# Enhanced diffusion over a periodic trap by hydrodynamic coupling to an elastic mode

Juliette Lacherez, Maxime Lavaud, Yacine Amarouchene, David S. Dean,
Thomas Salez

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Juliette Lacherez, Maxime Lavaud, Yacine Amarouchene, David S. Dean, Thomas Salez. Enhanced diffusion over a periodic trap by hydrodynamic coupling to an elastic mode. Communications Physics, 2025, 8, pp.465. hal-04934529

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# Enhanced diffusion over a periodic trap by hydrodynamic coupling to an elastic mode

Juliette Lacherez¹, Maxime Lavaud¹,², Yacine Amarouchene¹,* David S. Dean¹,† and Thomas Salez¹‡

¹Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33400, Talence, France and

²Univ. Bordeaux, CNRS, Bordeaux INP, CBMN, UMR 5248, F-33600, Pessac, France

(Dated: February 7, 2025)

In many physical systems, degrees of freedom are coupled *via* hydrodynamic forces, even in the absence of Hamiltonian interactions. A particularly important and widespread example concerns the transport of microscopic particles in fluids near deformable boundaries. In such a situation, the influence of elastohydrodynamic couplings on Brownian motion remains to be understood. Unfortunately, the temporal and spatial scales associated with the thermal fluctuations of usual surfaces are often so small that their deformations are difficult to monitor experimentally, together with the much slower and larger particle motion at stake. Here, we propose a minimal model describing the hydrodynamic coupling of a colloidal particle to a fluctuating elastic mode, in presence of an external periodic potential. We demonstrate that the late-time diffusion coefficient of the particle increases with the compliance of the elastic mode. Remarkably, our results reveal that, and quantify how: i) spontaneous microscopic transport in complex environnements can be affected by soft boundaries – a situation with numerous practical implications in nanoscale and biological physics; ii) the effects of fast and tiny surface deformations are imprinted over the long-term and large-distance colloidal mobility – and are hence measurable in practice.

Brownian motion refers to the random movement of a colloid at equilibrium, induced by its collisions with the molecules of the surrounding fluid. It has played a foundational role in modern statistical physics since its discovery [1] and theoretical description [2, 3] over a century ago. The celebrated experiments of Perrin on colloids in a gravitational field [4] validated further the theoretical description and confirmed the atomic nature of matter. Extending the classical framework of Brownian motion in bulk Newtonian fluids to more realistic systems, such as particles diffusing in complex environnements, opens up new challenges for nanoscale and biological physics. One common route is to study Brownian motion in viscoelastic fluids or intracellular media, as typically performed through microrheology [5], which revealed the emergence of anomalous diffusion [6]. Another source of complexity lies in the introduction of topological constraints and boundaries. For instance, a simple-yet-canonical situation where interfaces matter is the diffusion near rigid walls, that has been extensively studied both theoretically and experimentally [7–13]. Besides the practical interest of such a configuration towards the sensing of surface forces [14], one of the key aspects lies perhaps in the apparition of anisotropic and space-dependent mobilities [15] due to the frictional boundary condition at the walls. As a consequence, multiplicative noises emerge, leading to non-Gaussian diffusion and the probability enhancement of rare events [16–20].

However, in most practical situations where molecules or microorganisms diffuse near surfaces, the latter are typically soft. Therefore, the dynamical properties of Brownian particles near deformable boundaries must be

<figure><img src="image_2.png" /><figcaption>FIG. 1. Schematic of the system. A test Brownian particle (1) is trapped in a periodic energy potential of amplitude $\Delta U$ and spatial period $\lambda$. It is coupled via a hydrodynamic-like interaction of constant damping coefficient $\gamma_{12}$ to a thermally-fluctuating elastic mode, represented by another Brownian particle (2) trapped in a harmonic trap of compliance (i.e. inverse stiffness) $\kappa$. Besides, both particles feel individual hydrodynamic drag forces represented by their constant self-drag coefficients $\gamma_{11}$ and $\gamma_{22}$, and are described within the overdamped Langevin regime.</figcaption></figure>

understood. Along this line of thought, a few studies, mostly theoretical, have addressed the case of tracers diffusing near soft surfaces. Examples include liquid interfaces [21, 22], and fluctuating ones [23]. The case of elastic interfaces was also addressed in the contexts of contactless microrheology [24] and biological membranes [25]. Nevertheless, these works focused on the low-coupling regime, where the particles are distant or point-like, and thus unable to generate rectified flows that lead to surface deformation. Interestingly though, within a purely deterministic context, near-contact elastohydrodynamic couplings have been studied in the past decades. Examples include contactless colloidal-probe rheology of soft substrates in normal mode [26, 27], as well as the emergence of lift forces at low Reynolds numbers [28, 29], that were first revealed theoretically [30–34], and then confirmed experimentally [35–39]. The leading-order corrections induced by elasticity to the full mobility tensor

<footnote>* yacine.amarouchene@u-bordeaux.fr</footnote>

<footnote>† david.dean@u-bordeaux.fr</footnote>

<footnote>‡ thomas.salez@cnrs.fr</footnote>
near a rigid wall [7] were obtained for various geometries and rheologies [40–42], and revealed a zoology of counter-intuitive inertial-like solutions.

In view of the above state of the art, the arising challenge is now to understand the influence of soft boundaries on the spontaneous motion of nearby finite-sized Brownian particles. Molecular-dynamics simulations were performed in such a context [43]. Besides, a recent experimental study indicated that soft colloids diffusing near rigid walls feel transient, non-conservative forces that could be related to an intricate combination of thermal fluctuations, soft boundaries and viscous flows [44]. Finally, at a practical level, a natural question to ask is whether or not the effects of fast and tiny surface fluctuations can be observed over the long times and large distances associated with colloidal diffusion. In this Letter, we thus introduce a minimal, but generic, theoretical model in order to investigate Brownian elasto-hydrodynamic effects. Specifically, we combine the hydrodynamic coupling of a colloidal particle with an elastic mode [45, 46] to an external periodic trapping [47–50]. We solve the problem both analytically and numerically, and demonstrate that the late-time diffusion coefficient of the particle increases notably with the compliance of the elastic mode.

**The model.** As depicted in Fig. 1, we consider an overdamped system made of two coupled degrees of freedom: i) a principal degree of freedom $q_1(t)$, corresponding to the spatial position of a test colloid over time $t$, which is free to explore space while being subject to a periodic trapping potential $\phi(q_1)$; and ii) an auxiliary degree of freedom $q_2(t)$, corresponding to an elastic mode, which is harmonically confined with a potential energy $q_2^2/(2\kappa)$, where $\kappa$ is the compliance. Thus, the total energy of the system is given by the Hamiltonian

$$H(q_1, q_2) = \phi(q_1) + \frac{q_2^2}{2\kappa}. \quad (1)$$

At thermal equilibrium, the probability density function (PDF) of $q_1$ and $q_2$ is given by the Gibbs-Boltzmann distribution

$$P_{\text{eq}}(q_1, q_2) = \frac{\exp[-\beta H(q_1, q_2)]}{Z}, \quad (2)$$

where $Z$ is the normalising partition function, and where $\beta = 1/(k_B T)$ with $T$ the temperature and $k_B$ Boltzmann's constant. The equilibrium statistics of $q_1$ and $q_2$ are independent due to the additivity of the Hamiltonian, which implies in particular that $q_2$ does not affect the equilibrium distribution of $q_1$.

The system dynamics is described by the coupled overdamped Langevin equations

$$\gamma_{11} \frac{dq_1}{dt} + \gamma_{12} \frac{dq_2}{dt} = -\phi'(q_1) + \eta_1(t), \quad (3)$$

$$\gamma_{12} \frac{dq_1}{dt} + \gamma_{22} \frac{dq_2}{dt} = -\frac{q_2}{\kappa} + \eta_2(t), \quad (4)$$

where the constants $\gamma_{ij}$, with $i, j \in \{1, 2\}$, are the components of the friction tensor $\gamma$, which is related to the mobility tensor $\mu$ through $\mu = \gamma^{-1}$; and where the functions $\eta_i(t)$ are zero-mean Gaussian white noises, with a correlation function $\langle \eta_i(t) \eta_j(t') \rangle = 2k_B T \delta(t - t') \gamma_{ij}$, where $\langle \cdot \rangle$ is the ensemble average. This choice of noise correlation is self-consistent with the equilibrium distribution of $q_1$ and $q_2$ being given by Eq. (2).

**Theoretical analysis.** In the weakly-compliant limit ($\kappa \ll 1$), one can compute the late-time diffusion coefficient $D^*(\kappa)$ of the colloid using a perturbation analysis at $O(\kappa)$, together with an exact representation of the effective diffusion coefficient obtained via the Stokes-Einstein relation, and a multiscale analysis [51, 52]. Doing so (see details in SM [53]), we obtain

$$D^*(\kappa) \approx D^*(0)(1 + \alpha), \quad (5)$$

with

$$\alpha = \beta \kappa \frac{\gamma_{12}^2 \int_0^\lambda dq_1 \phi'(q_1)^2 \exp[\beta \phi(q_1)]}{\gamma_{11}^2 \int_0^\lambda dq_1 \exp[\beta \phi(q_1)]}, \quad (6)$$

and

$$D^*(0) = \frac{k_B T \lambda^2}{\gamma_{11} Z_+ Z_-}, \quad (7)$$

where

$$Z_{\pm} = \int_{0}^{\lambda} dq_1 \exp[\pm \beta \phi(q_1)]. \quad (8)$$

Since $\alpha \ge 0$, the hydrodynamic coupling of the colloid to the elastic mode always increases the late-time diffusion coefficient of the colloid. Moreover, the effect is larger at larger compliance $\kappa$. We also see that a non-flat periodic potential $\phi(q_1)$ is required for the effect to exist, within the assumptions of the model.

In the particular case of a sinusoidal potential, where $\phi(q_1) = \Delta U \sin(2\pi q_1/\lambda)$, with $\Delta U$ the amplitude of the potential, and $\lambda$ its spatial period, we have

$$\alpha = 4\pi^2 \epsilon \frac{\gamma_{12}^2 v I_1(v)}{\gamma_{11}^2 I_0(v)}, \quad (9)$$

with $v = \beta \Delta U$, where $I_n$ denotes the order-$n$ modified Bessel function of the first kind, and where we introduced the dimensionless compliance

$$\epsilon = \frac{\kappa k_B T}{\lambda^2}. \quad (10)$$

As such, the relative increase of the late-time diffusion coefficient is larger for smaller $\lambda$. This is attributed to the fact that for a smaller $\lambda$ the coupling to the elastic mode increases the jump rate of the colloid between neighbouring minima, instead of being averaged out as would be the case at large $\lambda$. We also see that the relative increase of the late-time diffusion coefficient depends linearly on the
function $vI_1(v)/I_0(v)$, which is a monotonically increasing function of $v$. This means that the relative magnitude of the effect increases with the depth of the trap – the obvious counterpart being that larger times are then needed to reach the late-time diffusive regime. Finally, the relative magnitude of the effect increases with the relative magnitude $\gamma_{12}/\gamma_{11}$ of the hydrodynamic coupling to the elastic mode, as expected.

The multiscale analysis can also be employed in the limit where $\epsilon \to \infty$, which corresponds to a very compliant elastic mode. In this case, we also find (see details in SM [53]) that the late-time diffusion coefficient is increased relatively to the rigid-case value, and saturates to a maximum value at large $\epsilon$, as

$$D^* \approx \frac{k_B T \mu_{11} (\mu_{11}\mu_{22} - \mu_{12}^2)}{\lambda^{-2} Z_+ Z_- (\mu_{11}\mu_{22} - \mu_{12}^2) + \mu_{12}^2} + O\left(\frac{1}{\sqrt{\epsilon}}\right). \quad (11)$$

Such a plateau value can be notably higher than the rigid-case value, indicating the potential strength of the effect in practical systems relevant to soft matter and biophysics. Besides, once again, we note that when the periodic potential $\phi(q_1)$ is flat, the late-time diffusion coefficient is not modified by the hydrodynamic coupling to the elastic mode.

**Heuristic argument.** In order to understand the softness-induced late-time diffusivity enhancement from a physical point of view, we now present a much simpler heuristic argument giving the same result as the rigorous analysis in the small-$\epsilon$ limit (see details in SM [53]). The intrinsic time scale $\tau_2 = \kappa\gamma_{22}$ of the stochastic process $q_2(t)$ is assumed to be small with respect to all the other time scales. In this limit, the effective zero-temperature overdamped Langevin equation for $q_1(t)$ is

$$\frac{dq_1}{dt} = -\mu_e(q_1)\phi'(q_1), \quad (12)$$

where we have introduced the dressed mobility

$$\mu_e(q_1) = \frac{1}{\gamma_{11} \left[ 1 + \frac{\kappa \gamma_{12}^2 \phi''(q_1)}{\gamma_{11}^2} \right]} \quad (13)$$

The equilibrium marginal PDF in position $q_1$ is $P_{\text{meq}}(q_1) = (Z_-)^{-1} \exp[-\beta\phi(q_1)]$. Therefore, assuming that the time-dependent marginal PDF $P_1(q_1, t)$ satisfies an effective Fokker-Planck equation, the latter must have the form

$$\frac{\partial P_1(q_1, t)}{\partial t} = \frac{\partial}{\partial q_1} \left\{ \mu_e(q_1) \left[ k_B T \frac{\partial P_1(q_1, t)}{\partial q_1} + \phi'(q_1) P_1(q_1, t) \right] \right\}, \quad (14)$$

where the T-independent term on the right-hand side is determined from the zero-temperature dynamics, while the T-dependent term on the right-hand side is constructed to ensure convergence of $P_1$ to $P_{\text{meq}}$ at infinite time. Note that we have implicitly assumed here that the dressed mobility $\mu_e(q_1)$ does not exhibit any temperature dependence.

The dressed diffusion coefficient is given by the Stokes-Einstein relation, $D_e(q_1) = k_B T \mu_e(q_1)$, which differs

<figure><img src="image_3.png" /><figcaption>FIG. 2. a) Typical trajectories $q_i(t)$ of the test particle ($i=1$, blue line) and the elastic mode ($i=2$, orange line), simulated by numerically solving Eqs. (3) and (4), for $\mu_{11}=1.0$, $\mu_{12}=0.5$, $\mu_{22}=0.5$, $\Delta U=1.0$, $\lambda=1.0$, $\kappa=1.0 \times 10^{-3}$, $\Delta t=1.0 \times 10^{-4}$, and $k_B T = 1.0$. b) Late-time marginal PDF $P_{\text{meq}}$ for the periodised position $q^*$, given by $q_1 = q^* \text{mod}(\lambda)$, of the test particle, obtained by binning $10^2$ trajectories of $10^7$ points. The symbols are the results from the numerical simulations. The solid blue line corresponds to the Boltzmann distribution given in Eq. (16).</figcaption></figure>

from the bare (or “microscopic”) diffusion coefficient $D_m = k_B T / \gamma_{11}$ of the colloid. Interestingly, the spatial dependence of the dressed diffusion coefficient implies that the effective Langevin equation must have a multiplicative noise, leading to a supplementary source of non-Gaussian displacements [14, 16–20, 54] in addition to the trivial one induced by the trapping potential.

We now invoke the Lifson-Jackson formula [55], providing the late-time diffusion coefficient in a periodic potential, which reads

$$D^*(\kappa) = \frac{\lambda^2}{\int_0^\lambda dq_1 \frac{\exp[\beta\phi(q_1)]}{D_e(q_1)} \int_0^\lambda dq_1 \exp[-\beta\phi(q_1)]} \quad (15)$$

From this expression, we can recover Eqs. (5) and (6). We emphasize that the heuristic argument we presented here is not rigorous. Nevertheless, it can also be applied to evaluate the effective diffusion coefficient of an underdamped Brownian particle near the overdamped limit (see details in SM [53]), for which we once again recover the results obtained via rigorous multiscale analysis [51, 52]. The same method also yields the first-order correction to the effective diffusion coefficient for a particle carrying a dipole moment which interacts with a random or periodic electric field when the dipole dynamics is assumed to be much faster than the particle diffusion [56].

**Numerical simulations.** In order to verify the above predictions, Eqs. (3) and (4) are numerically solved for a sinusoidal trapping potential $\phi(q_1)$, using the Euler-Maruyama scheme [57], through an optimised Cython [58] code (see details in SM [53]). The initial positions of the test particle and the elastic mode are sampled from Eq. (2). Typical raw trajectories are shown in Fig. 2a). We observe that the elastic mode $q_2(t)$ fluctuates around its equilibrium value ($q_2=0$), within the harmonic potential it is subject to, whereas the position $q_1(t)$ of the test colloid fluctuates at short times in a minimum of the sinusoidal potential $\phi(q_1)$ before randomly jumping from one minimum to the other by passing the
<figure><img src="image_4.png" /><figcaption>FIG. 3. Mean-squared displacements $\langle q_1^2 \rangle_\kappa$ and $\langle q_1^2 \rangle_0$ of the test particle as functions of the time increment $\tau$, for $\kappa = 1.0 \times 10^{-3}$ (blue disks) and $\kappa = 0.0$ (black crosses) respectively, as computed from $10^2$ trajectories of $10^7$ points each, for $\gamma_{11} = 5.0$, $\gamma_{12} = 22.0$, $\gamma_{22} = 104.0$, $\Delta U = 1.0$, $\lambda = 1.0$, $\Delta t = 1.0 \times 10^{-4}$, and $k_B T = 1.0$. The symbols are the results from the numerical simulations. The dash-dotted black line indicates $2D^*(0)\tau$, where $D^*(0)$ is the classical Lifson-Jackson late-time diffusion coefficient [55]. The solid cyan line indicates $2D^*(\kappa)\tau$, where $D^*(\kappa)$ is the late-time diffusion coefficient given by Eq. (5). The dashed pink line indicates $2D_m\tau$, where $D_m = k_B T / \gamma_{11}$ is the bare diffusion coefficient. The characteristic times $\tau_2 = \kappa\gamma_{22}$ and $\tau^* = \lambda^2/D^*(\kappa)$ are indicated with the dark-grey and light-grey vertical dashed lines, respectively.</figcaption></figure>

potential barrier. From such trajectories, we can first compute the late-time PDF in positions. As an example, Fig. 2b) shows the equilibrium marginal PDF of the periodised variable $q^*$, defined by $q_1 = q^*\text{mod}(\lambda)$, which satisfies

$$P_{\text{meq}}(q^*) = \frac{\exp[-\beta\phi(q^*)]}{\int_0^\lambda dq \exp[-\beta\phi(q)]} \quad (16)$$

In order to probe the dynamical effects of the hydrodynamic coupling of the trapped test particle to the elastic mode, we now turn to the analysis of the mean-squared displacement (MSD), defined as

$$\langle q_i^2 \rangle(\tau) = \langle [q_i(t + \tau) - q_i(t)]^2 \rangle, \quad (17)$$

where $\tau$ is a time increment. As shown in Fig. 3, the MSD of the test colloidal particle exhibits two distinct diffusive regimes separated by a crossover region. The short-time diffusive regime is characterized by the bare diffusion coefficient $D_m$. The late-time diffusive regime is characterized by the late-time diffusion coefficient $D^*$, and is reached when the particle has diffused over many periods of the trapping potential, i.e. when its MSD is large compared to $\lambda^2$. This sets a time scale $\tau^* = \lambda^2/D^*$, which must be exceeded to see the late-time diffusive regime.

Finally, we focus on the late-time diffusive regime of the simulated trajectories, and compute from the MSD

<figure><img src="image_5.png" /><figcaption>FIG. 4. Relative increase $\Delta D^*/D^* = [D^*(\epsilon) - D^*(0)]/D^*(0)$ of the late-time diffusion coefficient $D^*$ of the test particle as a function of the dimensionless compliance $\epsilon$, for $\mu_{11} = 1.0$, $\mu_{12} = 0.5$, $\mu_{22} = 0.5$, $\Delta U = 1.0$, $\lambda = 1.0$, and $k_B T = 1.0$. The symbols are the results from the numerical simulations. The dash-dotted blue line corresponds to the small-$\epsilon$ prediction of Eq. (5). The dash-dotted orange line corresponds to the large-$\epsilon$ prediction of Eq. (11). The solid black line corresponds to the Kubo formula of Eq. (S65) (see details in SM [53]).</figcaption></figure>

the relative excess in late-time diffusion coefficient, i.e. $\Delta D^*/D^* = [D^*(\epsilon) - D^*(0)]/D^*(0)$, as a function of $\epsilon$. The results are shown in Fig. 4 for a given set of parameters. We observe that the numerical-simulation data agree well with the small-$\epsilon$ and large-$\epsilon$ asymptotic predictions of Eqs. (5) and (11), respectively. Furthermore, the Kubo formula can be used to evaluate $D^*$ for any value of $\epsilon$ (see details in SM [53], and in particular Eq. (S65)). We see that the numerical-simulation data agree at all $\epsilon$ values with the Kubo formula, and that the latter interpolates well the two asymptotic predictions.

In conclusion, we theoretically and numerically studied the Brownian motion of colloid-like particles near elastic interfaces using minimal ingredients. We showed that a Brownian particle trapped in a periodic potential sees its late-time diffusivity dressed and significantly enhanced by hydrodynamic coupling to a hidden elastic mode. Despite being minimalist, our description reveals generic, original and large effects, relevant for the transport of colloidal particles near complex boundaries. In particular, our results suggest the intriguing possibility of understanding, and even tuning, the diffusivity of colloids near deformable and patchy interfaces – with key implications for biophysics and nanofluidics.

*Data availability.* All codes used in this study to produce data are available through the online repository https://github.com/EMetBrown-Lab/Lacherez2024_Codes.

*Acknowledgements.* The authors thank Nicolas Fares, Quentin Ferreira, Aditya Jha and Elodie Millan for interesting discussions. Computer time for this study was provided by the computing facilities of the Mésocentre
de Calcul Intensif Aquitain. The authors acknowledge financial support from the European Union through the European Research Council under EMetBrown (ERC-CoG-101039103) grant. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. The authors also acknowledge financial support from the Agence Nationale de la Recherche under EDIPS (ANR-23-CE30-0020), Softer (ANR21-CE06-0029) and Fricolas (ANR-21-CE06-0039) grants, as well as from the Interdisciplinary and Exploratory Research Program under a

MISTIC grant at the University of Bordeaux, France. They acknowledge as well the support from the LIGHT S&T Graduate Program (PIA3 Investment for the Future Program, ANR-17EURE-0027). Finally, they thank the RRI Frontiers of Life, which received financial support from the French government in the framework of the University of Bordeaux's France 2030 program, as well as the Soft Matter Collaborative Research Unit, Frontier Research Center for Advanced Material and Life Science, Faculty of Advanced Life Science, Hokkaido University, Sapporo, Japan, and the CNRS International Research Network between France and India on "Hydrodynamics at small scales: from soft matter to bioengineering".

[1] Robert Brown. XXVII. A brief account of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. *The Philosophical Magazine*, 4(21):161-173, September 1828. Publisher: Taylor & Francis.

[2] Albert Einstein. *Über die von der molekularinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen*. Annalen der Physik. Johann Ambrosius Barth, 1905.

[3] Marian Von Smoluchowski. Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. *Annalen der Physik*, 326(14):756-780, January 1906.

[4] Jean Perrin. Mouvement brownien et grandeurs moléculaires. *Le Radium*, 6(12):353-360, 1909.

[5] FC MacKintosh and CF Schmidt. Microrheology. *Current opinion in colloid & interface science*, 4(4):300-307, 1999.

[6] Ralf Metzler and Joseph Klafter. The random walk's guide to anomalous diffusion: a fractional dynamics approach. *Physics reports*, 339(1):1-77, 2000.

[7] Howard Brenner. The slow motion of a sphere through a viscous fluid towards a plane surface. *Chemical Engineering Science*, 16(3):242-251, 1961.

[8] Luc P. Faucheux and Albert J. Libchaber. Confined Brownian motion. *Physical Review E*, 49(6):5158-5163, June 1994.

[9] Eric R. Dufresne, Todd M. Squires, Michael P. Brenner, and David G. Grier. Hydrodynamic coupling of two brownian spheres to a planar surface. *Phys. Rev. Lett.*, 85:3317-3320, Oct 2000.

[10] E. A. J. F. Peters and Th. M. A. O. M. Barenbrug. Efficient Brownian dynamics simulation of particles near walls. *I. Reflecting and absorbing walls*. *Physical Review E*, 66(5):056701, November 2002.

[11] B. U. Felderhof. Effect of the Wall on the Velocity Autocorrelation Function and Long-Time Tail of Brownian Motion. *The Journal of Physical Chemistry B*, 109(45):21406-21412, November 2005.

[12] Guanglai Li, Lick-Kong Tam, and Jay X. Tang. Amplified effect of Brownian motion in bacterial near-surface swimming. *Proceedings of the National Academy of Sciences*, 105(47):18355-18359, November 2008.

[13] Pauline Simonnin, Benoît Noetinger, Carlos Nieto-

Draghi, Virginie Marry, and Benjamin Rotenberg. Diffusion under Confinement: Hydrodynamic Finite-Size Effects in Simulation. *Journal of Chemical Theory and Computation*, 13(6):2881-2889, June 2017.

[14] Maxime Lavaud, Thomas Salez, Yann Louyer, and Yacine Amarouchene. Stochastic inference of surface-induced effects using Brownian motion. *Physical Review Research*, 3(3):L032011, July 2021.

[15] Mpumelelo Matse, Mykyta V. Chubynsky, and John Bechhoefer. Test of the diffusing-diffusivity mechanism using near-wall colloidal dynamics. *Phys. Rev. E*, 96:042604, Oct 2017.

[16] Bo Wang, Stephen M. Anthony, Sung Chul Bae, and Steve Granick. Anomalous yet Brownian. *Proceedings of the National Academy of Sciences*, 106(36):15160-15164, September 2009.

[17] Eli Barkai and Stanislav Burov. Packets of diffusing particles exhibit universal exponential tails. *Phys. Rev. Lett.*, 124:060603, Feb 2020.

[18] Arthur Alexandre, Matthieu Mangeat, Thomas Guérin, and David S. Dean. How Stickiness Can Speed Up Diffusion in Confined Systems. *Physical Review Letters*, 128(21):210601, May 2022.

[19] Elodie Millan, Maxime Lavaud, Yacine Amarouchene, and Thomas Salez. Numerical simulations of confined Brownian-yet-non-Gaussian motion. *The European Physical Journal E*, 46(4):24, March 2023.

[20] Haolan Xu, Xu Zheng, and Xinghua Shi. Deciphering non-gaussianity of diffusion based on the evolution of diffusivity. *Physical Review Research*, 6(2):023014, 2024.

[21] Thomas Bickel. Brownian motion near a liquid-like membrane. *The European Physical Journal E*, 20(4):379-385, August 2006.

[22] G. M. Wang, R. Prabhakar, and E. M. Sevick. Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces. *Phys. Rev. Lett.*, 103:248303, Dec 2009.

[23] Sophie Marbach, David S. Dean, and Lydéric Bocquet. Transport and dispersion across wiggling nanopores. *Nature Physics*, 14(11):1108-1113, November 2018.

[24] Chen Bar-Haim and Haim Diamant. Correlations in suspensions confined between viscoelastic surfaces: Noncontact microrheology. *Physical Review E*, 96(2):022607, 2017.

[25] Abdallah Daddi-Moussa-Ider and Stephan Gekle. Brow-
nian motion near an elastic cell membrane: A theoretical study. *The European Physical Journal E*, 41:1–13, 2018.

[26] Samuel Leroy, Audrey Steinberger, Cécile Cottin-Bizonne, Frédéric Restagno, Liliane Léger, and Élisabeth Charlaix. Hydrodynamic interaction between a spherical particle and an elastic surface: A gentle probe for soft thin films. *Phys. Rev. Lett.*, 108:264501, Jun 2012.

[27] Yumo Wang, Charles Dhong, and Joelle Frechette. Out-of-contact elastohydrodynamic deformation due to lubrication forces. *Phys. Rev. Lett.*, 115:248302, Dec 2015.

[28] Lionel Bureau, Gwennou Coupier, and Thomas Salez. Lift at low reynolds number. *The European Physical Journal E*, 46(11):111, 2023.

[29] Bhargav Rallabandi. Fluid-elastic interactions near contact at low reynolds number. *Annual Review of Fluid Mechanics*, 56(Volume 56, 2024):491–519, 2024.

[30] K Sekimoto and L Leibler. A mechanism for shear thickening of polymer-bearing surfaces: elasto-hydrodynamic coupling. *Europhysics Letters*, 23(2):113, 1993.

[31] J. Beaucourt, T. Biben, and C. Misbah. Optimal lift force on vesicles near a compressible substrate. *Europhys. Lett.*, 67:676, 2004.

[32] Jan M. Skotheim and L. Mahadevan. Soft Lubrication. *Physical Review Letters*, 92(24):245509, June 2004.

[33] Javier Urzay, Stefan G. Llewellyn Smith, and Beverley J. Glover. The elastohydrodynamic force on a sphere near a soft wall. *Physics of Fluids*, 19(10):103106, October 2007.

[34] J. H. Snoeijer, J. Eggers, and C. H. Venner. Similarity theory of lubricated hertzian contacts. *Phys. Fluids*, 25(10):101705, 2013.

[35] Baudouin Saintyves, Theo Jules, Thomas Salez, and L. Mahadevan. Self-sustained lift and low friction via soft lubrication. *Proc. Nat. Acad. Sci.*, 113(21):5847–5849, 2016.

[36] Heather S. Davies, Delphine Debarre, Nouha El Amri, Claude Verdier, Ralf P. Richter, and Lionel Bureau. Elastohydrodynamic lift at a soft wall. *Phys. Rev. Lett.*, 120:198001, 2018.

[37] B. Rallabandi, N. Oppenheimer, M. Y. B. Zion, and H. A. Stone. Membrane-induced hydroelastic migration of a particle surfing its own wave. *Nature Phys.*, 14:1211, 2018.

[38] Pierre Vialar, Pascal Merzeau, Suzanne Giasson, and Carlos Drummond. Compliant surfaces under shear: Elastohydrodynamic lift force. *Langmuir*, 35(48):15605–15613, 2019.

[39] Zaicheng Zhang, Vincent Bertin, Muhammad Arshad, Elie Raphael, Thomas Salez, and Abdelhamid Maali. Direct measurement of the elastohydrodynamic lift force at the nanoscale. *Physical Review Letters*, 124(5):054502, 2020.

[40] Thomas Salez and L. Mahadevan. Elastohydrodynamics of a sliding, spinning and sedimenting cylinder near a soft wall. *Journal of Fluid Mechanics*, 779:181–196, September 2015.

[41] Vincent Bertin, Yacine Amarouchene, Elie Raphaël, and Thomas Salez. Soft-lubrication interactions between a rigid sphere and an elastic wall. *Journal of Fluid Mechanics*, 933:A23, 2022.

[42] Aditya Jha, Yacine Amarouchene, and Thomas Salez.

Capillary lubrication of a spherical particle near a fluid interface. *Journal of Fluid Mechanics*, 1001:A58, 2024.

[43] S Sheikh, B Lonetti, I Touche, A Mohammadi, Z Li, and Micheline Abbas. Brownian motion of soft particles near a fluctuating lipid bilayer. *The Journal of Chemical Physics*, 159(24), 2023.

[44] Nicolas Fares, Maxime Lavaud, Zaicheng Zhang, Aditya Jha, Yacine Amarouchene, and Thomas Salez. Observation of Brownian elastohydrodynamic forces acting on confined soft colloids. *Proceedings of the National Academy of Sciences of the USA*, 121:e2411956121, 2024.

[45] A. Bérut, A. Imparato, A. Petrosyan, and S. Ciliberto. Stationary and transient fluctuation theorems for effective heat fluxes between hydrodynamically coupled particles in optical traps. *Phys. Rev. Lett.*, 116:068301, Feb 2016.

[46] S. Dago, B. Besga, R. Mothe, D. Guéry-Odelin, E. Trizac, A. Petrosyan, L. Bellon, and S. Ciliberto. Engineered swift equilibration of brownian particles: consequences of hydrodynamic coupling. *SciPost Phys.*, 9:064, 2020.

[47] P. Reimann, C. Van den Broeck, H. Linke, P. Hänggi, J. M. Rubi, and A. Pérez-Madrid. Giant acceleration of free diffusion by use of tilted periodic potentials. *Phys. Rev. Lett.*, 87:010602, Jun 2001.

[48] S. Lee and D. G. Grier. Giant colloidal diffusivity on corrugated optical vortices. *Phys. Rev. Lett.*, 96:190601, May 2006.

[49] B. Lindner and I. M. Sokolov. Giant diffusion of underdamped particles in a biased periodic potential. *Phys. Rev. E*, 93:042106, Apr 2016.

[50] L. Bellando, M. Kleine, Y. Amarouchene, M. Perrin, and Y. Louyer. Giant diffusion of nanomechanical rotors in a tilted washboard potential. *Phys. Rev. Lett.*, 129:023602, Jul 2022.

[51] M. Hairer and G.A. Pavliotis. From ballistic to diffusive motion in periodic potentials. *The Journal of Statistical Physics*, 131:175, July 2008.

[52] G.A. Pavliotis and V. Voginannou. Diffusive transport in periodic potentials. *Fluctuation and Noise Letters*, 8:L155–L173, 2008.

[53] Juliette Lacherez, Maxime Lavaud, Yacine Amarouchene, David Dean, and Thomas Salez. See Supplementary Material at http://xxx for details on the analytical derivations and numerical simulations., 2025.

[54] V. Sposini, S. Nampoothiri, A. Chechkin, E. Orlandini, F. Seno, and F. Baldovin. Being heterogeneous is advantageous: Extreme brownian non-gaussian searches. *Physical Review Letters*, 132(11):117101, 2024.

[55] Shneior Lifson and Julius L. Jackson. On the Self-Diffusion of Ions in a Polyelectrolyte Solution. *The Journal of Chemical Physics*, 36(9):2410–2414, May 1962.

[56] C. Touya, C. Sire, and D.S. Dean. Dipole diffusion in a random electrical potential. *J. Phys. A: Math. Theor.*, 42:375001, 2009.

[57] Gisiro Maruyama. Continuous Markov processes and stochastic equations. *Rendiconti del Circolo Matematico di Palermo*, 4(1):48–90, January 1955.

[58] Stefan Behnel, Robert Bradshaw, Craig Citro, Lisandro Dalcin, Dag Sverre Seljebotn, and Kurt Smith. Cython: The best of both worlds. *Computing in Science Engineering*, 13(2):31–39, 2011.