numerical study of diffusion induced transport in 2d systems

7 February 2000

Ž .Physics Letters A 265 2000 337–345www.elsevier.nlrlocaterphysleta

Numerical study of diffusion induced transport in 2D systems

M. Kostur a,b, L. Schimansky-Geier a

a Institute of Physics, Humboldt-UniÕersity at Berlin, InÕalidenstr. 110, D-10115 Berlin, Germanyb Institute of Physics, Silesian UniÕersity, ul. Bankowa 14, 40-007 Katowice, Poland

Received 17 August 1999; received in revised form 9 December 1999; accepted 9 December 1999Communicated by C.R. Doering

Abstract

Ž .Transport in a two-dimensional 2D diffusive system with a ‘ratchet-like’ potential is studied. An additive constant forceis applied perpendicularly to the direction where the reflection symmetry of the potential is broken. It is demonstrated that aflow of Brownian particles orthogonally to the applied force is induced. Finite element method has been used as a tool forthe solution of the corresponding Smoluchowski-equation. The results have been compared to Monte Carlo simulations andshow good agreement. We also present the comparison of the 2D system with an 1D flashing ratchet. q 2000 ElsevierScience B.V. All rights reserved.

1. Introduction

Much theoretical work has been devoted to theproblem how a fluctuating force can induce a di-rected motion of Brownian particles in spatially peri-odic systems. Such devices have been called

w x‘Brownian ratchets’ or ‘Brownian motors’ 1–4 . Ithas been shown that in most cases 1 two ingredientsare indispensable to create the directed flow. Thefirst one is a broken symmetry in the system. Usu-ally, the potential is chosen to violate reflectionsymmetry within one spatial period. The secondingredient is the driving temporal force. In the non-trivial case the force vanishes in time-average but themotion of the Brownian particle is not equilibratedwith respect to the action of this force. Accordingly,

1 We consider non-interacting Brownian particles.

one has to assume an external force or nonequilib-rium fluctuations which drive the Brownian particles.

In such devices Brownian particles can have non-zero average velocities and macroscopic flows ap-pear. It was proven analytically and numerically invarious systems with different kinds of temporalforces, periodic and noisy, additive and multiplica-tive, with different potentials. A wide spectrum ofpossibilities to induce a directed motion at the meso-scopic level in non-equilibrium has been elaborated

w xin the near past 2,5–13 .However, most of these theoretical considerations

deal with one dimensional periodic structures. Hence,the time dependent forces acts in parallel to themotion of the induced flows. On the other hand, themajority of experiments on transport in sawtooth likepotentials concern with two-dimensional devices. Forexample, a directed motion of particles was observedin 2D-‘Christmas tree’-like structure of obstacles inw x14–16 . These 2D-obstacles have originated the

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00906-8

( )M. Kostur, L. Schimansky-GeierrPhysics Letters A 265 2000 337–345338

necessary break of the symmetry and non-equi-librium was obtained by switching on and off thepotential in time.

Recently, new ideas of a directed motion ofBrownian particles induced by a constant force havebeen developed. The motivation for constructing suchsystems comes from the need for an effective tech-

w xnique to separate macromolecules 17–20 . Generalaspects of this phenomenon have already been re-

w xported in 21 . It has been found that in 2D-systemsthe constant force in the presence of obstacles canexcite a transport orthogonally to the applied force.Moreover, if the shape of the obstacles is symmetricin the direction of the applied force the induced flux

w xis an even function of the driving force 21 . Thisfeature makes the phenomenon even more interesting

w xfor practical applications 20 .In this paper we present the detailed numerical

study of the continuous model of this 2D-ratchet. Wewill solve the corresponding 2D-Smoluchowski-equation and find stationary probability fluxes de-scribing the averaged flow of the Brownian particles.

ŽThe corresponding overdamped 1D-problem with.flow in parallel to the force was first solved in

connection with the phase locking effect of drivennonlinear oscillators. Stratonovich has given an ana-lytical stationary solution of the distribution and the

w xflux for the 1D-periodic problems 22 . The 1D-caseincluding effects of inertia can be found in the bookof Risken where numeric solutions of the corre-sponding Fokker–Planck-equation by matrix contin-

w xued fraction techniques are presented 23 .The symmetry of the 2D-sawtooth like potential is

broken in x direction. The additional constant biasacts in the y direction. Our main interest will befocused on the calculation of the induced stationarytotal flows in x-direction averaged over one period

Ž .in y direction see Sections 2 and 3 . But from amore general point of view we mention that theaction of a constant force combined with periodicboundary conditions gives one of the simplest situa-tions to drive a system out of equilibrium. Thetopology of the calculated local probability flows inx and y is of interest as a matter of principle.Conservation of probability on the 2D-plane allowsthe local probability flow to circulate, but without

w xsinks and sources 24 . It is similar to flows inconservative systems with centers, saddle points and

closed orbits, only. Most interestingly as will be seenin Section 4, in the case where the topology of theflow in x and y changes we will observe an unex-pected reversal of the total current in x.

The paper ends with a comparison of the consid-ered 2D-ratchet with the 1D-flashing ratchet. Indeedthere exist similarities between these both situations.The particles, driven by the constant force in ydirection return after one period to the ratchet poten-tial spanned in x direction. It remembers the flashing‘on’ and ‘off’ ratchet. But, as will be seen in Section4, the dynamics of the 2D-case is much richer andcannot be reduced to the 1D-ratchet.

2. Smoluchowski equation for periodic obstaclesin 2D

We consider a Brownian particle moving on a twoŽ .dimensional plane x, y . So far hatted values standˆ ˆ

for non-scaled variables. A periodic array of obsta-cles in x and y directions is assumed. These obsta-ˆ ˆcles should be originated by a continuous periodic

ˆ ˆ ˆŽ . Ž . Ž .potential V x, y s V x q L , y s V x, y q L .ˆ ˆ ˆ ˆ ˆ ˆ0 0 x 0 y

The potential is assumed to have a broken reflectionsymmetry in the xdirection, particularly a piecewiseˆlinear sawtooth-like shape has been chosen. On theother hand, the potential has reflection symmetry inthe y direction.ˆ

Moreover, a constant driving force acts on theparticle in the y direction. Therefore, the total forceˆ

ˆŽ .field is determined by the potential V x, y sˆ ˆˆ ˆŽ .V x, y yFy.ˆ ˆ ˆ0

It is convenient to start with the correspondingLangevin equations for motion of the Brownian par-ticle. We will consider the overdamped limit whichis given by the stochastic differential equations

ˆd x E V x , yŽ .ˆ ˆ ˆ ˆ ˆ'g sy q 2g kT j t ,Ž .xˆd t E x

ˆd y E V x , yŽ .ˆ ˆ ˆ ˆ ˆ'g sy q 2g kT j t , 1Ž . Ž .yˆd t E y

ˆ ˆand j and j are independent d-correlated Gauss-x y

ian white noises with zero average and the correla-ˆ ˆ X X² Ž . Ž .: Ž .ˆ ˆ ˆ ˆtion function j t j t sd ty t , isx, y.i i

( )M. Kostur, L. Schimansky-GeierrPhysics Letters A 265 2000 337–345 339

Ž .System 1 can be converted into dimensionlessform by

ˆts trt ,0

xsxrL ,ˆ x

ysyrL ,ˆ y

ˆ ˆV x , y sV x , y rDV . 2Ž . Ž .Ž .ˆ ˆHere L and L are the periods of the system in xx y

ˆand y directions, respectively, DV is the maximalelongation of the potential and t is a characteristic0

time in the system. Particularly, we took t to be the0

time a deterministic overdamped particle needs toˆmove under influence of the force DVrL over thex

distance Lx

g L2x

t s . 3Ž .0 ˆDVŽ .After simple algebra one obtains from 1

E V x , yŽ . ' w xxsy q 2 D j t , xg 0,1 ,Ž .˙ xE xE V x , yŽ . ' w xysy q 2 D j t , yg 0, L , 4Ž . Ž .˙ yE y

where the dimensionless diffusion coefficient readsˆDskTrDV and LsL rL stands for the aspecty x

ratio. j and j are again two uncorrelated Gaussianx y

white noise sources with intensity 1.The 2D-Smoluchowski equation for the evolution

Ž .of the probability density P x, y,t can be immedi-ately formulated in dimensionless variables

E P x , y ,t EŽ .sy f x , y P x , yŽ . Ž .xE t E x

Ey f x , y P x , yŽ . Ž .yE y

E 2

qD P x , yŽ .2E x

E 2

qD P x , y . 5Ž . Ž .2E y

Therein, f and f are the components of the dimen-x y

sionless force given by negative gradients of theŽ . Ž .V x, y . As a potential V x, y which fulfills our

Ž .requirements we use later on see Fig. 1 :

V x , y sV x , y yF yŽ . Ž .0

s 1yA sin 2p yrL U x yF y. 6� 4Ž . Ž . Ž .

Ž . Ž .Fig. 1. The 2D-potential V x, y sV x, y without external force0

Fs0. Periodicity is obeyed in both directions and reflectionsymmetry is broken in x-direction. Other parameters: As0.3,Ls1, and ks0.8.

Ž .U x is the one-dimensional piecewise linear ratchetpotential of unit height with the parameter of asym-

Ž . Ž .metry kg 0,1 , particularly if ks1r2 then U x isŽ .symmetric. U x is periodically modulated in y

direction with an amplitude A. The linear term standsfor the constant bias which drives the system out ofequilibrium.

Ž . Ž .The Smoluchowski Eq. 5 with the potential 6remains invariant under transformations:

x , y ™ xq1, y , x , y ™ x , yqL . 7Ž . Ž . Ž . Ž . Ž .Hence, if one chooses periodic initial conditionŽ . Ž .P x, y,ts0 , i.e. invariant under 7

P x , y ,0 sP xq1, y ,0 ,Ž . Ž .P x , y ,0 sP x , yqL,0 , 8Ž . Ž . Ž .and normalized in one periodic box

1 LP x , y ,0 dx dys1, 9Ž . Ž .H H

0 0

Ž .then the solution P x, y,t will be periodic andnormalized for arbitrary time. In particular, the sta-tionary state

P 0 x , y s lim P x , y ,tŽ . Ž .t™`

will also be periodic and normalized over one period.Due to above symmetries of the system we con-

Ž . w xsider the Smoluchowski Eq. 5 on the domain 0,1w x= 0, L with boundary conditions

P x , y ,t sP xq1, y ,t ,Ž . Ž .P x , y ,t sP x , yqL,t . 10Ž . Ž . Ž .

ŽUsing finite elements method see Appendix A theŽ . Ž .Eq. 5 with boundary conditions 10 can be solved

numerically. Because we investigate the stationary


transport in the long-time limit, the stationary solu-0Ž . Ž . 0Ž .tion P x, y of 5 is of special interest. P x, y

contains all relevant informations for determining thestationary flows in the system, namely

E0 0J x , y s f x , y P x , y yD P x , y ,Ž . Ž . Ž . Ž .x x x E x

E0 0J x , y s f x , y P x , y yD P x , y .Ž . Ž . Ž . Ž .y y y E y

11Ž .The total transport over one period in the x directioncan be quantified by determining the average veloc-

² :ity Õ . Taking into account the normalization con-xŽ .dition 9 the average velocity is expressed by the

total current via

1 Ltot² :Õ sJ s J x , y dx dy 12Ž . Ž .H Hx x x0 0

Ž .Insertion of 11 eventually yields

1 L 0² :Õ s f x , y P x , y dx dy , 13Ž . Ž . Ž .H Hx x0 0

Ž .since the second item in 11 does not contribute tothe total flow.

3. Numerical analysis of the induced current

When the bias F vanishes then the potentialŽ .V x, y is periodic and the solution is given by the

0Ž . yV Ž x , y.r DBoltzmann-distribution P x, y ,e . Obvi-Ž .ously, in this case the local flux vanishes, J x, y sx

Ž .J x, y s0. The addition of a tilt in y-directiony

drives the system out of equilibrium. No analyticalsolution is known in this case. Therefore, we havedecided to solve numerically the Smoluchowski Eq.Ž .5 in the stationary case. For this purpose we devel-oped a finite element solver based on a packageDiffpack 1.4. Details of the finite element method areincluded in the Appendix, here and in the nextsection we will discuss the results of the integration.

Non-zero tilt F/0 induces mean flows in thesystem. Apart from the expected flow in the ydirection, a current appears in x-direction which is at

Ž 2 . w xleast of the order O AF 21 . The probability fluxdensity is presented in Fig. 2. Directions of the flow

w x w xon the entire plane 0,1 = 0, L are drawn by arrowsof unit length and the background shows the surface

Fig. 2. Plot of directions of the probability currentJ x , y , J x , y for Ds0.2, As0.8, Ls1, Fs5 andŽ . Ž .Ž .x y

Ž .ks0.8. As the background surfaces of potential V x, y withouttilt F have been drawn.

of the potential V . One sees that the x-component0

of the local current adopts positive and negativevalues.

However, the main contribution to the total cur-rent comes from the flow through the saddle point of

Ž .the potential xs0.8 and ys0.75 . Therefore,globally a flux to the left side dominates. Fig. 3shows quantitatively the averaged velocity. Due to

Ž .the symmetry of V x, y in y-direction the flow0w x totŽ .does not change replacing F™yF 21 and J Fx

s0 at Fs0. Also the slope of the dependencetotŽ .J F also vanishes at Fs0. Increasing F inducesx

a negative flow and J tot reaches its maximal abso-x

lute value at F,5.5. It is near the critical value ofthe force, above which the potential monotonouslydecays in the y direction. Larger values of F sup-press again the effect.

In addition results of Monte Carlo simulations ofŽ .the Langevin Eq. 4 have been included in Fig. 3.

The good agreement of those two data sets comingfrom independent sources ensures the validity of theused algorithms. However, it should be underlinedthat the simulations are in general less precise andrequire much more computing power than the solu-tion of the Smoluchowski equation. Simulations hadto be run for a few hours on a 32 processor machineto get results with standard deviations shown aserrors bars in Fig. 3. In turn the numerical solution

Ž .of 5 required 20 minute on one single processor,


Fig. 3. The total flux in the system in x-direction versus drivingforce F. The error bars depict the results of Monte Carlo simula-

Žtions a size of the bar is a standard deviation of the result taken.from sample of 32 measurements .

what makes finite elements method about 100 timesfaster in this application. Nevertheless, the simula-tion method served as a valuable checking tool inour analysis.

The mentioned efficiency of the numerical methodhave allowed the investigation of the total flux in thefull parameters space. We have taken into account

Ž .three parameters L, D,F . They represent three vari-ous features of the system: the aspect ratio L modi-

Žfies its geometry the aspect ratio L extends or.shrinks the potential , D controls the relative tem-

perature and F is a measure of the external drivingforce. Because the change of L modifies the y-sizeof one period of the system, special care should betaken in order to compare fluxes for different de-vices. Since we define one obstacle to be one hill of

Ž . Ž .the potential V x, y or one period , the total fluxJ sÕ measures the average number of particlesx x

passing the right boundary per unit time from onerow of obstacles. In order to compare devices withdifferent L, the averaged current density should beconsidered. It is the ratio J totrL and gives the fluxx

of particles per unit of length in y. It would be justthe value of interest if one wanted to construct aseparation device using an array of microfabricated

w xobstacles 19,20 .In Fig. 4 the average flux density J totrL arex

Ž .shown as a function of L, D,F . One observes aglobal extremum for finite values of the parameters

Ž .Fig. 4. The flux density in parameter space D, L,F . The slicesŽ .through the space D,F for various L are shown. The global

minimum is for shrunk system at L,0.5. The optimal value oftemperature is always in the range of D,0.25. The optimal valueof the force changes, it is larger for small and big forces.


Ž .Fig. 5. The total flux density J F, L for fixed temperaturex

around its optimal value Ds0.208.

Ž .L, D,F . It is a crucial conclusion for practicalapplications. It means that there exists an optimalshape of the potential, optimal temperature and force,which maximize the effect. In other words, for a

Žgiven separation problem size and properties of.particles one can find a definite temperature, bias

force and shape of the obstacle to achieve maximalŽ .efficiency see Fig. 5 .

4. Comparison to flashing ratchets

The idea of the two dimensional ratchet system isan extension of the concept of the one dimensionalflashing ratchet. The action of the constant forcereplaces the periodic time dependent modulation ofthe potential. Particles moving in the y directionŽ .parallel to the force underly a periodic modulationsimilarly as a particle in periodically flashing 1Dpotential. This correspondence becomes exact if there

Žis no noise in y-direction starting from now wedistinguish between D and D being the intensitiesx y

of the noise sources in the dynamics for x and y,˙ ˙2 .respectively . In addition, the force F must be

2 There is also another reason to investigate the case of anon-symmetric diffusion constant. One can imagine that in aphysical system fast fluctuating fields are applied in one direction.Then, provided that the correlation time of this perturbation issufficiently small, such fields could independently contribute tothe effective diffusion constant in the chosen direction.

sufficiently strong that contributions to the force f yŽ .arising from the potential V x, y can be neglected.0

Ž .In this approximation, the second Langevin Eq. 4 issimply replaced by ysF.˙

We compare three situations: the 1D flashingŽratchet, the 2D ratchet with isotropic diffusion Dx

. ŽsD and with anisotropic diffusion D 4D sy x y.0.01 . This comparison is shown in Fig. 6. First, we

Ž .observe that for forces larger than a threshold F,5the value of the current of all three ratchets becomessimilar. The particles in both 2D cases move with

Žconstant mean velocities in the y direction when Fis of order of 10 then both D s0.3 and D s0.01y y

.are negligible and realize a periodically flashingsituation in the x direction.

However, for small F the behavior of the 2D-sys-tems differs from the flashing ratchet. The 2D ratchetwith isotropic diffusion gives a smaller current thanthe flashing one. It is due to the fact the motion ofthe particle in y direction is significantly disturbedby forces f coming from the potential V . They 0

regime with a constant mean velocity in the y direc-tion is left. The motion in y direction is hindered bythe high values of the potential and particles canonly move forwardly in y escaping barriers by ther-mal activation.

In the anisotropic case the behavior is even morecomplex. The motion in the y direction is limited for

ŽFig. 6. The comparison of the 1D flashing ratchet a numerical.simulation with 2D steric ratchet with symmetric and non-sym-

metric diffusion constant. The remaining parameters are D s0.3,x

ks0.8 and As0.8. The curves behave for forces F)5 simi-larly, however for small driving the discrepancy is dramatic.


small forces since particles are further on not ther-mally activated due to small D . An unexpectedy

phenomenon takes place: for small values of F thetotal current becomes positive. This effect is qualita-tively different from the behavior of the 1D flashingratchet.

In this situation the local probability current indi-cates a complicated dynamics of the Brownian parti-

Ž .cles see Fig. 7 . The current reversal is accompaniedwith the occurrence of circulating flows. The vectorfield has now four singularities: two centers and twosaddle points. We underline that such circulationsappear only in the case of an anisotropic diffusion.They persist for vanishing forces F but without totalcurrent. Oppositely, higher values of F remove thiscomplex pattern of the local current and the flowbecomes laminar, similar to Fig. 2.

In summary, we have investigated motion of aBrownian particle on 2D domain in a periodic poten-tial and under the influence of a constant drivingforce. Using numerical analysis we have shown thatthe bronek symmetry of the potential can induce aflux of particles perpendicular to the force. For giventype of obstacles the effect has its maximal effi-ciency at finite values of parameters. The system

Fig. 7. The local fluxes in a system with non-equal diffusionconstants D s0.3 and D s0.01. Arrows and lines show thex y

direction of the flow, at the background is a potential without a tiltterm. The domain was shifted in order to show the singularities.The remaining parameters are Fs1, Ls1, ks0.8 and As0.8.

with anisotropic diffusion constant has also beenconsidered. The pattern of flows in that case isnon-trivial, the field of the average velocity of parti-cles has two centers and saddle points.

Acknowledgements

We are grateful to the Komitet Badan NaukowychŽ .grant 2P03B 160 17 , Sfb 555 of the DeutscheForschungsgemeinschaft and Deutscher Akademis-

Ž .cher Austauschdient DAAD for funding. We wouldlike to thank to J. Luczka for his valuable remarksand discussions.

Appendix A. Finite element method

Ž .The finite element method FEM is a flexiblenumerical method for solving partial differential

Ž .equations PDE . Perhaps its most attractive featureis the straightforward handling of geometricallycomplicated domains. It is also easy to construct

w xhigher order approximations 25–27 . Here a briefoutline of its basic concepts is presented.

Suppose that we have a PDE in the form

Lusg , A.1Ž .where L is a linear operator, g is a given functionand u the unknown function. Then its weak formreads 3

Lu ,Õ s g ,Õ A.2Ž . Ž . Ž .Žfor an arbitrary test function Õ brackets denote a

.scalar product in the corresponding function space .The finite element method seeks for approxima-

tions of the unknown u by expanding into localŽ .function N x asj

M

us u N x . A.3Ž . Ž .ˆ Ý j jjs1

The way of finding the coefficients u and the choicejŽ .of the functions N x determines the variation ofj

the method. The most common one is to require thatŽ . Ž . Ž .the residual R u , . . . ,u , x s Lu,Õ y f ,Õ , van-ˆ1 M

3 Summation over repeated index is assumed.


ishes for M linearly independent weighting functionsW :i

R ,W s0, for is1, . . . , M . A.4Ž . Ž .i

It is referred to a the weighted residual method . Inthis work the choice W sN is used and it is calledi i

ŽGalerkin’s method the weighted residual methodwith W /N is also frequently called a PetroÕ–i i

. w xGalerkin formulation 27 .Ž .Let us specify to the Smoluchowski Eq. 5 .

Using the following notation:

EE s , A.5Ž .i E xi

fs f , f s f , f ,Ž . Ž .1 2 x y

Js J , J s J , J ,Ž . Ž .1 2 x y

x sx , x sy ,1 2

it can be written as

E J s0. A.6Ž .k k

The first step in FEM procedure is to obtain itsŽ .discrete form. For a test function N sN x, y thei i

Ž .scalar product A.4 reads:

R , N s E J , N s dxE J NŽ . Ž . Hi k k i k k iG

sy ds J N q dxJ E N , A.7Ž .H Hk k i k k iEG G

where s is a component of the normal vector to thatk

surface EG . The first integral contributes to naturalboundary conditions. In our case with periodicboundaries it is neglected. Substituting the explicitformula for J and requiring the solution to be in thek

0 Ž . Ž .form as P x , y s Ý p N x , y we getjs 1 j j

Galerkin’s method:

A p s0, wherei j j

A s dxf E N N yD dxE N E N . A.8Ž .H Hi j k k i j k k i k jG G

The next step is to choose the test functions Ni

and set up the matrix A. The standard FEM proce-Ždure is first to construct the grid we have taken a

.rectangular one on the domain of G . Then theŽ .function N x, y can be taken as a linear function oni

the element number i and zero outside this element.With such choice the algorithm of calculating one

matrix element A can be reduced to the integrationi j

of single elements instead of the whole domain G .0Ž . Ž .Coefficients in expansion P x, y sÝ p N x, yjs1 j j

have also a simple and useful meaning, p is a valuei0Ž . w x.of P x, y on the appropriate node of the grid 25 .

In the last step, the boundary conditions have tobe imposed. The considered case of a Smoluchowskiequation with periodic boundary conditions differsfrom the standard case. We have to impose theperiodicity and normalization of the solution:

p s1, p sp for some set of pairs a,b .Ž .Ý i a bi

A.9Ž .

It must be stressed that the size of the matrix can belarge. If one takes, for example, 2D problem with

4 Ž10 test functions is corresponds to 100=100 rect-. 4angular grid then the matrix A is of the size 10 =

104. Without periodic boundaries the matrix A has abanded structure. The requirement of the periodicityof solution destroys this feature. Nevertheless it ispossible to store this matrix in a structural sparse

Ž .format it means storing only non-zero diagonals .The normalization condition has to be treated in aspecial way. Due to linearity of the solution we canimpose the essential boundary condition on one nodeŽ .e.g. p s1.0 and normalize the solution at the end23

of the procedure. One can notice that the left handside of the linear system is zero, hence withoutabove procedure it would be impossible to obtain aunique solution. The essential boundary conditionŽ .like e.g. p s1.0 the left hand side of the linear23

system to be nonzero. The only effective way tosolve the resulting linear system is an must be theiterative method. For the problems in this work the

ŽBi-Conjugate Gradient algorithm which is of type of.Krylov method together with the preconditioning of

Žthe matrix A using RILU Relaxed Incomplete LU. w xfactorization algorithm have been applied 25,28,29 .

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numerical study of diffusion induced transport in 2d systems

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