We consider the motion of a overdamped Brownian particle in periodic
asymmetric potential with space dependent friction coefficient. In the presence
of external time periodic forcing, the system shows multiple current reversals
on varying the amplitude of the external forcing and the temperature of the
thermal bath. In the adiabatic regime we find a single reversal of current as a
function of noise strength which can only be accounted due to the presence of
space dependent friction coefficient. For very large forcing term, the current
does not go to zero, instead it asymptotically tends to a limiting value
depending on the phase shift between the potential and the friction. This fact
plays an important role in obtaining multiple current reversals.Comment: 4 pages, latex/revtex, 4 eps figures. Proceedings of CMDAYS2K, held
  at Guru Ghasidas University, Bilaspur, Chattisgarh, India, Aug 29-31, 2

Dan, Debasis

Jayannavar, A. M.

Mahato, Mangal C.

English

arXiv

Jayannavar, A M

Mahato, Mangal C

IACS Institutional Repository

\drnJ-PhysJ6k(\),51-m (2002)U P  Aan international journalAC driven thermal ratchetsDebasis Dan*, A M JayannaviirInsiiluic ol Physics, Sachivalaya Maip. Bhubancswar-751 ()()*», Orissa. IndiaandMangal C MahaloDcpaninciil of Physics, Guru Gluisulas Uiiiversily, Bilaspur-4^'/5 009, Madhya Pradesh, IndiaH-nuiil danfei’iopb res inhslracl We consider the motion of a overdamped Brownian paiticle in a periodic asymmeinc poteniial with space-dependent Iriciionrlliticni In the presence of external lime periodic foiciny. the s>Mcm shows miiliiple current rcveisals on varying the amplitude of the external umi' and the tcm|ieiatiitx.' of the thermal bath In the adiabatic regime, we find a single reversal ol ciirien* as a lunction of noise strength which can ilv be .iceounied due to the presence ol space dependent friction coefficient For very large forcing, the current does not go to zero, instead, it MnpiDiicjlIy (ends to a limiting value depending on the phase shift between the potential and the Iriction This fact plays an important lole in )i.lining inuliiple current reversalscvwords Brownian pariule, periodic asymmetric potential, space dependent friction coefficient\( 'S  Nos. 40 -a, 0*i bO k S2 20 -wIntixMiuctionc L 'tif ic iilio n  o f  noise leading to unidirectional motion in ratchet > sicms has been an active field of research over the last decade.1 ihcsc systems, directed Brownian motion ol particles is kluccd by nonequilibrium noise in the absence of any nei uiciosLopic forces and potential gradients. Several models have I'cn proposed to explain this transport mechanism under arious nonequilibrium situation, like (a) Fla.shing ratchets 11,2|, hcMcm the particles experience a lluctualing energy prolile, (b) ockiMg ratchets [3,4|, where the Brownian particles experiences spatially uniform, time peritxlic uniform force, F(i -l- T) = h( t ) ,) Diffusion ratchets (51, wherein particles are driven by a lime Cl iodic diffusion coefficient DU) -  Pit  + r) and (d) orrclulion ratchets [6], where particles arc driven by spatially njlorm but temporally correlated force ^(f) of zero average. In lose systems, to gel a unidirectional current, cither spatially symmetric periodic potentials or time asymmetric external lorces re necessary. The recent burst of work on ratchet systems is loiivated m part, to explain the unidirectional transport ol lolccuhu' motors in biological systems and a new prospect ol t>vcl techniques for separation or segregation of particles ofCl'W sponding Authormicro-mclcr size. This technique for particle separation is based on the properly of current reversal in ratchet by varying the sliength of thermal noise, amplitude ol the iorcing, size ol the particle or any other relevant variable in the problem.In the area of Brownian rectifiers or ratchets, the study of current reversal has become a subject by itself. Barlussck [41 showed the occurrence of current reversals in rocked thermal ratchet as a function of amplitude of rocking force as well as the temperature of thermal bath. Multiple current reversals have also been shown in the deterministic limit of these ratchets when the inertial term is taken into account. However, these multiple current reversals in inertial ratchets arc not robust in the presence of noise. Beside rocking ratchets, current reversals have also been observed in flashing ratchets [7,2]. We in this work, report that m u l t i p l e  reversals can be achieved even in rocked o v e r d a m p e d  ratchets in the presence of space dependent mobility, as a function of the noise strength and amplitude of the rocking force. In the deterministic overdamped case, we get current reversal as a function ol the amplitude of nicking force. It is to be noted that systems with space dependent friction are not uncommon. Brownian motion in confined geometries, for examples, show spacc-depcndcnl friclion[81. It is believed that molecular motor proteins move close along the pcritxlic structure/s*k / n / w n  T A5KDehasis Don, A M Jayamumv ami Mangal C Mahatoo t m ic ro tu b u le s  a n d  w i l l  th e re fo re , c x p c n e n c c  a p o s itio n  dependent m o b ility  I9J. F ric tio n a l inhom ogcneitics are com m on  in  super lattice  structures and Josephson ju n c tio n s  etc [1 0 ]2. ModelWe consider an overdamped Brownian particle moving in a asymmetric potential V(x) with space-dependent friction coefficient 77( A ) under the influence of external force field f  (/) at temperature T. Throughout our analysis, we lake the ratchet potential \/(a ) = --^(sin (2 ;r a)-I-“ sin (4;r A)j . Here, F is the asymmetry parameter with values taken in the range ( ) < / , / < ! ,  friction coefficient 77(A) =  ih)( \ -  Asin(27T a +<p)) , \ X\ <\  . <p determines the relative phase shift between friction coefficient and potential. The lorcing term is taken to be F{r) =A sm (ojt y O) , (o) = , where r is period of force) Withoutany loss of generality, 0 is taken to be zero, 'fhe Langevin equation for this system in the over-damped limit is given by [IU2]'(A) ^ h , T(A)) '  V ^ A )4 ( 0 . (I)w h e re to ) IS Gaussian thermal noise with correlation ( ^ ( t ) ^ ( f ' )) -  25(7 “  /') /niclHikker Planck Equation (PPLi) [111 corresponding to cq. (1) is given by( ) P( x J ) _  rl I Dt  dxr i ( x)k j [  + ( V ' ( v ) - F ( M ) l / ^ ( A , 7 ) .  (2)a XHere, Fix, t ) is the probability density at position a at lime 1  Eq. (2) can he recast to a continuity equation —^ ^ - =r  ^wherefixi(-t,7) = - 1rUx)(V'(a)-F (7 )) + A«T \P{xJ)  (3) a xIS the probability current density. Since the potential and the driving force have spatial and temporal periodicity respectively, therefore 7 ( a ,7) = y(A +1,7 + T) 14,13). The net currenlj in the system is given by j  = lim, -j: dt J( v, 7) dx . It shouldbe noted that for symmetric potential and A = (), ./ = 0 . Rectification of current is possible if the potential is either asymmetric or A ^ 0 with 1141. It is important to notethat in lhe.se systems, rectification is due to a symmetry breaking as the potential is asymmetric and A = 0. For the case when the potential is symmetric and A ^ 0, the symmetry breaking results from the dynamics of the system, it is the space-dependent mobility that breaks the symmetry in this nonequilibrium problem and the magnitude of the symmetry breaking is related to the phase shift between the friction and potential profile, j  is independent of the initial phase 0 of the driving force. Wesolve the FPE numerically by method o f  finite difference and calculate the current j.  All the physical quantities such ajj /, T, A,01 are in dimensionless units [13-15].3. Results and discussionIn the Figure (1 A), the average current J is plotted as a function | of temperature T for different values of o i . Here the asymmeir) j parameter ju = 1.0 and A = 0.0 .Figure 1. Mean currcnl / v\ Icinpcralurc / loi (a) A -  0 . / /=  1 0 ami (b A- 0 1 ,  7 i - 0  and 0 -O.27r Noic ihcic i.s no curicm reversal when ilic polcnlial IS symineirieIn this case, the current reverses its sign (only omr) tui' frequencies sufficiently large as shown in the to -  4.0, ai -51 1  case. In the absence of asymmetric potential and pre.sencc I'l space dependent friction (A = 1.0), there is no current rcvcisiil irrespective of ai and pha.se shift 0 as shown in Figure (IHi Hence, asymmetric potential is cs.scntial lor curreni revcisal However, the direction of the current depends on the phase la<:0 . Separately in both these cases, absolute value of curicni exhibits a maxima as a function of T, reminiscent of slochaUk resonance phenomena. In a purely asymmetric case (A -0) current vanishes rapidly when T exceeds the temperalurL associated with the barrier height Whereas, in the symmcliic ca.se due to space-dependent friction, absolute values of currenh arc significantly higher and decay slowly to zero in the large temperature regime. Naturally, in the presence of both asymmetn and space-dependent friction for the case under study, the lov temperature regime is dominated by the effect of asymmetn while the high temperature regime is dominated by space dependent friction. From this, one can qualitatively explain ihc current reversals from positive to negative side as a function of temperature even in the adiabatic limit. Hence, in the presence of both space-dependent friction and asymmetric potential, we can have current reversal with T even in the adiabatic re i^nit for a suitably chosen value of 0 as shown in Figure (2A), where we have plotted j  vs T with ft = 1.0, A = l.O and 0 = 0.2;r fof different values of co .AC driven thermal ratchets 59In the presence of  finite frequency drive, there are two current crsals as shown in the Figure. This phenomena of twice rent reversal with temperature T is the Ibremost feature of system, previously unseen in any overdamped system, cn the phase difference 0 is such that the current due to cc dependent friction alone is in the same direction as that he current due to potential asymmetry only, then we do not c cuncnl reversal in the adiabatic case as shown in Figure ) where (p = \2 n  , though a single current reversal due to le frequency drive may be present. In all, the cases studied tar we have observed that the current reversal do not lake :c above a critical frequency of driving, which in turn, lends on 0 and other parameters in the problem.ure 2. The mean cuncnl / vs icmpcraUirc / Tor 0 = O2n, /\ = 0 .S and 01 The drivinji frct|ucncics arc co -  M),4() and .SO The iighl hand .■ Iigiitc shows current j  vs I for 0 ~ I 2;rMultiple current reversals can also be seen when the iplilude (i4) of the forcing term is varied in a suitable parameter :irne ofour system. In Figure 3, the plot of / versus A is shown diUcrcni values of (o , keeping A, 0 and T fixed at 0.1,0.88 nand 0.05 respectively. For co = 4.0 curve, we can see as many as four current reversals. For very large value of A, the current asymptotically goes to a constant value depending on (he value of (p, as was previously shown for the adiabatic case f 14J. The inset in Figure 3 shows current reversal even for the deterministic case also.In addition, it exhibits an interesting phenomena of current quantization and phase locking. These features are washed out with increasing noise strength.In Figure (4) we have plotted j  versus A for 0 = 1.2;r . There is no current reversal in the adiabatic regime for both deterministic as well as finite temperature case. The observation of multiple current reversals can be attributed to a cooperative interplay between the spatial asymmetry of the potential, the friction inhomogcneily and the finite frequency drive. Depending on the system parameters we may have multiple euiTent reversal or no current reversal at all (see Figures 3 and 4).Rurt* 3. The mean current j  with amplitude A of (he forcing lerm for -0K8;r. 7 = 0.05 and A =0.1 with w'= 3 0. 4.0, 5.0. The inset shows i^ i^ versal of deterministic current v.r the amplitude of the forcing.Figure 4. Mean currcnl / vs lemperalurc /’ for (a) A = 0 . /v = 10 and lb) A =0.1 . /i = 0 and 0 = O2;r Note (here is no current reversal when the poicnlinl is symmeiric4. ConclusionIn conclusion, we have studied the transport properties of overdamped Brownian particles moving in an asymmetric potential with spacc-dcpendcnt friction coefficient and rocked by periodic force. We observe multiple current reversal with both temperature T and /I, the amplitude of the rocking force in the presence of finite frequency driving. Current reversal also occurs in the deterministic adiabatic regime for suitable values of phase difference 0 . All the above results can be understood qualitatively. The space-dependent friction plays an active role in these systems in determining several counter intuitive phenomena [14-16] observed in the transport processes.R e f e r e n c e s[I] P Hdnggi and R Barlus.sek Nonlineat Physics of Complex System -  Current Status and Futuiv Trends 476 (Berlin ; Springer) 294 (1996)6 0 Dehasis Dan, A M Jayannavar and Mangal C Mahato12 ]  A A jd a r i  a n d  J P rost  hAUOphys i.cil 3 2  373  ( 1 9 9 5 )[ 3 1 M O  M a g n a s c o  Phys Kev le n  71 1477 (1 9 9 3 )14 )  P Bar lu s. sck , P H a n g g i  a n d  J G  K is sn c i  hAuopliys Lett. 2 8  4 5 9  ( 1 9 9 4 )[51 P Rcimann. R Bartussck, R Haussicr and P Manggi Ph\.\ Lett, A215 26 (1996)f 6 |  C  R O otT in g ,  W  H fjr s ih i-m kc  an d  I R m r d a n  Plivs Rev Lett 7 2  2 9 8 4  ( 1 9 9 4 )(7 )  M  B ic r  and  R I )  A s t u m ia n  Pfi\.\ Rev Lett 3 2  3 73  (1 9 9 5 )|K |  L ul P F-aULheux  a n d  A J L i h c h a h c r  Phys Rev K 4 9  5 1 5 8  ( 1 9 9 4 )|9 1  Roll  II LucliMiigLM P h u  Rev F  6 2  2 7 2  (2()()0)[10] C M Falco Am. J Phys 44 733 (1976)I I I ]  M C Mahalo, T P Parcck and A M Jayannavar hit, J, Mtxi, Ph\^  BIU 3857 (1996)[12] A M  Jayannavar and M C Mahalo Pramana J Phys. 45 (1995)[13] H Risken The Fokkei Phimk Equation (Berlin . Springer Vcrbgj (1984)[14] [) Dan. M C Mahalo and A M Jayannavar hit. J Mod Phys B on press), Cond-Mai, ()()06244[15] I) Dan, M C Mahalo and A M Jayannavar Phys. Lett, A258 2P (1999)[ I6 | D Dan, M C Mahalo and A M Jayannavar Phys. Rev E60 6421 (1999)

AC Driven Thermal Ratchets

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6655/1/Ac%20Driven%20Thermal%20Ratchets_Debasis%20Dan.pdf

AC driven thermal ratchets

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