been demonstrated in both electromagnetic and acoustic wave systems [START_REF] Karimi | Classical entanglement?[END_REF][START_REF] Spreeuw | A Classical Analogy of Entanglement[END_REF], and also in single-particle systems, for example through consideration of the internal degrees of freedom of a single atom [START_REF] Karimi | Classical entanglement?[END_REF]. However, neither the classical wave states nor the internal degrees of freedom in the single-particle system can be spatially separated [START_REF] Karimi | Classical entanglement?[END_REF][START_REF] Spreeuw | A Classical Analogy of Entanglement[END_REF]. Thus, such classical states can be used for neither performing dynamic Bell tests nor building analogs to qubits and quantum computing [START_REF] Karimi | Classical entanglement?[END_REF][START_REF] Spreeuw | A Classical Analogy of Entanglement[END_REF]. Conversely, our bipartite system exhibits spatially separated non-factorizable states, and so introduces the possibility of exploring novel forms of quantum-inspired classical computing, and performing more sophisticated Bell tests.

We have devised a platform for performing static Bell tests on a classical bipartite pilot-wave system. The maximum violation was found to be 2.49 ± 0.04, and arose when the system geometries were chosen such that the droplet motion was marked by strongly synchronized tunneling for one measurement setting combination, moderate and weak synchronization for the others. A key step in the process was recognizing that the system geometry may serve as a proxy for analyzer settings. In our system, the barrier depths play the role of polarizer angles in the photonic Bell tests. We expect this conceptual advance to prompt and facilitate the numerical and experimental execution of Bell tests in pilot-wave hydrodynamics and other stochastic bipartite classical systems.

Long-range correlations have been reported in other bipartite hydrodynamic pilot-wave systems, most notably in the establishment of statistical indistinguishability of (lower panel). The tunneling events are highly correlated only in the upper and lower panels. B) Trajectories corresponding to M (a * , a ) with a = 0.0937 cm. When the barrier depth a is sufficiently small, the wave-mediated communication between droplets is diminished, and droplets tend to get trapped in one cavity, leading to minima of S and M (a * , a ) ≈ 0 when we average over the droplet's initial conditions. C) Another minimum of M (a * , a ) and S occurs when one of the barrier depths is too large, in which case one of the droplets tunnels continuously, unimpeded by the barrier, as if it were in a single cavity. Averaging over all initial conditions leads to a relatively low value of M (a * , a ) ≈ 0. Note that the correlation function is deduced by averaging over all initial conditions in both subsystems.

a pair of distant droplets [START_REF] Nachbin | Walking droplets correlated at a distance[END_REF] and in the recent analog of superradiance [START_REF] Papatryfonos | Hydrodynamic superradiance in wave-mediated cooperative tunneling[END_REF]. In these examples, if observers were unaware of the pilot wave field and observed only the droplets, they could only account for the observed correlations by inferring a nonlocal connection between the droplet pairs. The violation of Bell's Inequality (Eq. 1) in our static Bell test may be rationalized in terms of the wave-mediated coupling between the two subsystems. Specifically, the form of the pilot wave and the concomitant particle positions depend on the geometry of both subsystems, and so necessarily violate the assumption of Bell locality. The feasibility of adapting our platform to incorporate time-dependent topography has recently been demonstrated by Nachbin [START_REF] Nachbin | The effect of isolation on two-particle correlations in pilot-wave hydrodynamics[END_REF]. This adaptation will allow for future Bell tests in which communication between the two subsystems may be eliminated and analyzer settings changed dynamically.

METHODS

Numerical method

System parameters are chosen to correspond to a fluid bath of density 0.95 g/cm 3 , viscosity 16 cS and surface tension 20.9 dynes/cm vibrating vertically in a sinusoidal fashion with peak acceleration γ 0 and frequency ω 0 = 80Hz. The resonant bouncing of the particle at the Faraday frequency triggers a quasi-monochromatic damped wave pattern with a corresponding Faraday wavelength of λ F = 4.75 mm. Each of the four cavities has a fixed length of 1.2 cm, corresponding to approximately 2.5λ F . In all simulations, we set the coupling cavity depth to d c = 6.3 λ F which ensures strong inter-cavity coupling. We thus describe our bipartite tunneling system in terms of two coupled, two-level systems, as shown schematically in Figure 1b.

Nachbin et al. [START_REF] Nachbin | Walking droplets correlated at a distance[END_REF][START_REF] Nachbin | Tunneling with a hydrodynamic pilot-wave model[END_REF] formulated a theoretical model for the one-dimensional motion of walking droplets over a vibrating liquid bath with complex topography. Here we adjust this model in order to consider the cooperative tunneling of two identical particles in the geometry depicted in Figure 1b. The positions, x j (j = 1, 2), of the two identical particles of mass m evolve according to Newton's Law:

mẍ j + c F (t) ẋj = -F (t) ∂η ∂x (x j (t), t). (2)
The particle moves in response to gradients of the wave elevation η(x, t), which thus plays the role of a timedependent potential. The particle motion is resisted by a drag force proportional to its speed. The drag constant c follows from the modeling presented in Molacek & Bush [START_REF] Moláček | Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory[END_REF]. The time dependence of these propulsive and drag forces is prescribed by F (t), as arises in the walker system owing to the droplet's bouncing [START_REF] Bush | Pilot-wave hydrodynamics[END_REF][START_REF] Nachbin | Tunneling with a hydrodynamic pilot-wave model[END_REF][START_REF] Milewski | Faraday pilot-wave dynamics: modelling and computation[END_REF]. In terms of their lateral motion, the particles are viewed as horizontal oscillators that can transition unpredictably between two neighboring cavities. The dichotomic property X assessed for Bell's inequality is assigned according to the particle location x j = x j (t m ) (j = A, B) at the measurement time t m . Specifically, X = +1 if the drop is in the outer, excited state, and X = -1 if it is in the inner, ground state. The particles serve as moving wave sources that establish their own time-dependent wave potential that is computed as follows. The velocity potential of the liquid bath φ(x, z, t) is a harmonic function satisfying Laplace's equation. In the bulk of the fluid, the velocity field is given by (u, v) = ∇φ. The wave model is formulated in the bath's reference frame, where the effective gravity is g(t) = g + γ 0 sin(ω 0 t), where g is the gravitation acceleration, and γ 0 is the amplitude of the bath's vibrational acceleration. The wave field thus evolves according to [START_REF] Nachbin | Tunneling with a hydrodynamic pilot-wave model[END_REF][START_REF] Milewski | Faraday pilot-wave dynamics: modelling and computation[END_REF]:

∂η ∂t = ∂φ ∂z + 2ν ∂ 2 η ∂x 2 , (3)
∂φ ∂t = -g(t)η + σ ρ ∂ 2 η ∂x 2 +2ν ∂ 2 φ ∂x 2 - j=1,2 P d (x -x j (t)) ρ . (4)
The particles (j = A, B) generate waves on the free surface by applying local pressure terms P d . The wave forcing term P d (x -x j (t)) and the coefficient F (t) are activated only during a fraction of the Faraday period T F , corresponding to the contact time T c in the walkingdroplet system and approximated by T c = T F /4. The particle is assumed to be in resonance with the most unstable (subharmonic) Faraday mode of the bath [START_REF] Milewski | Faraday pilot-wave dynamics: modelling and computation[END_REF], a key feature of pilot-wave hydrodynamics [START_REF] Bush | Pilot-wave hydrodynamics[END_REF][START_REF] Bush | Hydrodynamic quantum analogs[END_REF][START_REF] Protière | Particle-wave association on a fluid interface[END_REF]. The numerical approach to simulating Eqs. ( 2)-( 3) is detailed in the Supplementary Information.

Measurement procedure and data collection

To initialize the runs, the wave and velocity fields of the bath are set to zero, and the particle positions are assigned random, uniformly distributed values. Then, the model runs for 2000 Faraday periods, a measurement is made, and all fields are reset back to zero to initialize the subsequent run. This cycle is repeated for each set of parameter settings until the relative error in the running average of M (α = a, β = b) is reduced to an acceptably small value. We set this tolerance to be 3% for parameters that violate the inequality and 7% for those that do not. While extremely accurate, this 'discrete' technique is computationally intensive; thus we have used it only for the most critical points of the parameter space, in which the maximal Bell violations occurred. To explore the parameter space more efficiently, we adopt an alternative, relatively expedient, 'continuous' approach, in which the final conditions of one run serve as the initial conditions of the next. We demonstrated the statistical equivalence of the two approaches as follows. For specific selected data points, we performed approximately 30 different runs using the two techniques, and found the results of the 'discrete' and 'continuous' runs to be in agreement to within 3%. We then executed continuous runs for 48,000 Faraday periods, during which measurements are performed frequently at uniformly distributed random times. After a sufficiently long run, the full range of initial conditions will have been effectively explored. The consistency of the results deduced with the discrete and continuous approaches demonstrates that the longtime emergent statistics are independent of the initial conditions.

Since the inequality involves four different correlation functions (three for the symmetric case considered here), finding the combinations of measurement settings that maximized S was not entirely straightforward. Figure 4 summarises the strategy we followed in seeking violations. We first investigated the evolution of a single correlation function M (α = a, β = a) as a function of a. This gave us a good sense of parameters that maximize the difference δM (a, a ) = M (a, a) -M (a , a ) (see Figure 4a). δM (a, a ) involves two of the correlation functions of Eq. 1, in the symmetric case of interest where a=b and a =b . Figure 4b shows a 2D plot of the optimisation of δM as a function of a and a . The black dashed lines highlight the domain in which (max a,a (δM ) -δM )/ max a,a (δM ) > 0.9. The other term in the inequality, specifically 2M (α = a, β = a ), represents a combination of measurements from unequal barrier depths at the two measurement stations. Figure 4c represents the dependence of 2M (α = a * , β = a ) on depth a for fixed a = a * , the S-maximizing value considered in Figures 2 and3. Finally, Figure 4d shows the evolution of the correlation functions (M (a, a ); M (a , a )) with increasing a and fixed a = a * .

I. DATA AVAILABILITY

The data that support the findings of our study are available upon request.

II. CODE AVAILABILITY

The code that generated the data is available upon request.

Figure 1 .

 1 Figure 1. Photonic and hydrodynamic Bell test arrangements. A) Schematic of an optical Bell test setup.Pairs of entangled photons are produced at the source and sent in opposite directions to two measurement stations, conventionally named A and B. At each station, one of two measurement (polarizer) settings is selected randomly and the measurement is performed. The measurement outcomes, which can take on two possible values, +1 or -1, are noted. B) Schematic of our hydrodynamic Bell test. The system consists of a pair of drops (red and green) walking on the surface of a vibrating liquid bath (blue) that spans the solid substrate (grey). Each drop is confined to its subsystem, a pair of wells separated by barriers across which they may tunnel unpredictably at a rate influenced by the barrier depths α and β, as may assume values of a, a or b, b , respectively. C) The four possibilities for the drop configuration space, (XA, XB), in the hydrodynamic Bell test.

Figure 2 .

 2 Figure 2. Violation of Bell's inequality. A) Bell parameter S(a, a, a , a ) as a function of the barrier depth, a , for the symmetric case of a = b, a = b . For the calculation of the corresponding correlation functions, M (a, a), M (a, a ) and M (a , a ), the barrier depth a = a * = 0.099 cm remains fixed. For each combination of measurement settings, runs continue until statistics converge. The maximum Bell violation appears at a * = 0.1033cm, where S = 2.49 ± 0.04. B) Typical curve showing the convergence of Smax, the Bell parameter taken at the maximum point of violation (a = b = a * = 0.099 cm; a = b = a * = 0.1033 cm). The error bars indicate ±3 standard deviations. C) Relative error δSmax/Smax of the estimation of the Bell parameter S with the number of runs, evaluated for the maximum point of violation (a = b = a * ; a = b = a * ). Inset: log-log scale; the dashed line indicates a -1/2 slope as expected from the convergence of an ensemble average.

Figure 3 .

 3 Figure 3. Trajectory analysis. A-C) Droplet trajectories for the symmetric case (α,β) ∈ (a, a ) with a = a * . Time evolves in the vertical direction. In (A), a = a * =0.1033 cm (the maximizing value for S); in (B), a = 0.0937 cm; in (c), a = 0.11 cm. A) Trajectories corresponding to the three correlation functions M (a * , a * ) = 0.94 (upper panel), M (a * , a * ) = 0.84 (middle panel), and M (a * , a * ) = 0.13(lower panel). The tunneling events are highly correlated only in the upper and lower panels. B) Trajectories corresponding to M (a * , a ) with a = 0.0937 cm. When the barrier depth a is sufficiently small, the wave-mediated communication between droplets is diminished, and droplets tend to get trapped in one cavity, leading to minima of S and M (a * , a ) ≈ 0 when we average over the droplet's initial conditions. C) Another minimum of M (a * , a ) and S occurs when one of the barrier depths is too large, in which case one of the droplets tunnels continuously, unimpeded by the barrier, as if it were in a single cavity. Averaging over all initial conditions leads to a relatively low value of M (a * , a ) ≈ 0. Note that the correlation function is deduced by averaging over all initial conditions in both subsystems.

Figure 4 .

 4 Figure 4. Strategy to optimize S(a, a, a , a ) = M (α = a, β = a) -M (α = a , β = a ) + 2M (α = a, β = a ) . S is optimized by searching a parameter regime (a, a ) near the maximum of δM (a, a ) = M (α = a, β = a) -M (α = a , β = a ) and a range of a that maximizes M (a, a ) with a fixed. A) Evolution of M (α = a, β = a) as a function of a. The indicated values a * = 0.099 cm and a * = 0.1033 cm are the S-maximizing values used in Figs. 2-3. Note δa = a -a. The difference between the corresponding correlation functions δM (a, a + δa) = δM (a, a ) = M (α = a, β = a) -M (α = a , β = a ) is marked in orange. B) Optimization of δM (α = a, β = a + δa) as a function of barrier depth a and δa. The domain for which (max a,a (δM ) -δM )/ max a,a (δM ) > 0.9 is bound by the black dashed lines. C) 2M (α = a * , β = a ) as a function of depth a for fixed a = a * . D) Evolution of S(a, a, a , a ) in the correlation representation space (M (a, a ); M (a , a )) with a = a * . The direction of increasing a is indicated by the blue arrow. The dots are colored with respect to their S values. A grey dashed line indicates the limiting case S = 2.

Yves Couder and Emmanuel Fort [1, 2] discov- * We acknowledge the generous financial support of the NSF through grants CMMI-1727565 and CMMI-2154151, the European Union's Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie project EnHydro, grant No. 841417, and the CNPq under (PQ1D) 307078/2021-3 and FAPERJ CNE, Project No. E-26/201.156/2021.