Driven anisotropic diffusion at boundaries: noise rectification and particle sorting

Driven anisotropic diffusion at boundaries: noise rectification and particle sorting

Stefano Bo    Ralf Eichhorn Nordita, Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
July 15, 2019

We study the diffusive dynamics of a Brownian particle in proximity of a flat surface under non-equilibrium conditions, which are created by an anisotropic thermal environment with different temperatures being active along distinct spatial directions. By presenting the exact time-dependent solution of the Fokker-Planck equation for this problem, we demonstrate that the interplay between anisotropic diffusion and hard-core interaction with the plain wall rectifies the thermal fluctuations and induces directed particle transport parallel to the surface, without any deterministic forces being applied in that direction. Based on current micromanipulation technologies, we suggest a concrete experimental set-up to observe this novel noise-induced transport mechanism. We furthermore show that it is sensitive to particle characteristics, such that this set-up can be used for sorting particles of different size.


The ability to manipulate, monitor and fabricate microscopic systems has witnessed a dramatic increase in recent years, opening the way towards accurate analysis of physical processes on scales where thermal fluctuations are lead actors. A key finding of these developments is that such fluctuations are not necessarily a detrimental nuisance, but rather may provide novel, noise-induced mechanisms for controlling motion and performing specific tasks at the microscale. Well-known examples are ratchets rectifying fluctuations by means of (spatial or time-reversal) asymmetries buttiker87 (); landauer88 (); reimann02 (); reimann02a (); hanggi09 (), Brownian engines extracting work from fluctuations blickle12 (); martinez16_carnot (); martinez16 (), and microscopic Maxwell demons exploiting fluctuations to convert information into energy parrondo15 (); lutz15 ().

In the present Letter we suggest a new strategy for rectifying thermal noise into directed movement of a microscopic particle, which requires only three simple ingredients: an anisotropic thermal environment, a plain, unstructured surface (or “hard wall”), and a static force oriented towards the surface so that particle motion is constrained to its close vicinity. The anisotropic thermal environment induces anisotropic diffusion, even for a spherical particle. As we will show below, the interplay between this diffusive behavior and the constraining boundary couples the drift components of the particle motion perpendicular and parallel to the surface. As a consequence, the particle on average moves along the surface, even though there is no deterministic force component applied in that direction. The drift velocity of that directed movement is essentially controlled by the anisotropy of the bath. Unlike common noise-rectifying ratchets, this mechanism does therefore not require any state-dependent diffusion buttiker87 (); landauer88 (), spatially asymmetric periodic force field or ratchet-like topographical structure, or any time-asymmetric driving forces reimann02 (); reimann02a (); hanggi09 ().

From an experimental viewpoint, an anisotropic thermal environment may be realized, for instance, by two different temperature “baths” acting along different directions, or by a superposition of the usual fluid bath of the Brownian particle with a second, “hotter” source of (almost) white-noise fluctuations applied along a specific direction filliger07 (). In a number of recent experiments, the latter alternative has been implemented by means of noisy electrostatic fields gomez10 (); martinez13 (); mestres14 (); berut14 (); dieterich15 (); martinez15 (); martinez16_carnot (); berut16 (); martinez16 (); dinis16 (); soni17 (). This technique has been shown to provide effective heatings to extremely high temperatures, and it has been applied to implement a microscale heat engine martinez16_carnot (); martinez16 (); dinis16 (). It thus appears to be an ideal candidate for creating the anisotropic thermal environment needed for establishing non-equilibrium conditions in our system. We analyze such an experimental set-up and demonstrate how the noise rectification can be exploited to transport differently-sized colloidal beads in opposite directions along the surface. This setup thus provides an elegant method for efficient particle sorting without feedback control, which can be implemented at low cost with state-of-the-art microfluidics technology martinez16_carnot ().

Stochastic particle motion on the microscale is commonly modeled in terms of an overdamped Langevin equation (neglecting inertia effects) mazo02 (); snook07 (). The presence of a reflecting wall requires additional care to correctly account for the hard-core interactions with these boundaries, as they cannot directly be introduced into the equation of motion as well-defined interaction forces behringer11 (); behringer12 (). However, in the equivalent representation of driven diffusion in terms of the Fokker-Planck equation risken84 (), governing the time evolution of the probability density of particle positions, hard walls are easily implemented as reflecting boundary conditions. In one dimension, such Fokker-Planck equation with a reflecting boundary (i.e. a “hard wall” constraining particle diffusion to the one-dimensional half-space) has been solved analytically by Smoluchowski smoluchowski17 (); chandrasekhar43 (), even in the presence of a constant external force. He obtained an expression for the time-dependent probability density in terms of exponential and error functions (see Eq. (5) below).

For isotropic thermal baths, Smoluchowski’s result is immediately generalized to particle diffusion close to a reflecting surface in three (and also higher) dimensions, because the different spatial directions are uncorrelated, so that the three-dimensional Fokker-Planck equation decouples into a set of three one-dimensional equations. However, for anisotropic diffusion with principal directions not being aligned with the surface, correlations between the spatial components perpendicular and parallel to the surface prohibit such a simple decomposition. To the best of our knowledge, the analytical solution for this situation—anisotropic particle diffusion driven by a constant force in the proximity of a hard surface—is not known in the literature. We here present the exact time-dependent analytical solution of the corresponding Fokker-Planck equation, and reveal that it is intimately connected to the original Smoluchowski solution for the one-dimensional case. The above mentioned theoretical predictions of noise-induced systematic particle motion along the surface are derived from the properties of this solution.


We model the particle’s diffusive motion in three dimensions using the Fokker-Planck equation risken84 () for the probability density of finding the particle at a given position at time ,


where summation over repeated indices is understood. The constant drift velocity is imposed by an externally applied constant force, and the constant symmetric diffusion tensor with components () characterizes the anisotropic thermal bath and, possibly, anisotropic particle properties. We consider the case of an impenetrable surface (reflecting boundary) being located at , such that measures the height above the surface note:a (). The presence of such hard wall implies a no-flux boundary condition for the 3-component of the probability flux at ,


General results.

We are looking for a solution of (1), (2) for a delta-distributed initial density , where we require that . Without the no-flux boundary condition (2), the “free” (i.e. unconstrained) solution note:t () is a trivariate Gaussian with mean and covariance . We can split off the component by rewriting it as . The conditional density is a two-dimensional Gaussian with mean


for the components , and covariance matrix proportional to the Schur complement of in ,

The part represents free diffusion (no boundaries) in one dimension with drift and diffusion coefficient (and, accordingly, is normalized over ).

In the SM SM () we show that the exact time-dependent solution of (1) on the half-space with the reflecting boundary condition (2) retains the (conditional) free diffusion in the and components , while the unconstrained component is replaced by the solution for one-dimensional diffusion on the half-line with a reflecting boundary at smoluchowski17 (); chandrasekhar43 (),


The explicit form of has been derived by Smoluchowski smoluchowski17 (). It reads


where is the same expression for free diffusion as before, but now applied only to the half-line , and where the additional terms account for the “collisions” of the particle with the wall behringer11 (); behringer12 () ( denotes the complementary error function).

First and second moments.

Exploiting the factorized form of the solution in (4), with being a Gaussian, it is possible to directly compute all the moments of the particle displacement as a function of time. Performing the Gaussian integrals over and and using (3) we obtain for the first moments

and for the second moments

with being 1 or 2. The remaining integrals over involve combinations of error functions, Gaussians and polynomials (see (5)), but still can be performed analytically. The resulting, explicit time-dependent expressions for the moments and are rather lengthy and are given in the SM SM ().

Yet, when the drift is pointing towards the wall, , the one-dimensional motion in direction reaches a stationary state for large times given by


with stationary mean and variance . Recalling that (for ), the long-time limits of the moments (6a) and (6b) then assume the compact form (with being equal to 1 or 2)


and define effective long-term velocities , and effective diffusion coefficients along the surface.

The net average particle velocity (8a) is the most striking consequence of our solution: the long-term average displacements parallel to the surface in the directions , are not only driven by the drift velocities , , but also by the perpendicular component in combination with the elements , , respectively, of the diffusion tensor. The origin of this motion can be understood as follows. The drift pushes the particle towards the surface and confines its motion to the close proximity of the plain wall in an exponential height distribution (see (7)). Being forced towards the surface, the diffusing particle experiences frequent “collisions” with this hard-wall boundary along a preferential direction which is determined by the anisotropic (but unbiased) thermal environment. In effect, the anisotropic thermal fluctuations become rectified, and induce directed particle motion. In absence of any drift components parallel to the surface, , the particle will therefore move over the surface at a speed and direction that are determined by its diffusion tensor. This effect can even induce particle migration against drift forces applied parallel to the surface.

Noise-rectifying particle motion without a systematic force being applied in the direction of motion is characteristic for ratchet systems reimann02 (); reimann02a (); hanggi09 (), and is usually generated by broken spatial or time-reversal symmetries in the applied (potential) forces. Here, no such asymmetric forces are present, but the overall spatial symmetry is broken by the principal axes of the anisotropic thermal bath not being aligned with the orientation of the surface. Indeed, as can be seen from (8a), the effect disappears if the principal axes of the bath are aligned with the surface ( diagonal) or if the bath is isotropic ( proportional to the identity). On the other hand, it might seem from (8a) that we can expect noise-rectification to occur even for isotropic thermal baths if the particle itself is anisotropic or if there are anisotropies in the viscous properties of the environment (like, e.g., in the intracellular medium), because then is generally non-diagonal. In fact, both cases, a non-spherical, anisotropic particle as well as anisotropic viscosity, are characterized by a friction tensor , resulting in a (generally non-diagonal) diffusion tensor . Anisotropic friction furthermore couples the various components of the external constant force resulting in the drift velocity . The appearance of in both, and , makes the terms proportional to in (8a) drop out (see SM ()), such that systematic long-term drift along the surface can only be induced by the force components , parallel to the surface. In other words, noise-rectification does not occur if the thermal environment is isotropic and anisotropic diffusion is only due to anisotropic particle properties or anisotropic viscosity; a finding reminiscent of the no-go theorem for ratchet systems with non-constant friction (see Sec. 6.4.1 in reimann02 ()), and consistent with the fact that an isotropic thermal environment corresponds to an equilibrium heat bath.

Experimental proposal.

In order to illustrate our main result (8a), we consider the experimentally realistic situation martinez13 (); mestres14 (); berut14 (); berut16 () of a colloidal particle in an aqueous solution at room temperature , which is “heated” anisotropically by randomly fluctuating forces applied along the direction . For the sake of generality, we formally keep the tensor properties of the particle friction when setting up the model in the following, even though we will exclusively consider spherical particles in the explicit examples below. The particle is pushed towards the plane surface at by a constant external force .

Using the model for the anisotropic heat bath, put forward and verified in martinez13 (), the equation of (unconstrained) motion for the Brownian particle can be written as


where collects three unbiased, mutually independent Gaussian white noise sources (with ) which represent the thermal fluctuations, and where is defined via , exploiting that is a positive definite tensor. Finally, denotes the amplitude of the anisotropic fluctuations . Note that in (9) dissipation effects connected to these fluctuations are assumed to be negligibly small. It has been demonstrated martinez13 () that can in very good approximation be represented as an unbiased delta-correlated white noise, . In that case, the isotropic thermal noise and the “synthetic” directional noise can be combined into an effective anisotropic thermal environment with different effective temperatures acting along different directions martinez13 (); martinez16 (); dinis16 () (for details, see the SM SM ()). Doing so, the equation of motion (9) turns into the equivalent form


with again being unbiased Gaussian white noise sources, and with


and .

For describing the motion of the Brownian particle close to the plain surface at , the Langevin equation (10) (or (9)) does actually not provide a complete model, because the hard-core interactions with the reflecting boundary are not specified, and can in fact not be included as well-defined interaction forces behringer11 (); behringer12 (). We can, however, make use of our exact solution of the associated Fokker-Planck equation presented above. In particular, if the deterministic drift is oriented towards the boundary, the particle’s long-term behavior is determined by (8) with the velocity field and the diffusion tensor given in (11). Specifically, we consider a spherical particle with a friction tensor proportional to the identity , . Moreover, we tilt the direction of the “synthetic” fluctuations by an angle with respect to the surface, i.e. we have , by convenient orientation of the and axes such that is the angle between the axis and (see Fig. 1a). For this set-up, the explicit expressions for the long-term drift velocities (8a) read


where we have introduced the standard definition for the “hot” kinetic (or effective) temperature martinez13 (); martinez16 (); dinis16 () of a spherical particle with radius and Stokes friction coefficient in an unbounded fluid. The “hot” temperature together with the direction characterize the anisotropy of the environment.

From the results (12) we can make a number of interesting observations (see also the SM SM (), where we provide plots of (12b)):

(i) The average drift velocity in direction has the trivial form , because the “synthetic” noise does not have an component and thus diffusion in - planes is isotropic and can not be rectified.

(ii) In an isotropic (equilibrium) thermal bath, when (implying ), net drift along the surface is only present if there are non-vanishing deterministic force components parallel to the surface, i.e.  or , as already inferred above on more general grounds.

(iii) Likewise, if the “synthetic” noise is applied in a direction parallel or perpendicular to the surface (i.e.  or ), an average drift velocity over the surface can be induced only by or .

(iv) The noise rectification effect, which is quantified in (12b) by the -term, depends on the friction coefficient . It is thus sensitive to particle shape and size, being stronger for particles with smaller . For appropriate choices of the “synthetic” noise parameters and , such that the two terms and in (12b) have the same sign, the rectification effect acts even opposite to the deterministic force component . Hence, the long-term velocity can change direction when varying particle size (and thus the friction coefficient ), with the smaller particles moving against the force (see Fig. 1c). In other words, we can always find a combination of and , which makes two different particle species with different move into opposite directions on the surface, such that they become separated with high efficiency (see Fig. 1b). Note that although the “synthetic” temperature of the experimental setup used in martinez13 (); martinez16 (); dinis16 () can be made extremely high, such large temperatures do not necessarily increase the sorting efficiency. Indeed, for too large , the term in (12b) becomes negligible such that the sensitivity of for particle properties gets lost. We furthermore remark that our theory does not take into account particle-particle interactions and thus describes the dilute limit of the sorting problem. Moreover, in practice, care has to be taken by appropriate choices of materials and coatings that the particles do not stick to the surface.

Figure 1: (a) and (a) Illustration of the setup. A Brownian colloid diffuses in proximity to a two-dimensional surface, the - plane at (illustrated by the gray areas). Its average, long-term velocity is indicated by the blue arrow. The “synthetic” noise is applied along the direction in the - plane (red bar). The external force points towards the surface (gray arrow). In (a) the full three-dimensional setup is sketched, (a) shows the - plane in which the “synthetic” noise is applied. (b) Two different spherical particles move along opposite directions on the surface; solid orange curve: Brownian sphere with radius ; dashed green curve: Brownian sphere with radius . Shown are the probability densities for the two particles, obtained from the exact solution (4) at time . (c) Net particle velocity , (12b), as a function of the particle radius . The two particle sizes from (b) are marked by corresponding solid orange and dashed green lines. Parameters used in (b) and (c): (room temperature), (“synthetic” kinetic temperature martinez13 (), specified for a reference spherical particle with radius and Stokes friction coefficient ), , . The friction coefficients of the spherical particles have been calculated—as an approximation—for an unbounded fluid, i. e. , where is the viscosity of water at room temperature.

Concluding remarks.

The main results of this Letter are the exact, time-dependent solution of the Fokker-Planck equation (1) with the no-flux boundary condition (2) note:gen (), and its implications for particle diffusion close to a plain surface. Most notably, we find (see (8a)) that an anisotropic thermal environment induces directed particle motion along the boundary even if no systematic forces are applied in this direction. To illustrate these results we analyze the average motion of a Brownian colloid close to a plain surface. The anisotropic thermal environment is created by superimposing externally applied, (almost white) random fluctuations to the thermal fluid bath, using a technique that has been established experimentally in recent years gomez10 (); martinez13 (); mestres14 (); berut14 (); dieterich15 (); martinez15 (); martinez16_carnot (); berut16 (); martinez16 (); dinis16 (). In modeling this system with the Fokker-Planck equation (1) and when applying our analytic solution, we tacitly assume that the friction coefficient of the Brownian particle is constant, i.e. independent of particle position. This idealization does not take into account the changes of viscous friction with the distance from the surface due to hydrodynamic interactions happel83 (). Close to the surface hydrodynamic friction becomes very large and will slow down the movements of the particle. Since the particle sorting mechanism we suggest here occurs in the vicinity of the surface, we therefore expect the sorting efficiency to decrease when properly taking into account hydrodynamic effects. Numerical simulations (see SM SM ()) confirm these expectations, but also show that our main qualitative finding—systematic noise-induced transport along the surface in a direction which depends on particle properties—seems to be robust. A detailed analysis of hydrodynamic effects will be subject of future work.



We thank Hans Behringer for many stimulating discussions, RE acknowledges financial support from the Swedish Science Council (Vetenskapsrådet) under the grants 621-2012-2982, 621-2013-3956 and 638-2013-9243.


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