Rectification and non-Gaussian diffusion in heterogeneous media

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.


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 [1]. 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, f 0 , such as a gravitational [2], optical [3] or entropic [4] force, the ratio is given by: where p(∆x, t|x 0 , t 0 ) is the probability of measuring a particle displacement of magnitude ∆x at time t, given the initial condition p(x 0 , t 0 ) and β −1 = k B T with k B the Boltzmann constant. Eq.(1) has been obtained using different theoretical frameworks [1,5] and for different observables such as entropy production rate [6] or mechanical work [7]. 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) [3,4], solution of the corresponding Smoluchowski equation, and of the potential nature of the force [8]. For a 1D dynamics, as is the case of Eq.(1), forces are always potential ensuring a Gaussian probability distribution function. For a 3D 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 [8]. 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 [2][3][4] [35]. 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.

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 x-direction, as provided e.g. by gravity [2], an optical trap [3] or an entropic force [4]. The particle is initially at position x = 0. The homogeneous nature of the medium leads to a constant diffusion coefficient, D 0 . 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 p( The corresponding ratio between positive, ∆x, and negative, −∆x, particle displacements reads The quantity f 0 ∆x represents the work done on the particle by the force.
Let us now consider that particles move under the action of an x-dependent force, is a potential periodic contribution of period L and zero average. A similar form is assumed for the diffusion coefficient: is periodic of period L and it is vanishing small once averaged over L. 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 p(x 0 , 0) is the probability distribution for t = 0. In the case of constant force and diffusion coefficient and for the initial condition p(x 0 , 0) = δ(x 0 ), this expression reduces to Γ = Γ 0 = e βf0∆x , i.e. to Eq. (4). An estimate of the changes in Γ due to deviations from Gaussianity can be obtained from Since analytical solutions of the Smoluchowski equation for x-dependent forcing and/or diffusion coefficient are in general difficult to obtain, we will assume that where p 0 (x, t|x 0 , 0) is given by Eq.(3) and p 1 is a perturbation. Accordingly, Substituting Eq.(9) in Eq. (6), and expanding up to the first order in ρ 1 we get therefore χ reduces to Symmetry enforces that [36], χ (t) = 1/Λ Λ/2 −Λ/2 χ(x, ∆x)d∆x = 1 from Eq. (11). It is useful to consider its averaged second moment where Λ is the subset over which Ω(t) and χ (t) are computed [37]. Ω quantifies the deviations of Γ from the homogeneous case for which Eq. (4) holds. For homogeneous systems, for which f 1 = D 1 = 0, one has Ω = 0, then Γ = e βf0∆x , i.e. it recovers the expression in Eq. (4). For both small local forcing, f 1 (x) ≪ f 0 , and small modulations of the diffusion coefficient, D 1 (x) ≪ D 0 , we can compute Ω(t) by using the expressions ρ 0 (−∆x, t) = ρ 0 (∆x, t)e −βf0∆x and ρ 1 (−∆x, t) = ρ 1 (∆x, t)e −β∆G . Expanding Eq. (12) to first order in |βf 0 ∆x − β∆G| ≪ 1, one obtains where a(x) = 1 Λ Λ/2 −Λ/2 a(x)dx and the time-dependent coefficients A(t), B(t), C(t), E(t) and F (t) are integration constants whose explicit forms are given in the Appendix.
Ω depends in general on the second moment of the force, f 2 1 , and on D 2 1 . Hence, to lowest order in both quantities, different physical mechanisms leading to comparable modulations may lead to similar values of Ω. As shown in the Appendix, the coefficients of the second moment of the forcing, A(t), and of the diffusion coefficient, B(t), are positive, while the cross terms like C(t), E(t) and F (t) can be positive or negative. Therefore, in the presence of both modulations, the deviations from the Gaussian behavior, that modulate the magnitude of Ω, can either increase or decrease. Ω also depends implicitly on the average force, f 0 , through the time-dependent coefficients.

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 [9][10][11][12][13]. As a second case, we will analyze the motion of a Brownian particle moving in a confined medium which induces x-dependent entropic forces [14][15][16][17][18][19][20][21][22][23]. Such a situation is typically observed in molecules moving through ion-channels or membrane pores [24][25][26][27][28][29] and for molecular motors in porous media [30][31][32] just to mention a few among others.

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 p(x, 0) = δ(x) and over a channel made by 10 identical units each of which is periodic with period L where we have assumed periodic boundary conditions at the channel ends, located at x = ±5L. To avoid the interference of periodic images we have followed the evolution of the particle displacement probability up to a maximum time T max defined as the time at which the ratio θ = p(±5L, T max )/p(0, T max ) 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. θ ≤ 10 −10 . For t < T max the contribution to p(x) from particles at x ± 5L is negligible. Fig.(1.A) shows the dependence of χ on ∆x for different values of D 1 . For D 1 = 0, Γ reduces to Eq.(4) and we recover the expected relation χ = 1. Increasing D 1 leads to a non-Gaussian density distribution [33] and χ = 1, as shown in Fig.(1.A). The overall departure from Gaussianity is captured better by Ω. As shown in Fig.(1.C) when increasing the diffusion coefficient modulation, Ω increases and behaves as Ω ∝ (D 1 /D 0 ) 4 in good agreement with Eq.(13). Fig.(1.A) shows a breakdown of the left-right symmetry superimposed to a smoother modulation of χ. Indeed, we can regard the system as being driven by an effective force f ef f = f 0 + δf where δf is a bias. Hence, δf /f 0 can be regarded as a dimensionless parameter that quantifies particle rectification arising from the interplay between the net force, f 0 , and the x-dependent diffusion coefficient [38].
The contribution of δf to χ is given byχ that accounts for the deviations from Gaussianity for a system under an effective force. The second moment ofχ quantifies both the overall departure from Gaussianity and also provides a route to obtain δf . The value of δf that better captures the breakdown of left-right symmetry in Fig.(1.A) can be obtained by minimizingΩ. Hence minimizing Eq. (17) leads to the following expression for δf : Interestingly, Eq.(18) predicts δf = 0 for χ− χ = 0, implying Ω = 0. 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 D 1 and f 1 we can approximate χ(∆x, t) ≃ χ (t) + ξ∆x, implying Ω ∝ ξ 2 . In the limit of ξ → 0, Eq.(18) reduces to Comparing the dependence of δf and Ω on D 1 , we notice that δf ∝ √ Ω, 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 Ω is shown in Fig.(1.E). At short times, Ω displays a remarkable dependence on time and reaches a plateau at longer times, for t ≥ t 0 ≃ 0.1L 2 /D 0 . Since L < Λ, Ω relaxes to its steady value faster than particle diffusion over the relevant length scale, Λ. Finally, Fig.(1.F) shows the dependence of Ω on the external constant force, f 0 , obtaining a quadratic dependence Ω ∝ f 2 0 and consequently a linear dependence of δf on f 0 (data not shown).

Diffusion in a periodic channel
We consider the diffusion of a particle in a channel of periodic half-section where L is the period and L z the width along the z-direction assumed to be constant. In the overdamped regime, the evolution of the probability density function, P (x, y, z, t), of the particle under the action of a constant force, f 0 , is governed by the 3D Smoluchowski equation: ∂ ∂t P (x, y, z, t) = Dβ∇ · [P (x, y, z, t)∇U (x, y, z) + D∇P (x, y, z, t)] where the potential U (x, y, z) is given by and involves both the external driving, f 0 , and the presence of boundaries. For smoothly varying channel amplitudes, ∂ x h(x) ≪ 1, the diffusing particles equilibrates much faster in the transverse direction than in the main transport direction. One can then assume where p(x, t) is the probability distribution in the coarse-grained description and A(x) is the corresponding free energy This quantity consists of an enthalpic contribution, f 0 x, and an entropic contribution, . This approximation shows that diffusion in 3D can be analyzed through 1D diffusion in the presence of entropic barriers [14][15][16][17]34]. Accordingly, we can define the dimensionless energy barrier that the particles experience along the channel, where h min and h max are the minimum and maximum channel aperture, respectively. Integrating Eq.(21) along the channel transverse section, we obtain the Fick-Jacobs equation where is an effective diffusion coefficient, with alpha α = 1/3(1/2) in three(two) spatial dimensions [16]. Comparison of Eq.(26) with Eq. (5) shows that the geometrical confinement enters through the potential A(x). 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. We have solved numerically Eq.(26) with the same numerical scheme used in the previous section. Fig.(2.A) shows the behavior of χ. Analogously to the results reported in the previous section, χ is strongly affected by the local drift and diffusion coefficient modulation. Larger values of ∆S, i.e. larger modulations, lead to a more involved dependence of χ on ∆x, and consequently to a larger departure from the Gaussian solution obtained for ∆S = 0. Moreover, comparing Fig.(2.A) and Fig.(1.A), we notice that the qualitative and quantitative behaviors of χ differ for bounded and unbounded diffusion. While the latter case is characterized by a smoothly modulated overall extra-tilt for χ, in the former larger modulations are overimposed to a smoothly-varying tilt, even for entropy barriers as large as ∆S = 10.
The dependence of Ω on ∆S 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, Ω shows a weaker dependence on ∆S. Disentangling the underlying mechanisms responsible for this lack of sensitivity is not straightforward, because modulations of D 1 and f 1 due to variations of the channel section compete with each other, as becomes clear in Eq. (13). Nonetheless, for ∆S > 1, using Eq.(27), we have implying that D 2 1 vary very smoothly for larger ∆S. Thus, the dependence of Ω on ∆S enters essentially through the entropic force. One can assume that D 2 1 is practically constant to obtain: where ∆S 0 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 ∆S ≃ 10. The deviation from the behavior Ω ∝ (∆S − ∆S 0 ) 4 observed for smaller values of ∆S is due to the time-dependence of Ω. As shown in Fig.(2.E), Ω reaches a quasi-steady state after a transient that depends on ∆S. For increasing entropy barriers, ∆S, 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 [33]. Smaller values of ∆S require longer relaxation times that cannot be considered in our numerical solution.
Even though the dependence of χ on ∆x 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 δf . It results that δf ∝ (∆S − ∆S 0 ) 4 as shown in Fig.(2.D). Since Ω ∝ (∆S − ∆S 0 ) 4 , we predict δf ∝ Ω, that differs from the behavior observed in the previous case in which variations of χ led to δf ∝ √ Ω. 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 Ω on the external force and shows that for decreasing forces the deviation in Ω become vanishing small recovering the equilibrium value Ω = 0, for βf 0 L = 0. Moreover, Ω ∼ f 2 0 (Fig.(2.F)), as also observed for a constant section channel (Fig.(1.F)). 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, Γ (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 Γ are insightful. The functional shape of the second moment Ω of Γ shows that the corrections to Γ, 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 Ω 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, δf , 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.