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the search for the possibility of unidirectional motion in a periodic system without the application of any obvious bias is of current research interest@xcite . such possibility requires the system to be out of equilibrium in order for the process to be consistent with the second law of thermodynamics . several physical models have been proposed to obtain such motion . in all the models noise plays the central role . one of the most discussed models is the one in which an asymmetric periodic potential system is adiabatically rocked@xcite by applying constant forces @xmath0 and @xmath1 at regular intervals of time . one obtains unidirectional motion because in such a system the current @xmath2 . eventhough the time averaged applied force over a period vanishes the averaged current @xmath3 $ ] becomes finite in the presence of noise ( thermal fluctuations ) . moreover , the average current @xmath4 was found to peak at an intermediate noise strength ( or temperature ) . in this model it has been further shown that by suitably choosing the asymmetric periodic potential one may obtain current reversal@xcite as a function of temperature provided the rocking frequency is high . similar results , however , can be obtained in the presence of a unbiased colored noise instead of the oscillating force . there are several other interesting models to obtain unidirectional motion including models where potential barriers themselves are allowed to fluctuate@xcite or models wherein symmetric potential system is driven by temporally asymmetric forces@xcite , etc . the result that thermal noise helps to obtain unidirectional current in a periodic system was quite important . but later on it was pointed out that obtaining mere current does not necessarily mean that the system does work efficiently@xcite . doing work involves flow of current against load and hence one must , in the spirit of the model , obtain current up against a tilted ( but otherwise periodic ) potential system . analysis shows , however , that the efficiency of an adiabatically rocked system ( ratchet ) monotonically decreases with temperature . therefore though such a ratchet system helps extract a large amount of work at an intermediate temperature ( where the current peaks ) the work is accomplished at a larger expense of input energy ; thermal fluctuation does not facilitate efficient energy conversion in this model ratchet system . in a subsequent work@xcite this deficiency was rectified but in a different model wherein the asymmetric potential oscillates in time , instead of its slope being changed ( rocked ) between @xmath5 and @xmath1 adiabatically . in both these models the friction coefficient was constant and uniform in space . the present work makes a detailed study of the rocked ratchet system with nonuniform friction coefficient which varies periodically in space@xcite .    in this work we take the friction coefficient to vary with the same periodicity as the potential but with a phase difference , @xmath6 . the phase difference @xmath6 , the amplitude @xmath7 of variation of friction coefficient , the amplitude @xmath8 of rocking , the load , etc . affect the functioning of the ratchet in an intricate and nontrivial manner . the two of the important results we obtain are : ( 1 ) the efficiency of the adiabatically rocked ratchet shows a peak as a function of temperature , though the peak ( which may or may not exist in case of spatially asymmetric potentials ) position does not coincide with the temperature at which the current peaks , and ( 2 ) the current could be made to reverse its direction as a function of noise strength and the amplitude @xmath8 even at low frequencies of rocking . these attributes are solely related to the medium being inhomogeneous with space dependent friction . it is worth noting that the introduction of space dependent friction , though does not affect the equilibrium properties ( such as the relative stability of the locally stable states ) , changes the dynamics of the system in a nontrivial fashion . recently it has been shown that these systems exhibit noise induced stability , it shows stochastic resonance in washboard potentials without the application of external periodic input signal,@xcite and also unidirectional motion in periodic symmetric potential ( non ratchet - like ) systems.@xcite    in the next section we describe our model and obtain an expression for current and efficiency in the quasi - static limit . in sec . iii we present our results . the nature of correct fokker - planck equation in the presence of space - dependent diffusion coefficient ( inhomogeneous medium ) was much debated earlier . later on the correct expression was found from a microscopic treatment of system - bath coupling . the motion of an overdamped particle , in a potential @xmath9 and subject to a space dependent friction coefficient @xmath10 and an external force field @xmath11 at temperature @xmath12 is described by the langevin equation @xcite @xmath13^{2 } } + \sqrt{\frac{k_{b}t}{\gamma        ( q ) } } \xi ( t ) , \ ] ] where @xmath14 is a randomly fluctuating gaussian white noise with zero mean and correlation : + @xmath15 . here @xmath16 denotes an ensemble average over the distribution of the fluctuating noise @xmath17 . the primes in eq . ( 1 ) denote the derivative with respect to the space variable @xmath18 . it should be noted that the above equation involves a multiplicative noise with an additional temperature dependent drift term . the additional term turns out to be essential in order for the system to approach the correct thermal equilibrium state . we take @xmath19 , where @xmath20 , @xmath21 being any natural integer . @xmath22 is a constant force ( load ) representing the slope of the washboard potential against which the work is done . also , we take the friction coefficient @xmath10 to be periodic : + @xmath23 , where @xmath6 is the phase difference with respect to @xmath24 . the equation of motion is equivalently given by the fokker - planck equation @xmath25 .\ ] ] this equation can be solved for the probability current @xmath26 when @xmath27 = constant , and is given by @xcite @xmath28 in the presence of space dependent friction and the phase lag @xmath29 and in the absence of load @xmath30 even for a spatially periodic symmetric potential . thus when the system is subjected to an external ac field @xmath11 the unidirectional particle flow ( or rectification of the current ) takes place . the phase lag @xmath6 brings in the intrinsic asymmetry in the dynamics of the system . when an externally applied ac force changes slowly enough ( quasi - static or adiabatic limit ) i.e. , when the time scale of variation of @xmath11 is much larger compared to any other time scales involved in the system we can readily obtain an expression for the unidirectional current . for a field @xmath11 of a square wave amplitude @xmath31 , an average current over the period of oscillation is given by , @xmath32 $ ] . this particle current can even flow against the applied load @xmath22 and thereby store energy in useful form . in the quasi - static limit following the method of stochastic energetics@xcite it can be shown @xcite that the input energy @xmath33 ( per unit time ) and the work @xmath34 ( per unit time ) that the ratchet system extracts from the external noise are given by @xmath35 $ ] and @xmath36 $ ] respectively . thus the efficiency ( @xmath37 ) of the system to transform the external fluctuation to useful work is given by @xmath38}{f_{0}[j(f_{0 } ) - j(-f_{0})]}.\ ] ]    henceforth all our variables like @xmath39 are made dimensionless . the amplitude of potential @xmath40 is set to unity as all other energy scales are scaled with respect to @xmath40 . we evaluate @xmath41 , and @xmath37 numerically@xcite using eq . first , we present our results for average ( net ) unidirectional current in a symmetric periodic potential induced by adiabatic rocking and in the absence of load . we emphasize here that to obtain these currents the system must be inhomogeneous . the phase lag @xmath6 ( except for @xmath42 , for which unidirectional current is not possible ) plays an important role in determining the direction and magnitude of @xmath4 . for @xmath43 , in the presence of external quasistatic force @xmath11 and in the absence of load @xmath22 , we have a forward moving ratchet ( current flowing in the positive direction ) and for @xmath44 we have the opposite . for instance , if we examine the effect of friction coefficient close to the minimum of the potential two different situations are encountered depending on the value of @xmath6 . when @xmath45 and @xmath46 the particle experiences lower friction near the barriers in the direction of acquired velocity . the situation is reverse when @xmath47 . from eq . ( 3 ) it follows that in a static force @xmath8 , @xmath48 , hence rectification of current occurs in the presence of external adiabatic drive . moreover , it should be emphasized that the magnitude of current or mobility , in the static field @xmath8 , depends sensitively on the potential and the frictional profile over the entire period . depending on the system parameters the current or mobility can be much larger or smaller than the current or mobility of a particle moving in a homogeneous medium characterised by the space averaged frictional coefficient . in fact , in the intermediate values of temperature and @xmath8 the mobility can be made much larger than their asymptotic limits . this leads to stochastic resonance in a washboard potential in the absence of ac signal @xcite .    in fig . 1 the average unidirectional current @xmath4 is presented as a function of @xmath6 and @xmath12 . for this figure the value of amplitude of square - wave ac field @xmath49 , and the amplitude of frictional modulation @xmath50 . it can be seen that @xmath4 changes sign as a function of @xmath6 and current exhibits either a minimum or a maximum as a function of temperature depending on the value of @xmath6 . at the two limits of temperature ( @xmath51 and @xmath52 ) the currents vanish . this is the case for the value of @xmath31 less than the critical value @xmath53 of @xmath31 where the barrier to motion in either direction vanishes . in our case the critical value of @xmath31 is equal to 1 . this stochastic resonance - like phenomenon has been observed in rocked ratchet systems characterised by asymmetric periodic potentials @xcite . from the contours of the plot it is clear that as we move away from phase shift @xmath54 positive ( or negative ) direction the temperature at which maxima ( minima ) occurs shifts to a larger value and the absolute value of the current at the peak decreases . however , the present symmetric potential situation does not lead to multiple current reversals as a function of temperature in the quasi - static limit of an external drive . the current as a function of @xmath12 and @xmath31 is shown in fig . 2 for @xmath55 and @xmath50 . for smaller fields @xmath8 compared to the critical field @xmath53 the current exhibits a maximum as a function of temperature . as we increase @xmath31 the temperature at which the peak occurs decreases . for fields larger than @xmath31 the current , however , decreases monotonically with temperature because the barrier to motion disappears . in this high field region mobility of a particle decreases with increase of temperature @xcite . however , as we increase the field the net current @xmath4 monotonically changes and saturates to a finite value . this is in contrast @xmath4 vanishes for large @xmath31 in the ratchet subjected to a rocking force in the absence of space dependent friction . in fig . 3 we have plotted @xmath4 versus @xmath56 and @xmath6 for fixed values of @xmath57 and @xmath58 . it can again be seen clearly that the current monotonically varies and saturates to a value given by @xmath59 independent of temperature . this result @xcite follows from the analysis of eq.(3 ) . as expected current reversal can be seen as a function of @xmath6 for large @xmath31 . it can also be verified that the current @xmath4 increases monotonically as a function of the amplitude @xmath7 of the friction coefficient and hence we do not present variation of @xmath4 , etc . , with @xmath7 . we now discuss the efficiency of a symmetric periodic potential system in the presence of space dependent friction driven by an adiabatic periodic field . the efficiency of such a system with uniform friction has been studied earlier and it has been shown that temperature does not facilitate the efficiency @xmath60 of energy conversion in the system @xcite . to calculate @xmath37 we make use of eqs . ( 3 ) and ( 4 ) . in our analysis the load @xmath22 is applied against the direction of net current ( in the absence of load ) . in this situation particle current can flow against the applied load @xmath22 less than some critical value @xmath61 thereby storing energy in useful form . for @xmath62 one can not talk meaningfully the concept of efficiency as the current flows in the direction of the load and hence no storage of useful energy takes place . 4 we have plotted efficiency @xmath37 , input energy @xmath63 and work done @xmath34 ( scaled up by a factor 60 for convenience of comparison ) as a function of @xmath12 for the parameter values , @xmath64 , @xmath65 , @xmath58 , and the load @xmath66 . the figure shows that the efficiency exhibits a maximum as a function of temperature indicating that thermal fluctuation facilitates energy conversion . this in contrast to the case of uniform friction coefficient where @xmath37 decreases monotonically with the increase of temperature in the same adiabatic limit @xcite . it is to be mentioned that the temperature corresponding to the maximum efficiency is not the same as the temperature at which the average current @xmath4 becomes maximum in the absence of load . the temperature at which the extracted work maximizes is not the same as the temperature at which the efficiency becomes maximum for the same parameter values . the input energy increases with temperature monotonically and saturates at the high temperature limit . @xmath37 , @xmath34 , and @xmath33 show similar qualitative behaviour for other parameter values . the above important observation of temperature facilitating the energy conversion is applicable for the spatially symmetric potential . in general in adiabatically rocked systems with frictional nonuniformities the increasing thermal noise need not increase the efficiency . the efficiency is sensitive to the qualitative nature of the periodic potential ( and also to the nonuniformity of friction ) . for instance , asymmetric potential exhibits quite complex behaviour of @xmath37 and @xmath4 . to illustrate this we take @xmath67 ( where @xmath68 lies between -1 and 1 , and is the asymmetry parameter ) . with this potential we discuss three separate cases : case a - system in a symmetric potential ( @xmath69 ) in an inhomogeneous medium ( @xmath70 ) , case b - system in an asymmetric potential ( @xmath71 ) in an inhomogeneous medium ( @xmath70 ) , and case c - system in an asymmetric potential ( @xmath71 ) in a homogeneous medium ( @xmath72 ) .    in fig.5 , we have presented results of @xmath37 versus @xmath12 for all the three cases described above . for this we have taken @xmath73 , @xmath74 , and @xmath55 . for case a @xmath58 , for case b @xmath75 and @xmath58 , and for case c @xmath75 . as discussed earlier for the case a temperature maximizes the efficiency . case c , where the medium is homogeneous , efficiency monotonically decreases with temperature . these observations have been emphasized in earlier literature . the case b , where potential asymmetry and frictional inhomogeneity are present , the efficiency decreases monotonically in this parameter regime . in general whether the temperature facilitates the energy conversion in case b depends sensitively on the system parameters . in some limited parameter range the peaking behaviour is seen as in fig.6 . the parameter values are indicated in the caption . the presence of asymmetry in the potential may or may not help in enhancing the efficiency of the system . this can be seen from figs . 5 and 6 . in all these cases the work done @xmath34 and the input energy @xmath33 show similar qualitative features as shown in the inset of fig . , we discuss the variation of efficiency as a function of load .    on general grounds it is expected that the efficiency too exhibits maximum as a function of load . it is obvious that the efficiency is zero when load is zero . at the critical value @xmath76 ( beyond which current flows in the direction of the load ) the value of current is zero and hence the efficiency vanishes again . in between these two extreme values of load the efficiency exhibits maximum . beyond @xmath77 the current flows down the load and therefore the idea of efficiency becomes invalid . in fig . 7 , we have plotted @xmath37 versus load for all the three cases for chosen values of parameters as mentioned in the figure . in all these cases current monotonically decreases as a function of load . the work done against load @xmath34 exhibits a maximum as a function of load . the load at which @xmath34 shows maximum does not coincide with the load at which @xmath37 becomes maximum . the input energy @xmath33 as a function of load varies non monotonically exhibiting a minimum . however , depending on the case under consideration the value of the load at which the minimum in the input energies observed may be larger than @xmath61 above which efficiency is not defined .    in fig . 8 , we have plotted the efficiency versus the amplitude of the adiabatic forcing @xmath31 for all the three cases . it can be seen from the figure that for the system in an inhomogeneous medium , namely for cases a and b , @xmath37 exhibits a maxima and saturate to the same value in the large amplitude limit . in contrast , for the case c after exhibiting maximum @xmath37 goes to zero . this follows from the simple fact that in the large amplitude limit in the absence of frictional inhomogeneities the net unidirectional current tends to zero . the peculiar feature of saturation of efficiency in inhomogeneous media is related to the fact that the average current saturates to a constant value in the high amplitude limit as discussed earlier . this somewhat counter - intuitive result is typical to inhomogeneous media . having discussed efficiency of energy conversion we now study the nature of net current @xmath4 in the presence of spatially asymmetric potential to examine if current reversals take place in the adiabatic limit in the absence of load . it is known from the earlier literature @xcite that in an adiabatically rocked asymmetric potential ratchet system net current does not exhibit reversals as a function of @xmath12 . in these systems current reversals are possible when the frequency of the applied ac field is large . we show here that in the presence of frictional inhomogeneities in addition to asymmetry in the potential one can observe current reversal as a function of thermal noise . in fig . 9 , we have plotted the magnitude of net current @xmath4 versus @xmath12 for all the three cases a , b , and c. the corresponding parameter values are mentioned in the caption of the figure . the cases a and c do not exhibit current reversal . this is a general result independent of parameter values . however , in case b current reversal is observed . to obtain current reversal both asymmetry in potential and nonuniform friction coefficient are essential , that is , current reversals arise due to the combined effect of @xmath6 and @xmath68 . moreover , it should be noted that to observe current reversals the parameter range should be such that the net current in case a is in the opposite direction to that in case c. for the case b for which current reversal is observed , the plot of efficiency separates into two disjoint branches as the load should be reversed keeping the magnitude same when the current reversal takes place . in the presence of @xmath71 and @xmath70 , where the current reversals are observed , the efficiency as a function of temperature is less than the maximum value of efficiency in either of the two cases a and c. that is , @xmath78 $ ] . to further analyze the nature of current reversals , in fig . 10 , we have plotted @xmath4 as a function of @xmath6 and @xmath12 for fixed values of @xmath79 , @xmath58 , and @xmath80 . from the contour plots it is clear that as a function of @xmath6 the current reverses sign twice in the intermediate temperature range . thus the current exhibits reentrant behaviour as a function of phase @xmath6 which is special to the case b. as we decrease the asymmetry in the potential the @xmath81 contour line shifts towards @xmath82 thereby enhancing the domain of current reversal to a lower value of temperature as a function of @xmath6 . as a function of temperature the current reversals occur in a definite range of phase @xmath6 which , in turn , depends on other material parameters . the qualitative behaviour of @xmath4 remains unaltered for different @xmath31 as long as @xmath31 is less than the critical value .    in fig . 11 , @xmath4 is plotted as a function of @xmath12 and @xmath8 for @xmath79 , @xmath83 , and @xmath58 . as opposed to the case of symmetric potential ( fig . 2 ) , currents in the small temperature regime do exhibit maxima and then saturate to a constant value as noted earlier . as a function of @xmath12 the current exhibits similar features as in the case a ( fig . figure 12 , shows @xmath4 as a function of @xmath8 and @xmath6 for @xmath84 , @xmath79 , and @xmath58 . as opposed to case a ( see fig . 3 ) @xmath4 shows current reversal as a function of @xmath8 in the range @xmath85 . however , in the asymptotic limit of @xmath8 current saturates to a value @xmath59 independent of the value of the asymmetry parameter @xmath68 . from the contour plot it follows that as a function of phase @xmath6 we observe the reentrant behaviour of current at high values of @xmath8 . as we decrease @xmath68 the @xmath81 contour shifts towards smaller values of @xmath31 thus making it possible to observe the double reversals at even smaller values of @xmath31 . this reentrant behaviour as a function of @xmath6 and the current reversal as a function of @xmath31 is very specific to the case b alone ( compare fig . thus we conclude from our studies that the dynamics of a particle in an inhomogeneous medium is rich and complex . in the presence of adiabatic forcing and asymmetry in the potential current reversals can be observed as a function of @xmath12 and @xmath31 . and depending on the system parameters thermal fluctuations facilitate the energy conversion . the above behaviour can not be seen in the homogeneous medium in the same adiabatic limit . however , it is possible to observe these in homogeneous media in the presence of finite frequency ac drive ( nonadiabatic regime ) . this seems to suggest that @xmath6 may play the characteristic role of frequency in our model in the absence of nonadiabatic ac drive . this has been noted earlier in the context of observation of stochastic resonance phenomena in inhomogeneous media in the presence of static tilt alone @xcite . as a function of phase @xmath6 , we observe reentrant behaviour for the current which arises because of interplay between asymmetry , inhomogeneity , thermal noise , and strength of the adiabatic forcing . some of the phenomena can be understood at best at a qualitative level only . the effect of nonadiabatic forcing may be of further interest . work on this line is under investigation . mcm acknowledges partial financial support and hospitality from the institute of physics , bhubaneswar . mcm and amj acknowledge partial financial support from the board of research in nuclear sciences , dae , india . p. hanggi and r. bartussek , _ nonlinear physics of complex systems - current status and future trends _ , lecture notes in physics , vol . 476 , ed . by j. parisi , s.c . mueller , and w. zimmermann ( springer , berlin , 1996 ) , pp . 294 - 308 .                                                                  fig . the net current @xmath4 as a function of @xmath6 and @xmath12 , for parameter values @xmath86 , @xmath58 . in the base plane contour of surface plot @xmath87 are given , dotted line indicates @xmath88 contour . efficiency @xmath89 and @xmath34 as a function of @xmath12 for @xmath65 , @xmath86 , @xmath50 , and @xmath90 . @xmath34 has been scaled up by a factor @xmath91 to make it comparable with @xmath37 and @xmath33 . y - axis is in dimensionless units .            fig . 9 . current @xmath4 versus temperature @xmath12 for ( i ) case a ( @xmath92 , and @xmath58 ) , ( ii ) case b ( @xmath97 , @xmath50 ) , and ( iii ) case c ( @xmath97 , @xmath94 ) , for fixed @xmath64 , @xmath99 , and @xmath100 .    fig . the net current @xmath4 as a function of @xmath6 and @xmath12 , for parameter values @xmath86 , @xmath58 , and @xmath101 . in the base plane contour of surface plot @xmath87 are given , dotted line indicates @xmath88 contour .      fig . @xmath4 as a function of the amplitude of the rocking force @xmath31 , and phase @xmath6 , for @xmath57 , and @xmath101 . in the base plane contour of surface plot @xmath87 are given , dotted line indicates @xmath88 contour .
we present a detailed study of the transport and energetics of a brownian particle moving in a periodic potential in the presence of an adiabatic external periodic drive . 
 the particle is considered to move in a medium with periodic space dependent friction with the same periodicity as that of the potential but with a phase lag . 
 we obtain several results , most of them arising due to the medium being inhomogeneous and are sensitive to the phase lag . when the potential is symmetric we show that efficiency of energy transduction can be maximised as a function of noise strength or temperature . however , in the case of asymmtertic potential the temperature may or may not facilitate the energy conversion but current reversals can be obtained as a function of temperature and the amplitude of the periodic drive . 
 the reentrant behaviour of current can also be seen as a function of phase lag . 
 + pacs number(s ) : 05.40+j , 05.60+w , 82.20mj    = 0.3 in = 0.6 in