Overdamped directed motion of Brownian motors in a spatially periodic
system, induced by Poissonian fluctuations of various statistics and driven
by thermal noise, is investigated. Two models of asymmetric as well as two
models of symmetric Poissonian fluctuations are considered. Transport
properties in dependence upon statistics of fluctuations imposed on the
system are analyzed

Czernik, T.

Niemiec, M.

Łuczka, Jerzy

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Brownian motors driven by Poissonian fluctuations  Author: T. Czernik, M. Niemiec, Jerzy Łuczka  Citation style: Czernik T., Niemiec M., Łuczka Jerzy. (2001). Brownian motors driven by Poissonian fluctuations. "Acta Physica Polonica. B" (2001, no. 2, s. 321-331). BROW NIAN M O TO RS D RIV EN  BY POISSONIAN FLUCTUATIONS*T .  CZERNIKDepartment of Applied Mathematics, Academy of Economics 1-go Maja 50, 40-226 Katowice, PolandM .  N i e m i e cInstitute of Physics, Opole University Oleska 48, 45-052 Opole, Poland e-mail: m ariusz. niemiecQuni . opo le . p iAND J . LUCZKAInstitute of Physics, Silesian University Uniwersytecka 4, 40-007 Katowice, Poland e-mail: luczkaQ us.edu.pl(Received November 15, 2000)Overdamped directed motion of Brownian motors in a spatially periodic system, induced by Poissonian fluctuations of various statistics and driven by thermal noise, is investigated. Two models of asymmetric as well as two models of symmetric Poissonian fluctuations are considered. Transport properties in dependence upon statistics of fluctuations imposed on the system are analyzed.PACS numbers: 02.50.Ey, 05.10.Gg, 05.40.-a1. In tro d u c tio nA subject which has lately been gaining interest is transport of Brow­nian particles (motors) moving in spatially periodic structures. In such systems directed m otion of particles can be induced by zero-mean deter­ministic and /o r random  forces (the so-called ratchet effect) [1]. W hy is this* Presented a t the XXIV International School of Theoretical Physics “Transport Phenom ena from Q uantum  to  Classical Regimes”, Ustron, Poland, September 25- October 1, 2000.subject so fascinating? For a t first glance it is counterintuitive. One expects th a t imposition of zero-mean forces on the system yields zero-mean reaction of the system. Now, the explanation of the transport phenomenon in such systems is rather obvious: the system is out of equilibrium and the detailed balance does not hold! The conditions for transport to  occur in periodic systems are known. The sym m etry should be broken. W hat kind of sym­m etry should be broken is a secondary question. For example, the reflection sym m etry of the spatially periodic structure can be broken [1] or the sta­tistical sym m etry of fluctuations can be broken [2]. The sym m etry can also be broken by correlation of various degree of freedom [3] and by correlation of various sources of noise [4]. Because real systems are rarely symmetric, the occurrence of transport should be rather a generic phenomenon than  exceptional and specific one.In this paper we study directed current of non-interacting particles in the system subjected to a spatially periodic potential. Particles are driven by both  zero-mean therm al equilibrium fluctuations and zero-mean nonequi­librium fluctuations modeled by Poissonian white shot noise. The problem sta ted  is not new since it has been previously studied [5,6] and it has been shown th a t indeed the ratchet effect can occur and preferential direction of Brownian m otion is induced by nonequilibrium fluctuations. However, our objective is to investigate the influence of various statistics of Poisso­nian fluctuations on properties of transport and eventually to find universal properties of transport in such systems.The article is organized as follows. In Sec. 2 we define a m athem ati­cal model of Brownian motors (a ratchet system) with all characteristics of its ingredients. In Sec. 3 we present two models of asymm etric Poissonian fluctuations and equations which determine the stationary  probability dis­tribution P ( x ) and the stationary  probability current J  of Brownian motors. Sec. 4 is devoted to symm etric Poissonian fluctuations. In Sec. 5 we analyze transport properties of Brownian motors and include main conclusions.2. M od elWe consider an ensemble of non-interacting Brownian particles moving in a one-dimensional spatially periodic potential V( x )  = V ( x  +  L)  of pe­riod L  and of the barrier height Vq = Vmdx — Vmm, and driven by random forces. The dynamics of the system is modeled by an overdamped stochastic Langevin equation. The equation of motion in the dimensionless form is (the dimensionless variables are discussed in detail in [7])=  /  ( ® )  +  r  (*)  +  v ( t ) , (1)where x  = x / L  is the dimensionless position, t  = t / r  the dimensionless time, r  =  j L 2/V q the characteristic time, and 7  is the friction coefficient. The determ inistic rescaled force=  < 2 >and V( x )  =  V ( x  +  1 ) is a rescaled periodic potential of unit spatial period and of unit barrier height. The stochastic force F(t )  is Gaussian therm al equilibrium noise of the first two moments<r(i)> =  0 , i m r ( s ) )  =  2DT 8(t -  s ) , (3)where the rescaled noise strength D t  = LbT / V o, fee is the Boltzmann con­stan t and T  tem perature of the system. Let us take a note th a t D t  is a relation between therm al energy of fluctuations and activation energy of the particle from the bottom  to the top of the potential V ( x ) .The random  force r/(t) models nonequilibrium fluctuations and is chosen to  be Poissonian shot noise [8 ]OOr](t) = Y 2  Zi6(t -  ti) -  \ { z i ) , (4)i=—oowhere ti are random  instants of 8 impulses, characterized by the Poissonian counting process n(t)  with the param eter A. It means th a t the probability for appearing of k  impulses in the time-interval [0, t) is given by the Poisson distribution, namely,P(n( t )  =  k) =  exp(—At ) . (5)k\The param eter A determines a mean number of the 8 impulses per unit time (a mean frequency of impulses). The amplitudes {zi}  of the 8 impulses are m utually independent random  variables and independent on the counting process n(t).  The amplitudes {Zi} are d istributed according to the common probability density p(z).  The process r/(t) is symmetric if the distribution p(z)  is symmetric, i.e. when p(z)  =  p(—z).  Otherwise, it is asymmetric noise. In the la tte r casea = X (zQ A 0 (6 )corresponds in (4) to  the negative value of the bias of Poissonian noise be­tween 8 spikes. The process r/(t) is a white noise (but nonequilibrated noise) with an average and correlation function given by(r](t)) =  0 , {n(t)r)(s)) = 2Ds S(t -  s ) . (7)The noise intensity reads:A (z2)D s  = ■ (8)The evolution equation for the probability distribution P ( x , t) of the process x(t )  has a form of the continuity equation [6],^ P ( x , t )  = —^ J ( x , t ) ,  (9)where the probability currentJ ( x , t )  = [f (x) — A (zi)\ P(x ,  t) — D T - ^ P ( x ,  t)OO 2+A J p(z) J P ( x  — y , t )  dy dz . (10)-o o  0The probability density P ( x , t) has to  obey the following conditions310 +  1P ( x , i ) > 0 .  P  (x. 0) =  P  (x + 1.0) / p ( x . i ) < i x  =  1, (11)310for arb itrary  x q .  In the stationary  state , when t —> oo,J  =  lim J ( x , t ) =  - /> /  / ’ (x) +  [f (x)  — A (zi)]P(x)t —¥ 0 OOO 2+A J  p(z) J  P ( x  — y ) d y d z , (12)— oo 0where P(x)  and J  are steady-state probability density and probability cur­rent, respectively; the prime denotes a derivative with respect to x.  The current J  is the most im portant characteristics of transport of Brownian motors. The relation between the steady-state (dimensionless) averaged ve­locity (v) of Brownian motors and the probability current is simple [7,9], namely,{v) = {x) = J .  (13)The dimensional averaged velocity (v) is given by the relation [7,9](v ) =  v q J ,  (14)where the characteristic velocity vq = L / t  =  Vo/ 7 L  is determined by three quantities: the barrier height Vo and period L  of the potential V( x )  and the friction coefficient 7  (which, in turn , depends on viscosity of the system in which Brownian motors move as well as on linear sizes of the motors).3. A sy m m etric  P o isso n ian  flu c tu a tio n sIn this section we specify asymm etric Poissonian fluctuations [5]. For asymm etric fluctuations, the probability density p(z)  of amplitudes {Z{} of S kicks is asymmetric. We consider two examples of such densities. Both are special cases of the Gam m a distribution [8], namely,p(z) = pi (z)  = A ^ l 0 ( z )  exp(—z / A ) , A > 0 ,  (15)andp(z) = p2(z) = A ^ 20  (z) z e x p  (—z / A)  , A > 0 ,  (16)where 0 ( z )  is the Heaviside function. For the first distribution pi(z) ,  the first two moments read(zi) = A ,  ( z2) = 2A2 . (17)For the second distribution p2(z),(zi) =  2 A , ( z 2 )  = 6A2 . (18)In these two cases, the amplitudes {Z{} are positive: from tim e to  tim e the S impulse kicks the particle to  the positive direction of x.  On the other hand, the negative bias, a =  A(zQ, pushes the particle to  the negative di­rection (let us remember th a t in sum m ary (r](t)) =  0). The main difference between these two distributions is (non)monotonicitv and in consequence qualitatively different most probable values of amplitudes {zi}  of S impulses. In the first case, very small amplitudes (close to zero) are more probable. In the second case, the amplitudes Z{ = A  are the most probable. Therefore, by m anipulation of the param eter A,  we have hoped for significant difference in transport properties for these two cases.The equation determining the stationary  probability distribution P(x)  = Pk(x),  k =  1,2, and stationary  current J  =  A; =  1,2, is Eq. (12).It is an integro-differential equation and it is very difficult to handle it. This is why in each case we convert it to the form of differential equations: In the first case the result isJ i  =  — D p A P l (x) — [Dp +  D s  — A f  (x)} Pl (x)+ [ f (x)  + A f ( x ) ] P i ( x ) .  (19)In the second case,J O 111 ,  sJ2 = —A  D t P2 (x)  +  A \ A f  (x) -  -  2DTj  P2 (x)In order to  obtain these equations, we have to change the integration order in the last term  of Eq. (12). Next, observe th a t Pk(z) (k =  1,2) obeys an ordinary differential equation of constant coefficients.4. S y m m etric  P o isso n ian  flu c tu a tio n sIn this section we introduce symm etric Poissonian fluctuations [6 ]. For symm etric fluctuations, the probability density p(z)  of amplitudes {zi}  of S kicks is symmetric. We present two examples of such distributions: The exponential distribution,p(z) = p3(z) =  0.5 A - 1  e x p ( — |z |/A ) , A > 0 ,  (2 1 )and a particular Gam m a distribution, namely,p (z) = p4(z) =  0.5 A - 2  \z\ exp ( — |z |/A ) , A > 0 .  (2 2 )The mean value of amplitudes (zQ =  0 and the second moments (z2) are the same as in the previous corresponding cases. The equations determining the stationary  probability distribution P(x)  = Pk(x ), k =  3 ,4  and stationary current J  =  Jfe, fc =  3 ,4  have the formJ 3 =  A2/> //'.. (x) — A2 [ f ( x)P3 (a;)]-  (DT +  D S ) P3 (x)  +  /  (x) P3 ( x )  (23)in the case (2 1 ) andJ4 = ^ A 4D T P {4 ] (x) + A 4 [f ( x ) P 4 (x)}{4]+ A 2 ^2D T +  P4(3) (x) -  2A2 [f (x) P4 (a;)](2)-  ( D t  +  D s ) P4(1) (x) +  /  (x) P4 (x)  (24)in the case (22). The superscripts (n), n  =  1,..., 5 denote the n-order deriva­tive with respect to  x.  One can notice th a t order of the differential equation for P(x)  depends strongly on the form of the probability distribution of am­plitudes {zi}.  E.g. for p4(z), the differential equation is of the 5-th order. It is interesting th a t from all these differential equations we can determine both  Pji(x) and J&, k =  1 .1 . and the same conditions ( 1 1 ) are suffi­cient to solve them  uniquely. In general, these equations cannot be solved by analytical means. Nevertheless, they can be solved numerically for anarbitrary  form of the potential V(x) .  However, exact analytical results can be obtained for a piecewise linear sawtooth-like potential. We choose the simplest form of such a potential, namely,v(x)  = f - 2f m >  x e [ - l / 2 , k ]  m o d i ,\  x e [ k ,  1/2] m o d i ,where k E ( — 1 /2 ,1 /2 ) determines the asymm etry of the potential: For k = 0 it is reflection-symmetric, i.e., V(x )  =  V ( —x); for k ^  0 the reflection symm etry of V(x)  is broken. In Fig. 1, we present V(x)  in its graphical form.Fig. 1. The rescaled periodic potential V(x) =  V(x  +  1) of asymmetry determined by k E (-1 /2 ,1 /2 ).5. A n a lysisWe analyze the stationary averaged velocity of Brownian motors or, equivalently, the steady-state probability current, J  =  (v), which is deter­mined by Eqs. (19), (20), (23) and (24) respectively. In order to solve these equations, we can proceed along the same way as in [5]. The m ethod of solution of equations like considered here is also presented in [9, 10]. In this m ethod, the probability density P(x)  and current J  is determined by a nonhomogeneous system of linear algebraic equations and evaluation of the current is a m atter of linear algebra. In each of four cases, J  is a quotient of two determinants: the fifth degree in the case (19), the seventh degree in the cases (20) and (23), and the eleventh degree for (24). The explicit form of J  is an extremely complex expression and therefore is not reproduced here. The detailed discussion of J  in cases (15) and (21) is performed in [5] and [6], respectively. Here, we mention main and the most interesting findings which are visualized in Figs. 2 to 6.(A) For the asymmetric distribution of amplitudes:— Generally, the properties of transport are qualitatively the same for both  distributions.— There are two regimes of transport: diffusive and non-diffusive. In the diffusive regime, Poissonian fluctuations rj(t) induces both  for­ward and backward transitions over the potential barrier. In the non-diffusive regime, Poissonian fluctuations rj(t) induces only for­ward transitions. This is the case when the bias a =  \ {zi )  is smaller than  the maximal value of the deterministic force f ( x ) .  In Figs. 2 and 3 we display the stationary probability density P(x)  in these twoxFig. 2. The stationary probability density P{x) in the diffusive regime for asymmet­ric Poissonian noise with exponentially distributed amplitudes (15) of S impulses.D T=1---------- D T=0.1_ ............D T=0 .01t  i ---------- D t= 0 .001a=1- / D s=1L m A r— -=-F-~-----T~___ 1____ .-----\ k=0-0 ,4  -0 ,2  0 ,0  0 ,2  0 ,4XFig. 3. The stationary probability density P(x)  in the non-diffusive regime for asymmetric Poissonian noise with exponentially distributed amplitudes (15) of S impulses.regimes. The main feature of P(x)  is its asymm etry even if the poten­tial V (x ) is symmetric. The reason is asymm etry of the deterministic part f ( x )  — X(zi) of the force, cf. (1) and (4).— J  > 0 independently of asymm etry of the potential. This is because of positive amplitudes {zi} of 5 kicks.— There are optim al values of param eters which maximize the current. It is shown in Fig. 4 with a generic dependence of the stationary current on param eters of fluctuations.Fig. 4. Dimensionless probability current (averaged stationary velocity) J  vs the bias of Poissonian noise in the case of the asymmetric Gamma distribution (16) of amplitudes of 8 impulses for selected values of temperature.(B) For the symmetric distribution of amplitudes:— The current J  =  0 if V(x)  is symmetric; J  ^  0 if V(x)  is asymmetric.— Generally, the properties of transport are qualitatively the same for both  distributions.— However, for specific choice of param eter set, one can notice a rem ark­able difference: for the exponential distribution (21), the current J  is monotonie function of Ds ,  while for the Gam ma distribution (22) it is non-monotonic (see Figs. 5 and 6).— The sign of J  depends on asymm etry of the potential. For asymmetric potentials, the distance dmm between a minimum and a neighboring maximum is different th a t the distance ¿Mm between the maximum and the next neighboring minimum. Let us consider the case of thepotential shown in Fig. 1 with asymm etry k > 0. Particles are sym­m etrically kicked by 6 impulses into the left and into the right direc­tions. However, the probability th a t the particle falls into the interval ( — 1/2, k) mod 1 is greater than  it falls into the interval (fc, 1/2) mod 1. As a consequence, motors move in the direction from the maximum to the minimum along both  slighter and longer slope of the potential V(x)  (in the situation shown in Fig. 1, in the right direction).Fig. 5. Dimensionless probability current J  vs the intensity Ds  of Poissonian noise in the case of the symmetric exponentially distributed (21) amplitudes of S impulses for selected values of temperature.Fig. 6. Same as in Fig. 5 but for the symmetric Gamma distribution (22) of ampli­tudes of S impulses.T he curren t is influenced by o ther p aram eters  of th e  m odel such as the  b ias of Poissonian w hite fluctuations, tem p era tu re  of th e  system  and asym ­m etry  of th e  po ten tia l. O ne can contem plate  o ther s ta tis tic s  of Poissonian fluctua tions to  be im plem ented to  th e  m odel considered. B u t th e  m ain conclusion is ra th e r d isappointing: differences caused by various sources of Poissonian fluctua tions are m irro r and th ey  drive th e  Brow nian m otors qualita tively  in th e  universal way. O n th e  o ther hand , we analyzed only one tran sp o r t coefficient, namely, th e  s ta tio n a ry  m ean velocity (v). O ther im p o rtan t tra n sp o r t characteristics like efficiency of th e  ra tch e t system , th e  diffusion coefficient, D =  lim t^ CX)({a;2 (i) — (x( t ) )2) / t ,  or fluctuations of velocity, {(An)2) =  (v2) — (v)2, have no t been analyzed. As usual, m ore unsolved problem s arise in th e  end of th e  p ap er th a n  we have s ta rted  w ith.J . L. would like to  th an k  DAAD for su pporting  his visit a t th e  U niversity of A ugsburg, w here p a r t of th e  research was perform ed.R E FE R E N C E S[1] M. Bier, Cont. Phys. 38, 371 (1997); F. Jiilicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69, 1269 (1997); P. Banggi, R. Bartussek, Nonlinear Physics of Com­plex Systems — Current Status and Future Trends, Eds. J. Parisi, S.C. Muller, W. Zimmermann, Springer Series Lecture Notes in Physics, Vol. 476, Springer, Berlin 1996, p. 294.[2] P. Banggi, R. Bartussek, P. Talkner, J. Luczka, Europhys. Lett. 35, 315 (1996).[3] M. Kostur, L. Schimansky-Geier, Phys. Lett. A265, 337 (2000).[4] A. Fulinski, T. Telejko, Phys. Lett. A152, 11 (1990); J.B. Li, Z.Q. Buang, Phys. Rev. E57, 3917 (1998).[5] T. Czernik, J. Kula, J. Luczka, P. Banggi, Phys. Rev. E55, 4057 (1997).[6] J. Luczka, T. Czernik, P. Banggi, Phys. Rev. E56, 3968 (1997).[7] J. Luczka, Physica A274, 200 (1999).[8] A. Papoulis, Probability, Random Variables and Stochastic Processes,Me Graw Bill, New York 1984.[9] J. Kula, M. Kostur, J. Luczka, Chem. Phys. 235, 27 (1998).[10] T. Czernik, J. Luczka, Ann. Phys. 9, 721 (2000).

Brownian motors driven by Poissonian fluctuations

https://rebus.us.edu.pl/bitstream/20.500.12128/7022/1/Czernik_Brownian_motors_driven.pdf

Brownian motors driven by Poissonian fluctuations

Abstract

Similar works

Full text

Available Versions

Repozytorium Uniwersytetu Śląskiego RE-BUŚ