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a longitudinal wave travelling through a fluid imparts a net drift motion to the suspended particles - an effect known as _ drift_. the _ classical _ stokes drift  @xcite refers only to the deterministic behavior , i.e. it does not account for the stochastic fluctuations or perturbations in the system . it has been extensively studied in various practical contexts like , for instance , the motion of tracers in meteorology and oceanography  @xcite and that of the ( doping ) impurities in crystal growth  @xcite . a simple intuitive explanation of classical stokes drift is that the suspended particles spend a longer time in the regions of the wave - train where the force due to the wave acts in the direction of wave s propagation than the time spent in those regions where the force acts in the opposite direction ; therefore particles are driven in the sense of wave s propagation . when several linearly superposed waves are present , the resulting drift velocity is simply the sum of the contributions from each wave  @xcite .    recently , there has been some interest in _ stochastic _ stokes drift  ( @xcite , see also  @xcite ) . it was found that the diffusive motion of the suspended particles can significantly alter the amplitude of the drift velocity . but it is the merit of  @xcite to open stochastic stokes drift to further investigation by putting it in its natural context , namely _ ratchet - like brownian motors _ in which the thermal motion of small ( microscopic ) particles is rectified by an asymmetric time - dependent potential  @xcite . this scenario is referred to as the _ flashing ratchet_. paper  @xcite is limited to the case when the motion of the overdamped suspended particles is diffusion - dominated , i.e. when the deterministic forcing due to the wave is very weak . an approximate perturbative technique is developed for calculating the drift velocity in arbitrary dimensions , which shows that the drift depends on particles diffusivity in magnitude and , in more than 1d , also in direction ( a very appealing feature , useful in particle selection devices - see , for example , in  @xcite doering et all , 1994 , and in  @xcite m. bier , 1996 ) . papers @xcite and , in a very different context  @xcite , give an exact integral expression for stokes drift of overdamped diffusive particles in 1d , for arbitrary wave forms . using a fokker - planck equation technique , it finds that white noise reduces stokes drift by comparison to its classical ( deterministic ) value , the flux reversal being absent . the purpose of this short communication is to show that the characteristics of the 1d stokes drift may become more complex ( including flux reversal or enhancement of the classical drift ) when the particles are subjected to stochastic forcing that is non - white ( i.e. it has a finite correlation time ) . we will focus on the case of dichotomic markov forcing , in which the particle is subjected to a randomly alternating force , in addition to the force exerted by the wave . for this choice of coloured noise , explicit analytic results can be obtained , while this is not the case for other types of coloured noise , including the ornstein - uhlenbeck process . even though the time - average of the dichotomic force is taken to be zero , its effect in conjunction with the asymmetry of the wave s perturbation is highly nontrivial . as such the model realizes in the stokes drift context another paradigm of brownian motors , namely that of the so - called _ rocking ratchet_. in this scenario , the systematic motion acquired under influence of an alternating zero - average force essentially revolves on the asymmetry of the nonlinear response in the presence of a steady but asymmetric potential  @xcite . the dichotomic markov forcing presents another important advantage in that the exact integral expression for stokes drift can be derived . furthermore , in some cases of interest , the resulting integrals can be evaluated explicitly . our analytic results are also complemented with computer simulations , which are found to be in full agreement with the theory . the starting point is the following stochastic differential equation :    @xmath0    where @xmath1 is a symmetric dichotomic noise ( a two - step markov process )  @xcite . it can take only two values , @xmath2 , with equal probability , jumping between them with a probability @xmath3 per unit time . it has zero mean and its autocorrelation function @xmath4 shows a finite correlation time @xmath5 .    with a suitable time scaling , eq.([eq1 ] ) models the 1d overdamped motion of a brownian particle subjected to a dichotomic forcing . @xmath6 is the particle position at time @xmath7 , while @xmath8 is the periodic forcing due to the wave travelling at the speed @xmath9 with wavelength @xmath10 : @xmath11 with @xmath12 as the case of physical relevance . the quantity of interest is the _ drift velocity _ , which represents the mean velocity of the particles in the long time limit : @xmath13 where the mean is to be taken over the realisations of the dichotomic noise . by defining a new variable @xmath14 one can rewrite eq.([eq1 ] ) as : @xmath15 with @xmath16 , and therefore : @xmath17\ ; . \label{eq9}\end{aligned}\ ] ] here @xmath18 represent the probability densities for the particle position @xmath19 and the value of the dichotomic noise equal to @xmath2 , respectively . their time evolution is described by @xcite : @xmath20   -k(\rho_{+}-\rho_{- } ) , \nonumber\\ \frac{\partial \rho_{-}(y , t)}{\partial t } & = &   -\frac { \partial}{\partial y}[(f - a)\rho_{- } ]    -k(\rho_{-}-\rho_{+ } ) . \label{eq10}\end{aligned}\ ] ] the first term in the r.h.s . of each of these equations corresponds to the deterministic flow in phase space ( between two jumps of the dichotomic noise ) , while the other term corresponds to the jumps of the dichotomic noise between @xmath2 . because of the periodicity ( [ eq3 ] ) of @xmath21 ( @xmath22 ) , in order to evaluate ( [ eq9 ] ) we only need to study the asymptotic behavior of the reduced probability density : @xmath23 . \label{eq11}\ ] ] from its definition , it is obvious that the reduced probability density is subjected to periodic boundary conditions : @xmath24 while the normalization condition reads : @xmath25 considering also the quantity @xmath26\ ; , \label{eq14}\ ] ] it follows from  ( [ eq10 ] ) that @xmath27 and @xmath28 obey a set of coupled evolution equations : @xmath29 the stationary solution @xmath30 can be easily found  @xcite : @xmath31\ , dy ' , \label{eq23}\ ] ] which is valid as long as @xmath32 , @xmath33 . here @xmath34 is defined by : @xmath35}{f^2(y')-a^2}dy ' . \label{eq20}\ ] ] and @xmath36 is the normalization factor according to eq.([eq13 ] ) . therefore the drift velocity can be finally expressed as : @xmath37 these results are valid for an arbitrary wave form , provided that the obvious necessary differentiability and integrability conditions are fulfilled . note that the drift depends nonlinearly on @xmath38 , so that the contributions of different waves are not additive . an explicit analytical expression for @xmath30 can be obtained in the case of a piecewise linear wave forcing : @xmath39 ,   & y \in [ \lambda/2 - 2\varepsilon,\lambda/2 [ \ ; \mbox{mod } \;\lambda \\ -   ( 1+b)v ,   & y \in   [ \lambda/2,\lambda-2\varepsilon [   \ ; \mbox{mod } \ ; \lambda \\ - [ ( 1+b)v   -bv(y-\lambda+2\varepsilon)/\varepsilon ] ,   & y \in [ \lambda-2\varepsilon,\lambda [    \ ; \mbox{mod } \ ; \lambda , \end{array } \right . \label{eq26}\ ] ] with @xmath40 . a square wave corresponds to @xmath41 , while @xmath42 represents the triangular - like wave . the final expressions for @xmath30 are rather lengthy and will not be reproduced here . but we performed numerical simulations , which show very good agreement with theoretical results , cf . ( [ fig : py ] ) . note that these results are valid for @xmath43 or for @xmath44 . the analytic results for the square - like wave , corresponding to the limit @xmath45 , are of more interest . they read as follows : @xmath46 \frac{2a^2b}{(1-b)^2v^2-a^2}\right\}\ ; , \label{eq28}\ ] ] for @xmath47 , and    @xmath48 \frac{2a^2b}{(1+b)^2v^2-a^2}\right\}\ ; , \label{eq29}\ ] ]    for @xmath49 , where we introduced the dimensionless variables : @xmath50}\ ; , \nonumber\\   v    & \equiv &   \frac{{\alpha}(1+b)}{2[(1+b)^2{\beta}^2 - 1]}\ ; , \nonumber \\   { \alpha } & \equiv & \frac{2kv\lambda}{a^2},\ \ \ \ \   { \beta}\equiv \frac{v}{a }   \label{eq30}\end{aligned}\ ] ]     stationary probability density for the piecewise linear wave , eq . ( [ eq26 ] ) , with @xmath51 , @xmath52 , @xmath53 and @xmath54 ( the arrows indicate the boundary points of the piecewise linear regions ) . dashed lines represent the theoretical results , while solid lines correspond to simulations of @xmath55 particles evolving for 200 time units . the insets show the two realisations of @xmath56 . [ fig : py](a ) : @xmath57 , @xmath58 . [ fig : py](b ) : @xmath59 , @xmath60.,scaledwidth=90.0% ]    the following expression for the drift velocity is obtained : @xmath61 where @xmath62 is the classical value of the stokes drift velocity . we mention some limiting cases of interest . first , the limit @xmath63 or @xmath64 ( i.e. @xmath65 and/or @xmath66 ) leads to the classical value of the stokes velocity @xmath67 . second , the drift velocity derived in  @xcite is recovered in the white noise limit @xmath68 , @xmath69 , with @xmath70 finite ( @xmath71 ) . finally , the quenched - noise limit @xmath72 ( @xmath73 ) ( i.e. half of the particles , chosen at random , are subjected to a constant external forcing @xmath74 , while the other half to an external forcing @xmath75 ) results in the following mean velocity : @xmath76 this expression reveals the possibility of a _ flux reversal _ : the drift velocity is negative for @xmath77 , i.e. for sufficiently large amplitude .    turning to the general case , we have represented in fig . [ fig : drift ] the results of eq . ( [ eq31 ] ) together with simulations for the square wave for two representative sets of parameter values . for small values of the forcing amplitude @xmath78 one finds that the drift is enhanced ( @xmath79 ) , but the enhancement is reduced upon increasing the frequency of the perturbation . in fact one can prove that enhancement occurs whenever @xmath80 , by noting that @xmath81 and making use of the inequality @xmath82 . on the other hand , a reduction of the drift or even a flux reversal ( @xmath83 ) can be induced at larger amplitudes ( for small @xmath3 , according to eq . ( [ eq36 ] ) ) , while the flux is now found to be an increasing function of the perturbation frequency . in conclusion , the rocking ratchet version for stokes drift reveals two new features , which are to be attributed to the colour ( finite correlation time ) of the applied force : the drift is enhanced by small amplitude perturbations , while a flux reversal occurs at sufficiently large amplitude and large correlation time . while our calculations are limited to the case of dichotomic markov noise , we expect that similar results hold for other coloured noises .    in @xcite it was reported that , in two dimensions , both the direction and amplitude of the stokes drift can change with the intensity of the brownian motion when several waves are present . a similar phenomenon will appear here when one considers , for example , two waves propagating in orthogonal directions @xmath84 and @xmath85 . a dichotomic forcing @xmath86 will induce different drifts along these directions depending on its projections @xmath87 and @xmath88 , as well as on its transition rate @xmath3 . in particular , by varying @xmath3 one is able to induce a significant change in the resulting drift direction , as can be seen in fig . [ fig:2d ] for the simple example of identical square waves . drift velocity according to eq . ( [ eq5 ] ) for a square wave with the same values of the parameters as in fig . [ fig : py ] , except here @xmath89 . symbols represent simulations with @xmath90 particles measured after @xmath91 time units . the lines represent the theoretical value , eq . ( [ eq31 ] ) . [ fig : drift](a ) : drift enhancement . [ fig : drift](b ) : flux reversal.,scaledwidth=90.0% ]    the resulting drift velocity ( in units @xmath92 ; thick arrows ) for two identical square waves ( with the same parameters as in fig . [ fig : py ] ) propagating respectively along the @xmath93 and @xmath94 axes , in the presence of a stochastic forcing with @xmath95 , @xmath96 ( applied along the direction of the thin arrows ) , for different transition rates @xmath3 . the dashed arrow corresponds to the classical limit @xmath97.,scaledwidth=65.0% ]          g. g. stokes , trans . * 8 * , 441 ( 1847 ) ; o. m. phillips , _ the dynamics of the upper ocean _ , cambridge university press , cambridge , 1977 ; j. lightwill , _ waves in fluids _ , cambridge university press , cambridge , 1978 .    a few recent papers from a rich bibliography on the subject : + a. m. bratseth , tellus * 50a * , 451 ( 1998 ) ; s. m. cox , fluid dyn . res . * 19 * , 149 ( 1997 ) ; h. c. graber , b. k. haus , r. d. chapman and l. k. shay , j. geophys . 102 * , 18749 ( 1997 ) ( experimental ) ; j. c. mcwilliams , p. p. sullivan and chin - hoh moeng , j. fluid mech . * 334 * , 1 ( 1997 ) ; m. nordsveen and a. f. bertelsen , int . j. multiphase flow * 23 * , 503 ( 1997 ) ; v. polonichko , j. geophys . res . * 102 * , 15773 ( 1997 ) ; s. a. thorpe , j. phys . oceanogr . * 27 * , 62 and 2072 ( 1997 ) . m. o. magnasco , phys . lett . * 71 * , 1477 ( 1993 ) ; m. bier , phys . a * 211 * , 12 ( 1996 ) ; p. hnggi and r. bartussek , in _ lect . notes in phys . _ , vol . * 476 * , ed . by j. parisi et al . , springer , berlin , 1996 ; r. d. astumian , science * 276 * , 917 ( 1997 ) ; f. julicher , a. ajdari and j. prost , rev . phys . * 69 * , 1269 ( 1997 ) .
we derive the explicit analytic expression for the stokes drift in one dimension in the presence of a dichotomic markov forcing . for small amplitudes of the forcing , 
 the drift is enhanced , but the enhancement is reduced with increasing frequency of the forcing . on the other hand , 
 a reduction of the drift or even a flux reversal can be induced at larger amplitudes , while the flux is now found to be an increasing function of the perturbation frequency . 
 pacs numbers : 02.50.-r , 05.40.+j , 05.60.+w