Flux reversal in a simple random-walk model on a fluctuating symmetric lattice

Flux reversal in a simple random-walk model on a fluctuating symmetric lattice Jesús Casado-Pascual Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, Sevilla 41080, Spain 共Received 15 November 2005; revised manuscript received 11 March 2006; published 11 August 2006兲 A rather simple random-walk model on a one-dimensional lattice is put forward. The lattice as a whole switches randomly between two possible states which are spatially symmetric. Both lattice states are identical, but translated by one site with respect to each other, and consist of infinite arrays of absorbing sites separated by two nonabsorbing sites. Exact explicit expressions for the long-time velocity and the effective diffusion coefficient are obtained and discussed. In particular, it is shown that the direction of the steady motion can be reversed by conveniently varying the values of either the mean residence times in the lattice states or the transition rates to the absorbing and nonabsorbing sites. DOI: 10.1103/PhysRevE.74.021112 PACS number共s兲: 05.40.Fb, 05.60.Cd, 87.16.Nn I. INTRODUCTION The study of the transport of Brownian particles along periodic structures has attracted considerable and ever growing interest in the past years due to its relevance in a wide variety of problems in physics, chemistry, biology, and nanotechnology 关1,2兴. As required by the second law of thermodynamics, Brownian motion cannot induce a steady directed motion in a system at equilibrium 共see, for instance, Ref. 关1兴 for a detailed proof兲. Nevertheless, a Brownian particle can display a directed motion in a periodic potential which switches randomly or periodically between two or more states 关3–7兴. For this to be possible, certain spatiotemporal symmetry and supersymmetry conditions must be broken 关8兴. An important phenomenon often observed in these systems is the reversal of the current direction upon variation of a system parameter 关9,10兴. Though in many cases the spatiotemporal symmetry breaking is achieved by using a spatially asymmetric periodic potential, in Ref. 关11兴it was shown that it can also be obtained by using a spatially symmetric periodic potential which fluctuates between three states, one of them being of free diffusion. As shown in Ref. 关12兴, if the symmetry of glide reflection is broken, a directed transport can also be induced in a spatially symmetric potential which switches randomly between two states. Besides the purely theoretical interest, the study of this kind of models is relevant because it may provide valuable information about the workings of the so-called molecular motors 关1,13兴. Molecular motors are able to travel along polymer filaments by utilizing chemical energy from ATP 共adenosine triphosphate兲hydrolysis, and are responsible for the transport of different substances inside the eukaryotic cells. Recently, it has been demonstrated experimentally that the Brownian motion of a colloidal sphere can be rectified using a periodic potential that alternates between three 关14兴or two 关15兴states which differ only by a discrete translation. Each state is spatially symmetric and consists of a large array of optical traps, created with dynamic holographic optical tweezers, each of which can localize a colloidal sphere. The pattern of discrete optical traps can be modeled as a periodic array of Gaussian wells of width ␴ and depth V0, uniformly separated by a distance Lsubstantially larger than ␴ .Inthe three-state case considered in Ref. 关14兴, the initial array of optical traps is extinguished after time Tand replaced immediately with a second array, which is displaced from the first by L/3. This process is repeated after time Twith an additional displacement of L/3. A cycle is completed when, after time T, the array of traps is displaced once again by L/3 and returns to its initial state. In the two-state case reported in Ref. 关15兴, the first step is the same as in 关14兴, i.e., the initial array of optical traps is extinguished after time T1and replaced immediately with a second array, which is displaced from the first by L/3. This second array is extinguished after time T2and replaced again by the first, thereby completing one cycle. One main result of the three-state case is that the direction of the motion depends sensitively on the duration of the states, T, relative to the time required for the particle to diffuse the intertrap separation 关14兴. By contrast, in the two-state case this direction depends on the relative durations of the two states T2/T1关15兴. Specifically, the flux is directed from each optical trap in the longer-duration state toward the nearest optical trap in the short-lived state, and vanishes for T1=T2. Thus, in both cases, by varying the value of the corresponding parameter the flux can be reversed. To gain a deeper insight into the behavior of these systems, it would be desirable to develop simplified models, amenable to analytical treatment, which capture the key features of the original complex systems. The main focus of this paper is to present such a simplified model for the experiment described in Ref. 关15兴. The paper is organized as follows. In the following section, we introduce the model and provide definitions of the quantities of interest, namely, the long-time velocity and the effective diffusion coefficient. In Sec. III, we obtain explicit expressions for these two quantities. The main results arising from these expressions are discussed in Sec. IV. Finally, we present conclusions for the main findings of our work. II. DESCRIPTION OF THE MODEL In our simplified model the motion of the colloidal sphere in the periodic potential is modeled by a random walk on a one-dimensional lattice with its sites located at xn=nh, with n苸Z, where it has been assumed that the distance between two consecutive sites xn+1−xn=his independent of n. The lattice can be in two possible states that will be indicated by PHYSICAL REVIEW E 74, 021112 共2006兲 1539-3755/2006/74共2兲/021112共5兲©2006 The American Physical Society021112-1 the label ␣ = ±1. In contrast with the approach described in Ref. 关15兴in which the potential switches periodically between its two states, here we will assume that the lattice fluctuates randomly between its two possible states. The evolution of the joint probabilities pn 共 ␣ 兲共t兲that the particle is at site xnand the lattice in state ␣ at time tare governed by the master equation p ˙n 共 ␣ 兲共t兲=−关Ln 共 ␣ 兲+Rn 共 ␣ 兲兴pn 共 ␣ 兲共t兲+Rn−1 共 ␣ 兲pn−1 共 ␣ 兲共t兲+Ln+1 共 ␣ 兲pn+1 共 ␣ 兲共t兲 + ␥ 共− ␣ 兲pn 共− ␣ 兲共t兲− ␥ 共 ␣ 兲pn 共 ␣ 兲共t兲,共1兲 where ␥ 共 ␣ 兲is the transition rate from the lattice state ␣ to − ␣ , and Ln 共 ␣ 兲and Rn 共 ␣ 兲are the transition rates from site xnto site xn−1 and xn+1, respectively, assuming that the lattice is in state ␣ . The initial condition is irrelevant since we are interested in the long-time behavior of the system. Retaining the main features of the model described in Ref. 关15兴, we will assume that both states of the lattice are spatially symmetric 共i.e., there exists a n⬘苸Zsuch that Ln⬘+n 共 ␣ 兲 =Rn⬘−n 共 ␣ 兲for all n苸Z兲and periodic with period N共i.e., Ln+N 共 ␣ 兲 =Ln 共 ␣ 兲and Rn+N 共 ␣ 兲=Rn 共 ␣ 兲for all n苸Z兲. This is an important distinction between our model and other similar models previously reported in the literature 关16–19兴, in which at least one of the lattice states is spatially asymmetric. For simplicity, we will restrict our study to the case N=3, which is the smallest value of the period for which a nonzero steady directed motion is possible under the above conditions of spatial symmetry and periodicity 关20兴. The lattice state ␣ =+1is obtained from the lattice state ␣ =−1 by translating it to the right one site, so that, Ln 共+1兲=Ln−1 共−1兲and Rn 共+1兲=Rn−1 共−1兲. When the lattice is in state ␣ =−1, there are absorbing sites at the locations x3j+1, for all j苸Z, and consequently, L3j+1 共−1兲=R3j+1 共−1兲 =L3j+2 共+1兲=R3j+2 共+1兲=0. These absorbing sites play the role of the optical traps in our simplified model. The transition rates to an absorbing and a nonabsorbing site will be denoted by g and k, respectively, so that L3j+2 共−1兲=R3j 共−1兲=L3j 共+1兲=R3j+1 共+1兲=gand R3j+2 共−1兲=L3j 共−1兲=R3j 共+1兲=L3j+1 共+1兲=k. A sketch of the model just described is shown in Fig. 1. It is worthwhile to point out that the model just described is formally equivalent to another random-walk model which mimics the motion of some molecular motors. In this model the walker switches randomly between two possible states, indicated by the label ␣ =±1, ␥ 共 ␣ 兲being the transition rate from the state ␣ to − ␣ 共see Fig. 2兲. These states are characterized by the position of two legs, each with a head-shaped object attached at one of their ends. The walker “walks” on a rod which possesses an infinite array of binding sites, depicted by gray squares in Fig. 2. These binding sites are separated from each other by a distance 3h, and translated by h/2 with respect to the sites of the lattice. Whenever any of the walker’s “heads” comes in contact with a binding site, the walker stays trapped at that site until a fluctuation of its state occurs. For each state of the walker, the sites of the lattice with this property 共absorbing sites兲are depicted by crosses in Fig. 2. When none of the walker’s heads are in contact with a binding site, the walker can hop to its nearestneighbor sites. Then, the transition rate from a site to its neighbors is g, if after hopping any of the heads comes in contact with a binding site, or k, otherwise. This interpretation of the model differs from the one considered at the end of Ref. 关15兴, in three important aspects. First, ours is discrete and exactly solvable. Second, in ours the walker fluctuates randomly between its two possible states, whereas in theirs it switches periodically between them. In this sense, our model may be more effective for describing real molecular motors, since in this class of molecules the directed motion arises as a consequence of a competition between two stochastic time scales 共the ATP binding rate and the ATP hydrolysis rate兲. Third, in 关15兴the direction of the steady motion depends only on the relative durations of the two states, whereas here, as we will see later on, it also depends on the sign of g−k. We are interested in the study of the long-time velocity or flux vand the effective diffusion coefficient Deff, defined as v= lim t→⬁ d dt具x共t兲典 共2兲 and Deff =1 2lim t→⬁ d dt关具x2共t兲典 −具x共t兲典2兴,共3兲 where the lth moment of the position is given by 具xl共t兲典 =hl兺 ␣ =±1兺n=−⬁ +⬁nlpn 共 ␣ 兲共t兲. FIG. 1. Sketch of the fluctuating lattice. The absorbing sites are depicted by crosses. FIG. 2. An alternative interpretation of the model sketched in Fig. 1. The binding and absorbing sites are depicted by gray squares and crosses, respectively. JESÚS CASADO-PASCUAL PHYSICAL REVIEW E 74, 021112 共2006兲 021112-2 There are certain properties of vand Deff that can be predicted by simple considerations. For instance, in the absence of fluctuations between the two lattice states, both the long-time velocity and the effective diffusion coefficient will be zero, as the particle ends up being trapped by an absorbing site. Even if the sites depicted by crosses in Fig. 1were not absorbing and, therefore, there were a nonzero transition rate from them 共the same to the left and to the right兲, the long-time velocity would be also zero in the absence of fluctuations between the two lattice states. This is so because, in that case, both lattice states would be invariant under reflections about any of the sites with crosses. Another interesting property is obtained if one takes into account that the fluctuating periodic lattice sketched in Fig. 1is invariant under the transformation 兵n, ␣ 其Þ兵−n,− ␣ 其. Consequently, the longtime velocity and the effective diffusion coefficient fulfill the symmetry relations v关 ␥ 共+1兲, ␥ 共−1兲兴=−v关 ␥ 共−1兲, ␥ 共+1兲兴and Deff关 ␥ 共+1兲, ␥ 共−1兲兴=Deff关 ␥ 共−1兲, ␥ 共+1兲兴, where we have written explicitly the dependence of vand Deff on ␥ 共+1兲and ␥ 共−1兲. Thus, if ␥ 共+1兲= ␥ 共−1兲, the long-time velocity will be zero. III. EXPLICIT EXPRESSIONS FOR THE LONG-TIME VELOCITY AND THE EFFECTIVE DIFFUSION COEFFICIENT In order to go further in the analysis vand Deff,itis necessary to obtain explicit expressions for them. To do that, we will extend the approach developed in Ref. 关21兴to the case of a fluctuating lattice. The procedure described here is quite general and can be applied to any fluctuating periodic lattice with an arbitrary period N. For this reason, in what follows it will be described in general terms. An alternative approach based on the gradient expansion of the probability distribution corresponding to a coarse-grained description of the random-walk model can be found in Ref. 关17兴. First, let us introduce the quantities qn 共 ␣ 兲共t兲=兺 l=−⬁ ⬁ pn+lN 共 ␣ 兲共t兲共4兲 and sn 共 ␣ 兲共t兲=h兺 l=−⬁ ⬁ 共n+lN兲pn+lN 共 ␣ 兲共t兲−具x共t兲典qn 共 ␣ 兲共t兲,共5兲 which are periodic in nwith period N, i.e., qn+N 共 ␣ 兲共t兲=qn 共 ␣ 兲共t兲 and sn+N 共 ␣ 兲共t兲=sn 共 ␣ 兲共t兲for all n苸Z. Substituting the expressions for the first and second moments of the position into Eqs. 共2兲 and 共3兲, and replacing p ˙n 共 ␣ 兲共t兲by the right-hand side of Eq. 共1兲, it is not difficult to prove that the long-time velocity and the effective diffusion coefficient can be expressed as v=h兺 ␣ =±1 兺 n=1 N 关Rn 共 ␣ 兲−Ln 共 ␣ 兲兴Qn 共 ␣ 兲共6兲 and Deff =h2 2兺 ␣ =±1 兺 n=1 N 关Rn 共 ␣ 兲+Ln 共 ␣ 兲兴Qn 共 ␣ 兲+h兺 ␣ =±1 兺 n=1 N 关Rn 共 ␣ 兲−Ln 共 ␣ 兲兴Sn 共 ␣ 兲, 共7兲 where Qn 共 ␣ 兲=limt→⬁qn 共 ␣ 兲共t兲and Sn 共 ␣ 兲=limt→⬁sn 共 ␣ 兲共t兲. Thus, for the evaluation of vand Deff it suffices to calculate the values of the 4Nnumbers Qn 共 ␣ 兲and Sn 共 ␣ 兲, with n=1,...,Nand ␣ =±1. Differentiating Eqs. 共4兲and 共5兲with respect to time, making use of Eq. 共1兲, as well as of the periodicity of Ln 共 ␣ 兲and Rn 共 ␣ 兲, and taking into account that limt→⬁q ˙n 共 ␣ 兲共t兲 =limt→⬁s ˙n 共 ␣ 兲共t兲=0, it is easy to see that the long-time limits Qn 共 ␣ 兲and Sn 共 ␣ 兲fulfill the set of 4Nalgebraic equations −关Ln 共 ␣ 兲+Rn 共 ␣ 兲兴Qn 共 ␣ 兲+Rn−1 共 ␣ 兲Qn−1 共 ␣ 兲+Ln+1 共 ␣ 兲Qn+1 共 ␣ 兲 + ␥ 共− ␣ 兲Qn 共− ␣ 兲− ␥ 共 ␣ 兲Qn 共 ␣ 兲=0, 共8兲 −关Ln 共 ␣ 兲+Rn 共 ␣ 兲兴Sn 共 ␣ 兲+Rn−1 共 ␣ 兲Sn−1 共 ␣ 兲+Ln+1 共 ␣ 兲Sn+1 共 ␣ 兲 + ␥ 共− ␣ 兲Sn 共− ␣ 兲− ␥ 共 ␣ 兲Sn 共 ␣ 兲=Fn 共 ␣ 兲,共9兲 for n=1,...,Nand ␣ = ±1, where Fn 共 ␣ 兲=h关Ln+1 共 ␣ 兲Qn+1 共 ␣ 兲−Rn−1 共 ␣ 兲Qn−1 共 ␣ 兲兴+vQn 共 ␣ 兲,共10兲 and where the periodic boundary conditions Q0 共 ␣ 兲=QN 共 ␣ 兲, QN+1 共 ␣ 兲=Q1 共 ␣ 兲,S0 共 ␣ 兲=SN 共 ␣ 兲, and SN+1 共 ␣ 兲=S1 共 ␣ 兲are assumed. Summing up the 2Nequations contained either in the set of equations 共8兲or 共9兲, it is easy to realize that in each set there are only 2N−1 independent equations 共excluding some pathological cases where the number can be even lower兲. Thus, to univocally determine the 4Nquantities Qn 共 ␣ 兲and Sn 共 ␣ 兲, it is also necessary to add the two additional equations 兺 ␣ =±1兺n=1 NQn 共 ␣ 兲=1 and 兺 ␣ =±1兺n=1 NSn 共 ␣ 兲=0, which can be straightforwardly proven from the definitions of Qn 共 ␣ 兲and Sn 共 ␣ 兲. To apply the above procedure to the model sketched in Fig. 1, first one has to solve this set of equations, with N =3 and the corresponding values for the transition rates Ln 共 ␣ 兲 and Rn 共 ␣ 兲. Replacing the results obtained for Qn 共 ␣ 兲and Sn 共 ␣ 兲in Eqs. 共6兲and 共7兲, one obtains after some lengthy simplifications that v=3h⌬ ␳ 共1−⌬ ␳ 2兲共g−k兲gk⌫2A−1 共11兲 and Deff =9 2h2共1−⌬ ␳ 2兲gk共g+k兲共2g+⌫兲⌫A−1 −v2B⌫−1A−1, 共12兲 where A=4g2共g+2k兲2+8g共g+k兲共g+2k兲⌫ +关4共g+k兲2−共1+3⌬ ␳ 2兲k2兴⌫2,共13兲 B=4g2共g+2k兲2+24g共g+k兲共g+2k兲⌫ +关28共g+k兲2−3共3+⌬ ␳ 2兲k2兴⌫2+8共g+k兲⌫3,共14兲 ⌫= ␥ 共+1兲+ ␥ 共−1兲, and ⌬ ␳ = ␳ eq 共−1兲− ␳ eq 共+1兲, ␳ eq 共 ␣ 兲being the equilib- FLUX REVERSAL IN A SIMPLE RANDOM WALK MODEL¼PHYSICAL REVIEW E 74, 021112 共2006兲 021112-3 rium population of the lattice state ␣ , i.e., ␳ eq 共 ␣ 兲= ␥ 共− ␣ 兲/⌫. This quantity can alternatively be expressed as ⌬ ␳ =关T共−1兲 −T共+1兲兴/关T共−1兲+T共+1兲兴, where T共 ␣ 兲=1/ ␥ 共 ␣ 兲is the mean residence time of the lattice in the state ␣ . IV. RESULTS An important result that follows from Eq. 共11兲is that the direction of the steady motion not only depends on the sign of ⌬ ␳ , as would be expected from the previous symmetry considerations, but also on the sign of the difference between the transition rates to an absorbing and a nonabsorbing site, g-k. Thus, for instance, if T共−1兲⬎T共+1兲, the steady directed motion is from left to right for g⬎k, whereas it is from right to left for g⬍k. Obviously, if T共−1兲⬍T共+1兲, the behavior is just the opposite. Finally, no steady directed motion is possible if either T共−1兲=T共+1兲or g=k. This subtle dependence of the direction of the steady motion on the values of gand k cannot be inferred a priori from symmetry considerations, and is one of the main results of this work. It is important to point out that this model predicts correctly the direction of the steady motion of the experiment described in Ref. 关15兴. Indeed, in that case, the values of gand kcan be estimated as the inverses of mean times required for the particle to diffuse a length L/3− ␴ /2 and L/3, respectively, where Lis the intertrap distance and ␴ is the width of an optical trap. Therefore, g=2D/共L/3− ␴ /2兲2and k=2D/共L/3兲2,Dbeing the diffusion coefficient of the colloidal sphere in the fluid. Since g⬎k, the motion is from left to right if T共−1兲⬎T共+1兲and from right to left if T共−1兲⬍T共+1兲, i.e., from each absorbing site in the longer-duration state toward the nearest absorbing site in the short-lived state. The above expressions for gand kalso show that the finite width of the optical traps plays an important role in the appearance of the steady directed motion. Some limiting values of the expressions 共11兲and 共12兲can be easily interpreted. For instance, as suggested by the previous simple considerations, vand Deff tend to zero both as ⌫→0共absence of fluctuations between the two lattice states兲 and as ⌬ ␳ →±1 共absence of fluctuations between the two lattice states in the long-time limit兲.Asg→0, vand Deff also go to zero because, in this limit, the particle stays trapped between two consecutive absorbing sites, since it is not allowed to jump to them. The expressions for vand Deff also tend to zero both as k→0 and as g→⬁. This is due to the fact that, in these limits, the motion of the particle consists of a series of forward and backward jumps from a nonabsorbing site to its nearest absorbing site, which arise as a consequence of the fluctuations in the lattice state. Finally, the limiting values lim⌫→⬁v=v⬁and lim⌫→⬁Deff=Deff,⬁, which can be easily calculated from Eqs. 共11兲and 共12兲, also have a simple interpretation. Using the results in Ref. 关21兴,itisnot difficult to check that v⬁and Deff,⬁are the long-time velocity and the effective diffusion coefficient corresponding to a random walk with the average transition rates Ln= ␳ eq 共+1兲Ln 共+1兲 + ␳ eq 共−1兲Ln 共−1兲and Rn= ␳ eq 共+1兲Rn 共+1兲+ ␳ eq 共−1兲Rn 共−1兲. In contrast with what happens in continuum models, the limiting value v⬁ does not vanish, in general. This difference between discrete and continuum models has been previously pointed out in the literature 共see, for example, Ref. 关18兴兲. From the expressions for v⬁and Deff,⬁, it can be easily proved that limg→⬁v⬁ =3h⌬ ␳ 共1−⌬ ␳ 2兲k/4 and limg→⬁Deff,⬁=9h2共1−⌬ ␳ 2兲k/8 which, in general, are not equal to zero. This reflects the fact that the limits ⌫→⬁and g→⬁of Eqs. 共11兲and 共12兲do not commute. In order to simplify the graphical presentation of results, it is convenient to introduce dimensionless quantities by defining characteristic length and time scales. The characteristic length scale Lis chosen to be the distance between two consecutive absorbing sites, i.e., L=3h. The characteristic time scale ␶ is chosen to be the mean time required for the particle to diffuse the length Lin the absence of absorbing sites. Taking into account that in the absence of absorbing sites the effective diffusion coefficient is Deff,NA=h2k, it follows that ␶ =L2/共2Deff,NA兲=9/共2k兲. Then, the number of independent dimensionless parameters is reduced to only three 共the values of ␶ ⌫,⌬ ␳ , and ␶ g兲, since the other two take the fixed values L−1h=1 3and ␶ k=9 2. In Fig. 3, we depict the dependence of the dimensionless long-time velocity L−1 ␶ v共top panels兲and effective diffusion coefficient L−2 ␶ Deff 共bottom panels兲on ⌬ ␳ 共left panels兲and log10共 ␶ g兲共right panels兲. We have taken ␶ g=9 共left panels兲, ⌬ ␳ =0.6 共right panels兲, and three values of ␶ ⌫, namely, ␶ ⌫ =0 共solid lines兲,10共dotted lines兲, and 100 共long-dashed lines兲. With dash-dotted lines we have plotted the limiting values L−1 ␶ v⬁and L−2 ␶ Deff,⬁. Note that both the absolute value of L−1 ␶ vand L−2 ␶ Deff are increasing functions of ␶ ⌫, except at those points at which these functions vanish. This behavior is confirmed by a detailed study of the signs of the derivatives ⳵ v/ ⳵ ⌫and ⳵ Deff/ ⳵ ⌫. The reversal of the steady directed motion is clearly illustrated in panels 共a兲and 共b兲. These panels also reveal that, for fixed values of the remaining parameters, there exist two optimal values of ⌬ ␳ 共opposite to each other兲and of ␶ gwhich maximize the absolute value of the long-time velocity in each direction. In panel 共b兲, one can also see that the location of the second maximum is shifted towards larger values of ␶ gas ␶ ⌫is increased, and that this maximum completely disappears in the limit ⌫→⬁. Panel 共c兲shows that the dimensionless diffusion coefficient as a function of ⌬ ␳ is maximized for ⌬ ␳ =0, i.e., FIG. 3. Dependence of the dimensionless long-time velocity L−1 ␶ v共top panels兲and effective diffusion coefficient L−2 ␶ Deff 共bottom panels兲on ⌬ ␳ 共left panels兲and log10共 ␶ g兲共right panels兲. The dimensionless parameter values are L−1h=1 3, ␶ k=9 2, ␶ g=9 共left panels兲,⌬ ␳ =0.6 共right panels兲, ␶ ⌫=0 共solid lines兲, ␶ ⌫=10 共dotted lines兲, and ␶ ⌫=100 共long-dashed lines兲. The limiting values L−1 ␶ v⬁ and L−2 ␶ Deff,⬁are plotted with dash-dotted lines. JESÚS CASADO-PASCUAL PHYSICAL REVIEW E 74, 021112 共2006兲 021112-4 when the mean residence times in each lattice state are equal. Analogously, there is also an optimal value of ␶ gwhich maximizes the dimensionless diffusion coefficient as a function of ␶ g关see panel 共d兲兴. The location of this last maximum exhibits a similar behavior with ␶ ⌫as the second maximum in panel 共b兲. V. CONCLUSIONS In summary, we have proposed a simple random-walk model which allows the analytical study of the flux reversal in a spatially symmetric fluctuating lattice. We have obtained explicit analytical expressions for the long-time velocity and the effective diffusion coefficient and made a study of them. In spite of its simplicity, by appropriately choosing the transition rates to an absorbing and a nonabsorbing site, our model is able to reproduce some important features of the experiment described in Ref. 关15兴. In particular, it provides the correct direction of the induced flux and shows that this direction is closely related to the finite width of the optical traps. Furthermore, our model predicts the behavior of the effective diffusion coefficient that would be observed in the experiment reported in Ref. 关15兴. Finally, it predicts that similar results to those contained in 关15兴would be observed if the array of optical traps switched randomly between its two possible configurations instead of periodically. We have also presented an alternative interpretation of our model which mimics the motion of some molecular motors. In contrast to the one considered at the end of Ref. 关15兴,we have proven that, in ours, the direction of the steady motion depends not only on the relative durations of the two states, but also on the sign of the difference between the transition rates to an absorbing and a nonabsorbing site. ACKNOWLEDGMENTS The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain 共FIS2005-02884兲and the Junta de Andalucía. 关1兴P. Reimann, Phys. Rep. 361,57共2002兲. 关2兴R. D. Astumian and P. Hänggi, Phys. Today 55共11兲,33共2002兲. 关3兴M. Bier and R. D. Astumian, Phys. Rev. Lett. 76, 4277 共1996兲. 关4兴J. Prost, J. F. Chauwin, L. Peliti, and A. Ajdari, Phys. Rev. Lett. 72, 2652 共1994兲. 关5兴J. F. Chauwin, A. Ajdari, and J. Prost, Europhys. Lett. 27, 421 共1994兲;32, 373 共1995兲. 关6兴J. Rousselet, L. Salome, A. Ajdari, and J. Prost, Nature 共London兲370, 446 共1994兲. 关7兴L. P. Faucheux, L. S. Bourdieu, P. D. Kaplan, and A. J. Libchaber, Phys. Rev. Lett. 74, 1504 共1995兲. 关8兴P. Reimann, Phys. Rev. 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Consequently, from Eq. 共6兲one has that v=0. Analogously, if N=1, v=0 trivially. 关21兴B. Derrida, J. Stat. Phys. 31, 433 共1983兲. FLUX REVERSAL IN A SIMPLE RANDOM WALK MODEL¼PHYSICAL REVIEW E 74, 021112 共2006兲 021112-5