aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n EPJ manuscript No. (will be inserted by the editor)
Ratchet transport with subdiffusion
S. Denisov a Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany
Abstract.
We introduce a model which incorporate the subdiffusive dynamicsand the ratchet effect. Using a subordination ideology, we show that the resultingdirected transport is sublinear, h x ( t ) i ≃ Jt β , β <
1. The proposed model may berelevant to a phenomenon of saltatory microbiological motility.
Thermal fluctuations alone cannot create a steady transport in an unbiased system. Luck-ily, microbiological realm operates far from equilibrium [1], where directed motion can appearunder nonequilibrium conditions [2]. The corresponding ratchet effect has been proposed asa physical mechanism of a microbiological motility [3]. The nonequilibrium conditions mightinduce another intriguing peculiarity of microbiological transport; namely, the anomalous dif-fusion [4]. The quasi-random wandering at the molecular scale can be characterized by a meansquare displacement (msd), σ ( t ) = h x ( t ) i − h x ( t ) i , which, in many cases, follows a power law, σ ( t ) ∼ t α , α = 1, rather than the linear time dependence for a Brownian particle [5,6]. α > α < x ( t ), and characterizedby the first and the second moments of the same process being therefore strongly conjugated.In the case of the normal diffusion and the superdiffusion, due to finiteness of all statisticalmoments, a directed transport, if any, corresponds to a linear grow of the mean displacement, h x ( t ) i ∼ t , with the corresponding current J = lim t →∞ x ( t ) /t [8].For the subdiffusion a situation is less obvious. In this regime the motion is made up ofperiods of sticking events separated by fast jumps to a new position (see, f.e., Ref. [6]). Formally,the broad power-law distribution of sticking times, ψ ( t ) ∼ t − − β , < β < , (1)leads to the subdiffusion with an exponent α = β [7]. The saltatory molecular motor’s trans-port [9], rapid bursts of directed movements interrupted by pauses of variable duration, hasbeen tracked within a cell by using microinjected fluorescent beads [10]. The abovementionedobservations call for a study of a ratchet transport mediated by anomalous trapping events.In this paper we propose a simple model which naturally incorporates both mechanisms,subdiffusion and ratchet effect, thus bridging two research lines that were so far basically dis-connected from one another. Using subordinated ideology [11] we show that the subdiffusion ina ratchet potential results in the sublinear directed transport, < x ( t ) > ≃ Jt β . The subordinateformalism enables us to reformulate the problem within the circle map’s theory and to derivenecessary and sufficient conditions for the directed current appearance.We start with the model which describes a dynamics of the overdamped particle exposedto the shot-noise, ˙ x = X a j ( x ) δ ( t − t j ) (2)where a j gives the length and the direction of the corresponding step taking place at thetime instant t j . We assume that the length a j depends locally on a periodic potential U ( x ), a e-mail: [email protected] Will be inserted by the editor U ( x + L ) = U ( x ), and a noise ξ , such that a j = − U ′ ( x ( t j )) + ξ ( t j ) . (3)The model (2-3) can be treated as the overdamped limit of the standard model [12]. Thecorresponding process, x ( t ), can be considered not as a function of time t , but rather as afunction of the number of steps, n . The dynamics of the system (2) can be represented as thenoised circle map [12], x n +1 = x j + f ( x n ) + ξ n , (4)where f ( x ) = − U ′ ( x ) stands for the acting force. The process x ( t ) is subordinated [11] to themap (4), such that the time is governed by the linear map, t n +1 = t n + △ t n . (5)In addition, here we assume that (i) the dispersion is finite, h a j i < ∞ , and (ii) the timebetween steps, △ t j = t j +1 − t j , is a random stationary process with the probability densityfunction (pdf) ψ ( △ t ). If ψ ( △ t ) has a finite first moment, h△ t i < ∞ , and a j is independenton x and has a symmetrical distribution (Gaussian, Poisonian, etc) then for the time scale t ≫ h△ t i we get normal Gaussian diffusion, < x ( t ) > ∼ t , which can be described by anordinary Langevin equation [11]. If ψ ( t ) has a divergent first moment, which is the case of thedistribution (1), then we deal with the subdiffusion, where the corresponding msd’s exponentis α = β [7].The role of the sticking time reduces to the fact that the actual number of steps made upto the time instant t fluctuates, so the operational time n is a random function of the physicaltime t . This function, however, is monotonously nondecaying with t and thus allows a causalordering of the events. For the pdf p ( x, t ) one has [11] p ( x, t ) = X n W ( x, n ) χ n ( t ) , (6)where W ( x, n ) is the pdf for the iterated process (4), and χ n ( t ) is the probability to makeexactly n steps up to time t .The asymptotic transport is h x ( t ) i = J ·h n ( t ) i , where h n ( t ) i = P ∞ n =0 nχ n ( t ), and the currentvalue J follows from transport properties of the map (4), h x ( n ) i ≃ Jn . By using the Laplacetransform in the time domain, it can be shown that h ˜ n ( s ) i = ˜ ψ/s (1 − ˜ ψ ), where ψ ( t ) is the pdffor sticking time. For the Poissonian process, ˜ ψ ( s ) = ν/ ( s + ν ), one can easily get h n ( t ) i = νt .For the power-law pdf ψ ( t ) follows that h ˜ n ( s ) i ≈ τ − β s − − β and, finally, h n ( t ) i ≈ t β τ β Γ (1+ β ), sothat h x ( t ) i ≈ Jτ β Γ (1 + β ) t β , (7)where Γ ( x ) is the Gamma function.The subordination approach allows us to separate transport properties, precisely the valueof the generalized current J , which follows from the map (4), from the sublinear asymptotic(7), which is governed by the sticking time pdf ψ ( t ).As an illustrative example we consider here the two-harmonics potential force, f ( x ) = E sin(2 πx ) + E sin(4 πx + θ ) , (8)which transforms the system (4) into a ratchet version of a climbing circle map [12]. Withoutloss of generality, we chose here the Gaussian noise ξ with the dispersion η .Certain symmetries of the Eqs.(2, 8) need to be broken in order to fulfill necessary conditionsfor a directed transport appearance [15]. Suppose that there is a transformation which leaves theequation (2) invariant and changes either the sign of x , x → − x , or invert time, t → − t (but notboth operations simultaneously!). Such a transformation maps a given trajectory into another ill be inserted by the editor 3 E -101 J -0.4-0.200.20.4 x (a)(b) Fig. 1. (a) The bifurcation diagram and (b) the current value J as functions of E for the map (4-8)( E = 0 . η = 0, and the grey line corresponds tonoised case, η = 0 .
1. Note that the current value in the last case is scaled by the factor 40. one, with the opposite velocity. If at least one of such symmetries exists, then contributions ofthe trajectory and its symmetry-related counterpart will cancel each other and the asymptoticcurrent J will be equal to zero [15].In order to fulfill the necessary condition for the dc-current appearance, the following sym-metries of the potential force should be violated: f ( x ) = − f ( − x ) (or U ( x ) = U ( − x )) , (9) f ( x ) = − f ( x + L/
2) (or U ( x ) = − U ( x + L/ , (10)which are the reflection- and the shift-symmetry following to Ref.[15] and termed as ”symmetry”and ”supersymmetry” in Ref.[16]. Both the symmetries are violated when E = 0 and θ = kπ .The symmetry violation is the necessary condition for the current appearance. The currentvalue is determined by microscopic dynamical mechanisms [17]. It is reasonable then to startthe analysis of map’s transport properties from the deterministic limit, ξ = 0. Fig.1 showsthe bifurcation diagram and the current value as functions of the second harmonic amplitude, E . There is an evident relationship between multiple current reversals and different kinds ofbifurcations, which is a general property of underdamped deterministic ratchets [18]. Here, atthe overdamped limit, this relationship can be explained qualitatively. The current value isequal to the average force, J = h f ( x ) i = Z L dxf ( x ) ˆ P ( x ) , (11)where ˆ P ( x ) = P ∞ n = −∞ P ( x + nL ), n ∈ Z , is the reduced invariant density for the map (4-8). Since any critical bifurcation, like a tangent bifurcation (transition from a limit cycle to achaotic regime) [12], always accompanied by drastic changes of the invariant density P ( x ), sucha bifurcation leads to a ”jump” of the current value.The addition of a noise changes the rectification dynamics, but still the system generates anon-zero current (gray curve in Fig.1b). However, the invariant density even at a weak noiselimit converges to the Boltzmann pdf [19], ˆ P ( x ) = N exp( − U ( x ) /σ ), so that the integral inrhs of Eq.(11) goes to zero. Thus, the current decays rapidly with the increasing of the noisestrength (Fig.1b). Will be inserted by the editor t < x > (b) Fig. 2.
The evolution of the ensemble mean displacement h x ( t ) i versus t for the different values ofsubdiffusive exponent: α = 0 . • ), α = 0 . (cid:4) ), and α = 1 . (cid:7) ). Lines correspond to asymptoticsfrom Eq. (12). The parameters are N = 10 , E = 0 . E = 0 . θ = − . π , and η = 0 . We furthermore assume that a time between subsequent kicking events is a random station-ary process with the pdf (1) . The asymptotic displacement can be written as h x ( t ) i = J ·h N ( t ) i ,where the current value J follows from transport properties of the map (4-8), h x ( n ) i ≃ Jn .Formally, we get h x ( t ) i ≃ Jt α . (12)We consider now a large ensemble of noninteracting ratchets. The dynamics of each particlein the operational time frame is governed by the same map (4-8), but the physical time t isdifferent for different particles. Nevertheless, the subordination formalism allows to make casualordering of events [11]. Thus we can calculate the displacement h x ( t ) i by using an ensembleaveraging. In Fig.2 we shown the evolution of the mean displacement, h x ( t ) i , for different valuesof the waiting time exponent α .Spatial distributions for different times are shown on Fig.3. For the case of normal dif-fusion (Fig.3a), α >
1, the evolution follows an universal Gaussian scaling, which has beenfound for weakly underdamped ratchets [20]. This Galilei invariant Brownian process, p ( x, t ) ≃ √ t g ( x − Jt √ t ), where g ( x ) stands for the Gaussian pdf, is very different from the Galilei variantsubdiffusive ratchet regime [7]. The corresponding pdf is asymmetric with respect to its cusp-like maximum which stays fixed at the origin, and the plume stretches more and more intotransport direction (Fig.3b). This behavior is reminiscent of a subdiffusive dispersive transportunder a constant tilting force [21].Summing up, we have considered the new model which yields anomalous ratchet dynam-ics. Transport properties, such as direction and value of the generalized current stem from theratchet-like periodic potential. The anomalous character of a kinetics is governed by the waitingtime pdf. The proposed model provides further contribution to the studies of the microbiologicaltransport. It sets up the link between ratchets and a power-stroke approach to a microbiolog-ical transport, still existing dichotomy [22]. The model can also be useful for an analysis ofexperimentally detected saltatory motility in a cell by using of microinjected fluorescent beads[10]. For the generation of the random variable △ t with the pdf (1) we have used the random variable ξ with the uniform distribution on the unit interval, [0 , △ t = △ t c ξ − /β .ill be inserted by the editor 5 P ( x ) -1e+03 -5e+02 0 5e+02 1e+03 x P ( x ) t x (a)(b) t=500 t=50000t=5000 t=50000 Fig. 3.
Spatial distributions for the ensemble of N = 10 single ratchets for (a) α = 1 . α = 0 . t . The inset shows single particle’s trajectory. Other parameters are as in Fig.2. References
1. S. Phillips and S. R. Quake, Phys. Today , 38 (2006).2. M. O. Magnasco, Phys. Rev. Lett. , 1477 (1993); P. H¨anggi and R. Bartusek, Lect. Notes. Phys. , 294 (1996).3. F. J¨ulicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. , 1269 (1997); R. D. Astumian and P.H¨anggi, Physics Today , 33 (2002); P. Reimann, Phys. Rep. , 57 (2002).4. M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature , 31 (1993); J. Klafter, M. F. Shlesinger,and G. Zumofen, Phys. Today , 33 (1996).5. A. Caspi, R. Granek, and M. Elbaum, Phys. Rev. Lett. , 5655 (2000); A. Caspi, R. Granek, andM. Elbaum, Phys. Rev. E , 011916 (2002); J. Suh, D. Wirtz, and J. Hanes, Biotechnol. Prog. 20,598 (2004).6. I. Golding and E. Cox, Phys. Rev. Lett. , 098102 (2006).7. R. Metzler and J. Klafter, Phys. Rep. , 1 (2000).8. S. Denisov, J. Klafter, M. Urbakh and S. Flach, Physica D , 131 (2002); S. Denisov, J. Klafterand M. Urbakh, Phys. Rev. Lett. , 194301 (2003).9. L. Rebhun, The J. of Gen. Physiol. , 223 (1967); M. Sheetz, Eur. J. Biochem. , 19 (1999).10. M. Beckerle, The J. of Cell Biol. , 2126 (1984); M. Hamaguchi, Y. Hamaguchi and Y. Hiramoto,Devel. Growth and Diff. , 461 (1986); Yildiz et al , Science , 2061 (2003); C. Kural, H. Balchi,and P. R. Selvin, J. Phys: Cond. Matt. 17, S3979 (2005).11. I. M. Sokolov and J. Klafter, Chaos , 026103 (2005).12. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1992).13. P. Reimann, Phys. Rep. , 149 (1997).14. The appearance of the Ito’s form is consistent with the map in Eq.(15).15. S. Flach, O. Yevtushenko, and Y. Zolotaryuk, Phys. Rev. Lett. , 2358 (2000).16. P. Reimann, Phys. Rev. Lett. , 4992 (2001).17. S. Denisov et al , Phys. Rev. E , 041104 (2002).18. J. Mateos, Phys. Rev. Lett. , 258 (2000); J. Mateos, Physica D , 205 (2002).19. P. Talkner and P. H¨anggi, in Noise in Nonlinear Dynamical Systems , V.2 /Cambridge UniversityPress, 1989) 87.20. P. Jung, J. G. Kissner and P. H¨anggi, Phys. Rev. Lett. , 3436 (1996).21. H. Scher and E. W. Montroll, Phys. Rev. B , (1975) 2455.22. H. Qian, Phys. Rev. Lett. , 3063 (1998); T. C. Elson, Biophys. J.82