Numerical study of diffusion induced transport in 2D systems

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<ul><li><p>7 February 2000</p><p> .Physics Letters A 265 2000 337345www.elsevier.nlrlocaterphysleta</p><p>Numerical study of diffusion induced transport in 2D systemsM. Kostur a,b, L. Schimansky-Geier a</p><p>a Institute of Physics, Humboldt-Uniersity at Berlin, Inalidenstr. 110, D-10115 Berlin, Germanyb Institute of Physics, Silesian Uniersity, ul. Bankowa 14, 40-007 Katowice, Poland</p><p>Received 17 August 1999; received in revised form 9 December 1999; accepted 9 December 1999Communicated by C.R. Doering</p><p>Abstract</p><p> .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.</p><p>1. Introduction</p><p>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</p><p>w xBrownian ratchets or Brownian motors 14 . 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,</p><p>1 We consider non-interacting Brownian particles.</p><p>one has to assume an external force or nonequilib-rium fluctuations which drive the Brownian particles.</p><p>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</p><p>w xin the near past 2,513 .However, most of these theoretical considerations</p><p>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 x1416 . These 2D-obstacles have originated the</p><p>0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. .PII: S0375-9601 99 00906-8</p></li><li><p>( )M. Kostur, L. Schimansky-GeierrPhysics Letters A 265 2000 337345338</p><p>necessary break of the symmetry and non-equi-librium was obtained by switching on and off thepotential in time.</p><p>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-</p><p>w xnique to separate macromolecules 1720 . Generalaspects of this phenomenon have already been re-</p><p>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</p><p>w xis an even function of the driving force 21 . Thisfeature makes the phenomenon even more interesting</p><p>w xfor practical applications 20 .In this paper we present the detailed numerical</p><p>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.</p><p>The corresponding overdamped 1D-problem with.flow in parallel to the force was first solved in</p><p>connection with the phase locking effect of drivennonlinear oscillators. Stratonovich has given an ana-lytical stationary solution of the distribution and the</p><p>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 FokkerPlanck-equation by matrix contin-</p><p>w xued fraction techniques are presented 23 .The symmetry of the 2D-sawtooth like potential is</p><p>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</p><p> .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</p><p>w xsinks and sources 24 . It is similar to flows inconservative systems with centers, saddle points and</p><p>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.</p><p>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 flashingon 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.</p><p>2. Smoluchowski equation for periodic obstaclesin 2D</p><p>We consider a Brownian particle moving on a two .dimensional plane x, y . So far hatted values stand </p><p>for non-scaled variables. A periodic array of obsta-cles in x and y directions is assumed. These obsta- cles should be originated by a continuous periodic</p><p> . . .potential V x, y s V x q L , y s V x, y q L . 0 0 x 0 yThe potential is assumed to have a broken reflectionsymmetry in the xdirection, particularly a piecewiselinear sawtooth-like shape has been chosen. On theother hand, the potential has reflection symmetry inthe y direction.</p><p>Moreover, a constant driving force acts on theparticle in the y direction. Therefore, the total force</p><p> .field is determined by the potential V x, y s .V x, y yFy. 0</p><p>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</p><p>d x E V x , y . g sy q 2g kT j t , .x</p><p>d t E x</p><p>d y E V x , y . g sy q 2g kT j t , 1 . .y</p><p>d t E y</p><p> and j and j are independent d-correlated Gauss-x yian white noises with zero average and the correla-</p><p> X X . .: . tion function j t j t sd ty t , isx, y.i i</p></li><li><p>( )M. Kostur, L. Schimansky-GeierrPhysics Letters A 265 2000 337345 339</p><p> .System 1 can be converted into dimensionlessform by</p><p>ts trt ,0xsxrL , xysyrL , y</p><p> V x , y sV x , y rDV . 2 . . . Here L and L are the periods of the system in xx y</p><p>and y directions, respectively, DV is the maximalelongation of the potential and t is a characteristic0time in the system. Particularly, we took t to be the0time a deterministic overdamped particle needs to</p><p>move under influence of the force DVrL over thexdistance Lx</p><p>g L2xt s . 3 .0</p><p>DV .After simple algebra one obtains from 1</p><p>E V x , y . w xxsy q 2 D j t , xg 0,1 , . xE xE V x , y . w xysy q 2 D j t , yg 0, L , 4 . . yE y</p><p>where the dimensionless diffusion coefficient readsDskTrDV and LsL rL stands for the aspecty x</p><p>ratio. j and j are again two uncorrelated Gaussianx ywhite noise sources with intensity 1.</p><p>The 2D-Smoluchowski equation for the evolution .of the probability density P x, y,t can be immedi-</p><p>ately formulated in dimensionless variablesE P x , y ,t E .</p><p>sy f x , y P x , y . .xE t E xE</p><p>y f x , y P x , y . .yE yE 2</p><p>qD P x , y .2E xE 2</p><p>qD P x , y . 5 . .2E yTherein, f and f are the components of the dimen-x ysionless force given by negative gradients of the . .V x, y . As a potential V x, y which fulfills our</p><p> .requirements we use later on see Fig. 1 :V x , y sV x , y yF y . .0</p><p>s 1yA sin 2p yrL U x yF y. 6 4 . . .</p><p> . .Fig. 1. The 2D-potential V x, y sV x, y without external force0Fs0. Periodicity is obeyed in both directions and reflectionsymmetry is broken in x-direction. Other parameters: As0.3,Ls1, and ks0.8.</p><p> .U x is the one-dimensional piecewise linear ratchetpotential of unit height with the parameter of asym-</p><p> . .metry kg 0,1 , particularly if ks1r2 then U x is .symmetric. U x is periodically modulated in y</p><p>direction with an amplitude A. The linear term standsfor the constant bias which drives the system out ofequilibrium.</p><p> . .The Smoluchowski Eq. 5 with the potential 6remains invariant under transformations:</p><p>x , y xq1, y , x , y x , yqL . 7 . . . . .Hence, if one chooses periodic initial condition . .P x, y,ts0 , i.e. invariant under 7</p><p>P x , y ,0 sP xq1, y ,0 , . .P x , y ,0 sP x , yqL,0 , 8 . . .and normalized in one periodic box</p><p>1 LP x , y ,0 dx dys1, 9 . .H H</p><p>0 0 .then the solution P x, y,t will be periodic and</p><p>normalized for arbitrary time. In particular, the sta-tionary stateP 0 x , y s lim P x , y ,t . .</p><p>t</p><p>will also be periodic and normalized over one period.Due to above symmetries of the system we con-</p><p> . w xsider the Smoluchowski Eq. 5 on the domain 0,1w x= 0, L with boundary conditions</p><p>P x , y ,t sP xq1, y ,t , . .P x , y ,t sP x , yqL,t . 10 . . .</p><p>Using finite elements method see Appendix A the . .Eq. 5 with boundary conditions 10 can be solved</p><p>numerically. Because we investigate the stationary</p></li><li><p>( )M. Kostur, L. Schimansky-GeierrPhysics Letters A 265 2000 337345340</p><p>transport in the long-time limit, the stationary solu-0 . . 0 .tion P x, y of 5 is of special interest. P x, y</p><p>contains all relevant informations for determining thestationary flows in the system, namely</p><p>E0 0J x , y s f x , y P x , y yD P x , y , . . . .x x x E x</p><p>E0 0J x , y s f x , y P x , y yD P x , y . . . . .y y y E y</p><p>11 .The total transport over one period in the x directioncan be quantified by determining the average veloc-</p><p> :ity . Taking into account the normalization con-x .dition 9 the average velocity is expressed by the</p><p>total current via1 Ltot : sJ s J x , y dx dy 12 . .H Hx x x</p><p>0 0</p><p> .Insertion of 11 eventually yields1 L 0 : s f x , y P x , y dx dy , 13 . . .H Hx x</p><p>0 0</p><p> .since the second item in 11 does not contribute tothe total flow.</p><p>3. Numerical analysis of the induced current</p><p>When the bias F vanishes then the potential .V x, y is periodic and the solution is given by the</p><p>0 . yV x , y.r DBoltzmann-distribution P x, y ,e . Obvi- .ously, in this case the local flux vanishes, J x, y sx</p><p> .J x, y s0. The addition of a tilt in y-directionydrives the system out of equilibrium. No analyticalsolution is known in this case. Therefore, we havedecided to solve numerically the Smoluchowski Eq. .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.</p><p>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</p><p> 2 . w xleast of the order O AF 21 . The probability fluxdensity is presented in Fig. 2. Directions of the flow</p><p>w x w xon the entire plane 0,1 = 0, L are drawn by arrowsof unit length and the background shows the surface</p><p>Fig. 2. Plot of directions of the probability currentJ x , y , J x , y for Ds0.2, As0.8, Ls1, Fs5 and . . .x y</p><p> .ks0.8. As the background surfaces of potential V x, y withouttilt F have been drawn.</p><p>of the potential V . One sees that the x-component0of the local current adopts positive and negativevalues.</p><p>However, the main contribution to the total cur-rent comes from the flow through the saddle point of</p><p> .the potential xs0.8 and ys0.75 . Therefore,globally a flux to the left side dominates. Fig. 3shows quantitatively the averaged velocity. Due to</p><p> .the symmetry of V x, y in y-direction the flow0w x tot .does not change replacing FyF 21 and J Fx</p><p>s0 at Fs0. Also the slope of the dependencetot .J F also vanishes at Fs0. Increasing F inducesx</p><p>a negative flow and J tot reaches its maximal abso-xlute 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.</p><p>In addition results of Monte Carlo simulations of .the Langevin Eq. 4 have been included in Fig. 3.</p><p>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</p><p> .of 5 required 20 minute on one single processor,</p></li><li><p>( )M. Kostur, L. Schimansky-GeierrPhysics Letters A 265 2000 337345 341</p><p>Fig. 3. The total flux in the system in x-direction versus drivingforce F. The error bars depict the results of Monte Carlo simula-</p><p>tions a size of the bar is a standard deviation of the result taken.from sample of 32 measurements .</p><p>what makes finite elements method about 100 timesfaster in this application. Nevertheless, the simula-tion method served as a valuable checking tool inour analysis.</p><p>The mentioned efficiency of the numerical methodhave allowed the investigation of the total flux in thefull parameters space. We have taken into account</p><p> .three parameters L, D,F . They represent three vari-ous features of the system: the aspect ratio L modi-</p><p>fies its geometry the aspect ratio L extends or.shrinks the potential , D controls the relative tem-</p><p>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</p><p> . .the potential V x, y or one period , the total fluxJ s measures the average number of particlesx xpassing 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 t...</p></li></ul>