Rectification and non-Gaussian diffusion in heterogeneous media

Rectification and non-Gaussian diffusion in heterogeneous media

P. Malgaretti Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, D-70569 Stuttgart, Germany IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany Departament de Fisica de la Matèria Condensada, Facultat de Fisica, Universitat de Barcelona, carre Martí i Franques 1, Barcelona 08028, Spain    I. Pagonabarraga Departament de Fisica de la Matèria Condensada, Facultat de Fisica, Universitat de Barcelona, carre Martí i Franques 1, Barcelona 08028, Spain    J. M. Rubi Departament de Fisica de la Matèria Condensada, Facultat de Fisica, Universitat de Barcelona, carre Martí i Franques 1, Barcelona 08028, Spain

We show that when Brownian motion takes place in a heterogeneous medium, the presence of local forces and transport coefficients leads to deviations from a Gaussian probability distribution that make that the ratio between forward and backward probabilities depends on the nature of the host medium, on local forces and also on time. We have applied our results to two situations: diffusion in a disordered medium and diffusion in a confined system. For such scenarios we have shown that our theoretical predictions are in very good agreement with numerical results. Moreover we have shown that the deviations from the Gaussian solution lead to the onset of rectification. Our predictions could be used to detect the presence of local forces and to characterize the intrinsic short-scale properties of the host medium, a problem of current interest in the study of micro and nano-systems.

I Introduction

The symmetry of the probability distribution of a system in equilibrium, expressed through the detailed balance condition breaks down when a driving force is applied Campisi et al. (2011). The ratio of probabilities between forward and backward particle displacements is in this case independent of time, equal to a Boltzmann factor. For a Brownian particle under a constant conservative force, , such as a gravitational Astumian (2007a), optical Astumian (2007b) or entropic Reguera and Rubi (2010) force, the ratio is given by:


where is the probability of measuring a particle displacement of magnitude at time , given the initial condition and with the Boltzmann constant. Eq.(1) has been obtained using different theoretical frameworks Campisi et al. (2011); Ciliberto et al. (2010) and for different observables such as entropy production rate Gallavotti and Cohen (1995) or mechanical work Crooks (1999).

The peculiar form of the ratio between probabilities given by Eq.(1) is a consequence of the Gaussian nature of the probability distribution function (pdf) Astumian (2007b); Reguera and Rubi (2010), solution of the corresponding Smoluchowski equation, and of the potential nature of the force Vainstein and Rubi (2007). For a dynamics, as is the case of Eq.(1), forces are always potential ensuring a Gaussian probability distribution function. For a dynamics, as in the case of Brownian motion in a shear flow, the ratio between probabilities depends on time due to the fact that the shear flow is not potential Vainstein and Rubi (2007). In a variety of situations, such as for particles diffusing in porous media or displacing through ion channels or membrane pores, the assumption of a constant force and/or transport coefficients is not justified. For these local transport scenarios Eq.(1) cannot be applied.

In this work, we will characterize the dynamics of Brownian particles when they displace in an heterogeneous environment in which transport coefficients and forces may depend on position. The local nature of these quantities leads to a non-Gaussian pdf for particle displacement due to a coupling between particle advection and diffusion. This behavior has not been reported in previously studied systems, where particles diffuse in a homogeneous medium and are subjected to uniform forces Astumian (2007a, b); Reguera and Rubi (2010) 111A case of nonuniform forces where an analytical solution of the associated Smoluchowski equation exists is the Ornstein-Uhlenbeck process. However, in this case the probability distribution converges rapidly to a Gaussian due to the confining nature of the potential Risken (1988).. Hence, for local transport, the time evolution of the particle pdf cannot be regarded as a diffusion process with respect to a moving mean value, rather convection and diffusion affect each other non-trivially leading to the appearance of new dynamical regimes. In particular our results highlight the presence of a rectification regime in which particle transport benefits from the heterogeneity of the medium.

The article is organized as follows. In Section 2, we derive the equivalent of Eq. (1) for local transport, in which the diffusion coefficient and the driving forces may depend on position, and derive an analytic perturbative expression that measures the deviation from the standard fluctuation relations. In Sections 3 and 4, we present the cases of transport in an inhomogeneous medium and in a confined system. Finally, in the last Section, we present our main conclusions.

Ii Diffusion in heterogeneous systems

To show how Eq.(1) is obtained, let us consider a particle moving in a homogeneous medium subjected to a constant force acting on the -direction, as provided e.g. by gravity Astumian (2007a), an optical trap Astumian (2007b) or an entropic force Reguera and Rubi (2010). The particle is initially at position . The homogeneous nature of the medium leads to a constant diffusion coefficient, . Therefore, it is enough to analyze the particle displacement distribution in the direction of the applied force. In the overdamped regime, the particle dynamics is governed by the Smoluchowski equation


whose conditional solution, given the initial condition , reads


The corresponding ratio between positive, , and negative, , particle displacements reads


The quantity represents the work done on the particle by the force.

Let us now consider that particles move under the action of an -dependent force, , where is a potential periodic contribution of period and zero average. A similar form is assumed for the diffusion coefficient: where is periodic of period and it is vanishing small once averaged over . The corresponding Smoluchowski equation is given by:


In this case, the solution of Eq.(5) is not Gaussian and Eq. (4) is no longer fulfilled.

To analyze the symmetry of the probability distribution function, we extend Eq. (4) by introducing


where is the probability distribution for . In the case of constant force and diffusion coefficient and for the initial condition , this expression reduces to , i.e. to Eq. (4). An estimate of the changes in due to deviations from Gaussianity can be obtained from


Since analytical solutions of the Smoluchowski equation for -dependent forcing and/or diffusion coefficient are in general difficult to obtain, we will assume that


where is given by Eq.(3) and is a perturbation. Accordingly,


Substituting Eq.(9) in Eq.(6), and expanding up to the first order in we get


therefore reduces to


Symmetry enforces that222We note that , from Eq.(11). It is useful to consider its averaged second moment


where is the subset over which and are computed333The choice of does not affect significantly the value of and and their dependence on becomes vanishing small at long time intervals.. quantifies the deviations of from the homogeneous case for which Eq. (4) holds. For homogeneous systems, for which , one has , then , i.e. it recovers the expression in Eq. (4).

For both small local forcing, , and small modulations of the diffusion coefficient, , we can compute by using the expressions and . Expanding Eq.(12) to first order in , one obtains


where and the time-dependent coefficients , , , and are integration constants whose explicit forms are given in the Appendix.

depends in general on the second moment of the force, , and on . Hence, to lowest order in both quantities, different physical mechanisms leading to comparable modulations may lead to similar values of . As shown in the Appendix, the coefficients of the second moment of the forcing, , and of the diffusion coefficient, , are positive, while the cross terms like , and can be positive or negative. Therefore, in the presence of both modulations, the deviations from the Gaussian behavior, that modulate the magnitude of , can either increase or decrease. also depends implicitly on the average force, , through the time-dependent coefficients.

Iii Results

In order to study the accuracy of the perturbative expression Eq. (13), we will consider two scenarios where different physical mechanisms lead to a local force and diffusion coefficient. In the first example, we will study the diffusion of particles in an inhomogeneous medium under the influence of a constant force. This case is frequently observed in colloidal suspensions in which particles interact through direct or hydrodynamic interactions and in diffusion in complex systems Pagonabarraga (1994); Höfling and Franosch (2013); Maes and Steffenoni (2015); Bénichou et al. (2015); Marini Bettolo Marconi et al. (2015). As a second case, we will analyze the motion of a Brownian particle moving in a confined medium which induces -dependent entropic forces Jacobs (1967); Zwanzig (1992); Reguera and Rubi (2001); Kalinay and Percus (2008, 2010); Dagdug et al. (2011); Malgaretti et al. (2013a); Chacón-Acosta et al. (2013); Kalinay (2013, 2016). Such a situation is typically observed in molecules moving through ion-channels or membrane pores Chinappi et al. (2006); U. Marini Bettolo Marconi and Pagonabarraga (2013); Malgaretti et al. (2014a); Malgaretti et al. (2015, 2016); Bianco and Malgaretti (2016) and for molecular motors in porous media Malgaretti et al. (2012); Malgaretti et al. (2013b, 2014b) just to mention a few among others.

iii.1 Diffusion in an inhomogeneous unbounded medium

We consider the motion of a Brownian particle moving under the action of a constant force in a medium characterized by a spatially varying diffusion coefficient


The corresponding Smoluchowski equation reads


We have solved Eq.(15) numerically, by means of a Lax-Wendroff method, with initial condition and over a channel made by identical units each of which is periodic with period where we have assumed periodic boundary conditions at the channel ends, located at . To avoid the interference of periodic images we have followed the evolution of the particle displacement probability up to a maximum time defined as the time at which the ratio between the probability of particles at the system edges and the corresponding probability in the middle of the channel overcomes a threshold value, i.e. . For the contribution to from particles at is negligible.

Fig.(1.A) shows the dependence of on for different values of . For , reduces to Eq.(4) and we recover the expected relation . Increasing leads to a non-Gaussian density distribution Risken (1988) and , as shown in Fig.(1.A). The overall departure from Gaussianity is captured better by . As shown in Fig.(1.C) when increasing the diffusion coefficient modulation, increases and behaves as in good agreement with Eq.(13).

Figure 1: Fluctuation relation for a particle in a constant-amplitude channel and inhomogeneous medium, under a constant force . A: as function of the displacement, , at for different values of the diffusion coefficient modulation: for dotted (brown), dashed (red) and solid (orange) lines, respectively. B: as a function of for the same parameters as in panel A, where has been calculated using Eq.(18). C: as a function of for with and , dotted line stands for . D: as function of for the same parameters as in panel C, dotted line stands for . E: as a function of time normalized by with , for ; the darker the line, the larger the ratio . F: as a function of the driving force for .

Fig.(1.A) shows a breakdown of the left-right symmetry superimposed to a smoother modulation of . Indeed, we can regard the system as being driven by an effective force where is a bias. Hence, can be regarded as a dimensionless parameter that quantifies particle rectification arising from the interplay between the net force, , and the -dependent diffusion coefficient 444For no rectification occurs while for the sign of identifies the rectification direction..

The contribution of to is given by


that accounts for the deviations from Gaussianity for a system under an effective force. The second moment of


quantifies both the overall departure from Gaussianity and also provides a route to obtain . The value of that better captures the breakdown of left-right symmetry in Fig.(1.A) can be obtained by minimizing . Hence minimizing Eq. (17) leads to the following expression for :


Interestingly, Eq.(18) predicts for , implying . Therefore, in the present regime, no rectification occurs either at equilibrium or for systems leading to a Gaussian distribution of particle displacements. For vanishing values of and we can approximate , implying . In the limit of , Eq.(18) reduces to


Fig.(1.B) displays the dependence of on , showing the absence of any net tilt. Therefore, the linear approximation for given by Eq.(18) properly captures the departure of , and consequently of , with respect to their values obtained for homogeneous diffusion, . Fig.(1.D) shows the dependence of on the modulation in the diffusion. While for larger values of a steeper dependence is observed, for smaller modulations of the diffusion reaches an asymptotic behavior . Comparing the dependence of and on , we notice that , as predicted by Eq.(19). The regime of validity of Eq.(13) is captured in Fig.(1.C), where the good agreement with the numerical solution of Eq.(15) highlights the wide range of reliability of Eq.(13). The temporal evolution of is shown in Fig.(1.E). At short times, displays a remarkable dependence on time and reaches a plateau at longer times, for . Since , relaxes to its steady value faster than particle diffusion over the relevant length scale, . Finally, Fig.(1.F) shows the dependence of on the external constant force, , obtaining a quadratic dependence and consequently a linear dependence of on (data not shown).

iii.2 Diffusion in a periodic channel

We consider the diffusion of a particle in a channel of periodic half-section


where is the period and the width along the -direction assumed to be constant. In the overdamped regime, the evolution of the probability density function, , of the particle under the action of a constant force, , is governed by the Smoluchowski equation:


where the potential is given by

and involves both the external driving, , and the presence of boundaries. For smoothly varying channel amplitudes, , the diffusing particles equilibrates much faster in the transverse direction than in the main transport direction. One can then assume


where is the probability distribution in the coarse-grained description and is the corresponding free energy


This quantity consists of an enthalpic contribution, , and an entropic contribution, . This approximation shows that diffusion in can be analyzed through diffusion in the presence of entropic barriers Jacobs (1967); Zwanzig (1992); Reguera and Rubi (2001); Kalinay and Percus (2008); Martens et al. (2011). Accordingly, we can define the dimensionless energy barrier that the particles experience along the channel,


where and are the minimum and maximum channel aperture, respectively. Integrating Eq.(21) along the channel transverse section, we obtain the Fick-Jacobs equation




is an effective diffusion coefficient, with alpha in three(two) spatial dimensions Reguera and Rubi (2001). Comparison of Eq.(26) with Eq.(5) shows that the geometrical confinement enters through the potential . Its spatial derivative gives rise to an effective force; therefore, we can understand the impact of the channel corrugation as providing a spatially-varying force acting on the Brownian particle.

Figure 2: Fluctuation relation for a Brownian particle in a varying-amplitude channel, , that induces a modulation in the effective diffusion coefficient, for a constant force . A: as function of the displacement, , for , with and different values of the entropic barrier for dotted (brown), dashed (red) and solid (orange) line respectively. B: as function of for the same values of the parameters as in panel A. C: , at , as a function of for standing bigger points for larger , with . Inset: as a function of the reduce entropic barrier with , for the same value of the parameters in the main figure; dotted line stands for . D: as function of for the same value of the parameters in panel C. E: as a function of time normalized by for , standing darker lines for larger values of . F: as a function of the forcing for standing darker lines for higher values of , being .

We have solved numerically Eq.(26) with the same numerical scheme used in the previous section. Fig.(2.A) shows the behavior of . Analogously to the results reported in the previous section, is strongly affected by the local drift and diffusion coefficient modulation. Larger values of , i.e. larger modulations, lead to a more involved dependence of on , and consequently to a larger departure from the Gaussian solution obtained for . Moreover, comparing Fig.(2.A) and Fig.(1.A), we notice that the qualitative and quantitative behaviors of differ for bounded and unbounded diffusion. While the latter case is characterized by a smoothly modulated overall extra-tilt for , in the former larger modulations are overimposed to a smoothly-varying tilt, even for entropy barriers as large as .

The dependence of on is also modified with respect to the behavior observed for a constant channel section, as can be appreciated comparing Figs.(1.C) with Fig.(2.C). In the system analyzed, shows a weaker dependence on . Disentangling the underlying mechanisms responsible for this lack of sensitivity is not straightforward, because modulations of and due to variations of the channel section compete with each other, as becomes clear in Eq.(13). Nonetheless, for , using Eq.(27), we have


implying that vary very smoothly for larger . Thus, the dependence of on enters essentially through the entropic force. One can assume that is practically constant to obtain:


where accounts for the contribution coming from the modulation in the diffusion coefficient. The inset of Fig.(2.C) shows the good agreement of the theoretical prediction with the numerical results, up to . The deviation from the behavior observed for smaller values of is due to the time-dependence of . As shown in Fig.(2.E), reaches a quasi-steady state after a transient that depends on . For increasing entropy barriers, , the effective forces acting on a Brownian particle increase leading to a reduction of the relaxation time, as it happens for particles in a potential well Risken (1988). Smaller values of require longer relaxation times that cannot be considered in our numerical solution.

Even though the dependence of on is quite involved and does not show a clear breaking of the left-right symmetry, we have used Eq.(18) to compute the rectification parameter . It results that as shown in Fig.(2.D). Since , we predict , that differs from the behavior observed in the previous case in which variations of led to .

We can then conclude that different local transport mechanisms lead to different relationships between the rectification parameter and the deviations from Gaussianity inherent to Omega. Fig.(2.F) displays the dependence of on the external force and shows that for decreasing forces the deviation in become vanishing small recovering the equilibrium value , for . Moreover, (Fig.(2.F)), as also observed for a constant section channel (Fig.(1.F)).

Iv Discussion

We have shown that the diffusion of particles is strongly affected by heterogeneities resulting from irregularities of the boundaries or from the intrinsic nature of the host medium. The presence of local forces or of a local diffusion coefficient breaks down the Gaussian form of the probability distribution for the particles and leads to an effective rectification.

For small modulations of the spatial heterogeneities it is possible to analyze the consequences of a non-Gaussian probability distribution. We have found that the ratio between the probabilities of forward and backward moves depends on the heterogeneities of the medium and also on time. We have derived an expression for their ratio, (Eq.(6)), that is valid for small modulations both in the forcing and/or in the diffusion coefficient. In order to quantify the average deviation from Gaussian behavior, higher moments of are insightful. The functional shape of the second moment of shows that the corrections to , induced by local transport are proportional to the dispersion of the modulation. When both force and diffusion coefficient are modulated, Eq.(13) predicts that different regimes can be achieved depending on the constructive or destructive interaction between the two mechanisms.

To test our predictions, we have checked Eq.(13) for two different scenarios, namely a particle moving in an inhomogeneous medium with a position-dependent diffusion coefficient and a particle in a channel of varying cross section in the presence of entropic forces. In the first case, the force exerted is constant whereas the diffusion coefficient depends on position. In the latter case, both the geometrically-induced effective force and the local particle diffusion coefficient depend on particle’s position along the channel. In both situations we observe a remarkable agreement between the numerical results and our prediction for in the case of mild variations in the forcing and/or medium heterogeneities.

The coupling between local forcing and diffusion can also lead to particle rectification; our analysis predicts when rectification emerges and identifies an effective parameter, , that quantifies the effective rectification. In particular, our analysis reveals how the dependence of rectification on the departure from Gaussianity is affected by the physical mechanism responsible for local transport. These results suggest a possible way to characterize the intrinsic properties of the host medium and of the confinement based on the use of the new fluctuation relation and the tracking of the particles.


We acknowledge MINECO and DURSI for financial support under projects FIS 2015-67837-P and 2014SGR-922, respectively. J.M. Rubi and I. Pagonabarra acknowledges financial support from Generalitat de Catalunya under program Icrea Academia. P.M. thanks Marco Ribezzi for useful discussions.

Appendix A

Here we derive the expression for given by Eq.(13). Substituting Eq.(10) into Eq.(12) and remembering that when calculate by Eq.(11) we obtain:


Remembering that and assuming we obtain:


Expanding for we get:


where we have used the fact that due to the even character of in the limit and to the odd character of . Finally we expand both and as a power series of the local modulation of the forcing, , and/or diffusion coefficient, . Since both and are periodic with zero mean, see Eq.(14),(24) the first non vanishing contribution to and is provided their second moment:


When the modulations are vanishing small we have that and, remembering , we have . Using the last expressions we can expand and up to first order:


where , , and are to be determined numerically or by a path integral solution of Eq.(26). Substituting Eq..(35),(36) into Eq.(32) we get:


and finally we can define the coefficient that appears in Eq.(13) as:


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