Rocking ratchets with stochastic potentials

Physica A 312 (2002) 109 – 118

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Rocking ratchets with stochastic potentials Tongjun Zhaoa;∗ , Tianguang Caob , Yong Zhana , Yizhong Zhuob; c a Institute

of Art and Science, Hebei University of Technology, Tianjin 300130, People’s Republic of China b China Institute of Atomic Energy, P.O. Box 275(18), Beijing 102413, People’s Republic of China c Institute of Theoretical Physics, Academia Sinica, Beijing 100080, People’s Republic of China Received 15 October 2001

Abstract An analysis has been made for the motion of an overdamped Brownian particle in a periodic potential subjected to a position-dependent stochastic perturbation and a sinusoidal external force. In the presence of the stochastic potentials, it is noticed that with the increasing intensity of the stochastic potentials, the maximum of the current in general decreases, while it is shifted to the higher temperature, and moreover, the correlation length also strongly in2uences the magnitude c 2002 Elsevier Science B.V. All rights reserved. of the current.  PACS: 05.40.−y; 87.10.+e; 82.20.Mj Keywords: Stochastic Potential; Rocking; Brownian particle

1. Introduction Various ratchet models have been proposed to study the directed motion of a Brownian particle in a periodic potential. The recent experimental results on molecular motors have inspired a lot of theoretical work on such models. Within the framework of these models, a Brownian particle (molecular motor) can move along a one-dimensional “track” where it is subject to a periodically asymmetric potential which corresponds to a completely ordered track. One of the most intriguing questions concerning the directed motion of Brownian particles is investigated on the basis of the time-dependent external noises [1–9]. In ∗

Corresponding author. E-mail address: [email protected] (T. Zhao).

c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter  PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 9 6 0 - 3

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reality, the track on which the Brownian particle moves cannot be completely ordered, it should contain certain amount of disorder due to complicated landscape of the potential arising from the many-body dynamics involved in chemical and mechanical cycle during the movement of a molecular motor. The purpose of this paper is to discuss the eEects of the disordered tracks on the transport properties of the Brownian particle. Based on the rocking ratchet model, we consider the stochastic potentials which are constructed from a piecewise linear potential with a superposition of dichotomous or Ornstein–Uhlenbeck potentials [10 –12] and replace the two-states rocking force by a sinusoidal function of time to learn about the in2uence of such stochastic potentials on the properties of the current of the Brownian particle. It should be mentioned that the kind of disorder tracks considered in this work is diEerent from the one in Ref. [13]. It was noticed that zero-average temporary oscillation or 2uctuation of the external force with particular period causes net 2ux [14 –17]. The studies of thermal diffusion on a piecewise linear potential surface where either the force or the barrier height 2uctuation between two states reveal that the 2uctuation of the barrier with time can cause directed motion even though the net average microscopic force is zero [17]. In the presence of 2uctuation of the barrier, the potential inclines in a manner jumping from one state to another, namely, the particle moves in a rocking potential. In this case, the 2uctuations are relevant to the variation of time but not to that of position.

2. Stochastic potentials We now consider a Brownian particle that moves under the simultaneous action of the applied force F(t), the force due to the potential U (x), which is a spatially stochastic potentials as mentioned above, and the temporally stochastic force (t). In the high-friction limit, the motion of the Brownian particle can be described by the following Langevin equation: dx dU (x) =− + F(t) + (t) : dt dx

(1)

The equation is dimensionless. The Gaussian white noise (t), which represents the heat bath at temperature T , is a random force with zero mean and with a correlation function that is proportional to a function (t) = 0;

(t)(t
) = 2kT (t − t
) :

(2)

The applied force is sinusoidal function changing very slowly with the time, which means that the process is adiabatic, F(t) = F0 sin(2t=T0 ) ;

(3)

where F0 is the amplitude of the applied force, and T0 is the period of the external force function.

T. Zhao et al. / Physica A 312 (2002) 109 – 118

111

Corresponding to the Langevin equation (1), the Fokker–Planck equation for the probability distribution P(x; t) of the particle takes the form
   dU 9P 9 9 9j = − F(t) P + kT P =− ; (4) 9t 9x dx 9x 9x where j is the probability current and it can be expressed as follows:  
 9 dU (x) P: − F(t) + kT j=− dx 9x

(5)

If F(t) changes very slowly, there exists a quasi-stationary state. In this case, the average velocity of the particle can be solved by evaluating the constants of integration under the normalization condition and the periodicity condition of P(x), and the current can be obtained and expressed as j(t) = (1 − e−F(t)L=kT )  L 0

kT dy exp(−F(t)y=kT )C(L; y)

where the space correlation function is given by    U (x) − U (x + y) 1 L : d x exp − C(L; y) = kT L 0

;

(6)

(7)

Considering that the external force F(t) is slowly changing with the time, the average probability current J over the time interval of a period can be expressed by  T0 1 J= j(F(t)) dt : (8) T0 0 It is clear that the space correlation function is essential for us to obtain the relationship between the current and the characteristic parameters in the model. The calculation of the correlation function depends on the form of the stochastic potentials. We assume that the potential U (x) can be divided into two separate contributions: the deterministic potential U0 (x) that is periodic (with the period of L = 1:0) and the stochastic potential (x). The deterministic part U0 (x) is taken to be a piecewise linear potential with the asymmetric parameter a=0:8. The stochastic potential (x) 2uctuates with position x. For the single dichotomous potential, we have U (x) = U0 (x) + (x);

(x) = (−1)n(x; 0) :

(9)

The random function n(x; 0) counts the number of jumps in the interval of (0; x), which follows that n(x2 ; x1 ) = |x2 − x1 |=l ;

(10)

where the correlation length l is the mean distance between jumps. Furthermore, it is easy to veriKed that (x)=0, and the probability distribution function of n(x; 0) follows the Possionian distribution: p(n) = exp(−n) L nLn =n! :

(11)

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It is straightforward to calculate the spatial correlation function for the dichotomous potential [18] K(x1 ; x2 ) = (x1 )(x2 ) = 2 exp(−2|x1 − x2 |=l) : 2n

(12) 2n

Since the two-state nature of the Potential ([(x)] =  and [(x)] facilitates the calculation, it is noticed that [11]   n  ∞  1 U (x) n 1 −U0 (x)=kT =e [(x)]n (−1) exp − kT n! kT n  (x) =e−U0 (x)=kT cosh(=kT ) − sinh(=kT ) : 

2n+1

2n

=  (x))

(13)

The correlation function averaged over the stochastic potential (x) can be read 
 L U0 (x) − U0 (x + y) exp − C(L; y) = kT 0


 (x) − (x + y) × exp − dx : (14) kT  For the case of simple dichotomous potential, we have 


(x) − (x + y) exp − kT       
2y   2 2 − exp − sinh : = cosh kT l kT So the correlation function is      
2y   − exp − sinh2 C(L; y) = cosh2 kT l kT 
 L U0 (x) − U0 (x + y) × dx : exp − kT 0

(15)

(16)

For the case of Ornstein–Uhlenbeck potential, (x) is a sum of N independent dichotomous potentials N  (x) = i (x) ; (17) i

where the random potential i (x) obeys the following relations: i (x) = 0;

i (x1 )j (x2 ) = ij 2i exp(−2|x1 − x2 |=li ) :

(18)

We can furthermore obtain the following formula:     N i (x) U (x) = e−U0 (x)=kT exp − exp − kT kT i=1

=e

−U0 (x)=kT

N  i=1

i (x) cosh i − sinh i ; i

(19)

T. Zhao et al. / Physica A 312 (2002) 109 – 118

with i = i =kT , and the correlation function can be given by 
 L U0 (x) − U0 (x + y) C(L; y) = exp − kT 0


 (x) − (x + y) × exp − dx : kT 

113

(20)

If we√consider the sum of dichotomous potentials all with the same i (each equal to = N ) and the same li (each equal to l), we get an Ornstein–Uhlenbeck process. When N tends to inKnity, we have

     


2 2y  (x) − (x + y) : (21) 1 − exp − = exp exp − kT l kT  So the correlation function is

     2 2y  C(L; y) = exp 1 − exp − kT l 
 L U0 (x) − U0 (x + y) × dx : exp − kT 0

(22)

Eqs. (16) and (22) are the main results of this paper. Within the linear piecewise rocking model they have analytical expressions and the numerical results can be carried out exactly.

3. Directed motion of the Brownian particle We have calculated the net current about the motion of the Brownian particle in the periodic potential with the superposition of the simple dichotomous stochastic potential and the Ornstein–Uhlenbeck potential, respectively. As the results from these two cases are very similar, for the convenience of physical discussion, we now mainly investigate the current in the O–U case and compare that with the case in the absence of the stochastic potential. The current J is plotted in Fig. 1 as a function of the stochastic potential intensity  for given temperature and diEerent amplitudes of external force in O–U case. For each given external force the current has a maximum at almost the same intensity . With the increasing of the amplitude of external force the current increases monotonically. For very large intensity of stochastic potential the net current vanishes, while for very small intensity the current does not vanish and tend to a nonzero value. This is easy to understand when the intensity  is much larger than the height of the asymmetric rocking potential, the stochastic potential becomes predominant and in this case one would expect that the current should tend to vanish. On the other hand, for very small intensity  the system should approach to a standard rocking ratchet system without stochastic potential in which the directed motion still exists and the current is positive [16].

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Fig. 1. The relationship between the stationary current J and the intensity  of stochastic potential with Kxed correlation length and diEerent amplitude of the external force for the simple O–U stochastic potential, where  = 1, U0 = 1, kT = 0:5, l = 1 and a = 0:8.

Fig. 2 displays the behavior of the directed motion for the Kxed correlation length l with the change of the amplitude of external rocking force. For a given  the increasing of the current as a function of the amplitude of the external rocking force is monotonically and the relationship is not linear which is consistent with the nonlinear response theory [11,12]. It is noticed that the current also increases with the increasing of the intensity . When  reaches the value of about 0.6 (for a given temperature at kT = 0:5), the current approaches to the maximum for diEerent value of F0 , which means that the maximum value of the current is independent of the value of F0 . The most appealing fact of the stochastic potential is whether the randomness of the potential can improve the directed motion or reduce it. Figs. 3 and 4 show qualitatively the behavior of the current as a function of temperature with the diEerent correlation length or diEerent intensity of the stochastic potential separately. Fig. 3 reveals the main characteristics of the relationship between the net current and the temperature: for small temperature, as well as for very high temperature the current vanishes just like that of most of the thermal ratchet models [1–3]. For an intermediate value of the temperature, the current reaches a maximum. DiEerent curves are correspondent to diEerent intensity values of the stochastic potential. By all appearances, the maximum value of the current in the case without stochastic potential is the largest (see Fig. 3). With the increasing of the intensity of the stochastic potential  the maximal current decreases, but the corresponding temperature Tm at which the current takes the maximum is shifted to the higher value of . However, this

T. Zhao et al. / Physica A 312 (2002) 109 – 118

115

Fig. 2. The stationary current J versus the intensity of external force F0 with Kxed correlation length and diEerent strength of the stochastic potential, where  = 1, U0 = 1, kT = 0:5, l = 1 and a = 0:8.

Fig. 3. The net current J as a function of temperature with Kxed correlation length and diEerent intensity  of the external O–U stochastic potential, where  = 1, U0 = 1, F0 = 0:3, l = 1 and a = 0:8.

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T. Zhao et al. / Physica A 312 (2002) 109 – 118

Fig. 4. The net current J as a function of temperature with Kxed intensity and diEerent correlation length of the external O–U potential, where  = 1, U0 = 1, F0 = 0:3,  = 0:3 and a = 0:8.

fact reveals that in the presence of the stochastic potential the parameter range where there exists a substantial current is enlarged, in other words, the net current becomes much larger than the case without stochastic potential in some range of temperature. In this range of temperature the stochastic potential increases the net current tremendously. This Knding is quite surprising at the Krst sight, however, it can be intuitively understood. The stochastic potential superposed on the standard rocking ratchet plays a role as the traps for movement of the Brownian particle, in order for the particle to get out of the traps it needs to be at the higher temperature as compared to the standard rocking ratchet, this is why the Tm is shifted to the higher value of the temperature as soon as the stochastic potential is present. In Fig. 4 the current is plotted as a function of the temperature kT for the simple Ornstein–Uhlenbeck potential and the given intensity of the stochastic potential . It is shown that as the correlation length increases, the maximal current increases. In the limit l → ∞, the curve tend to the case just like that without the stochastic potential. In Fig. 5, as a function of the correlation length of the stochastic potential, the current of the particle is depicted at a Kxed temperature with diEerent intensity of stochastic potential . From this picture it provides us with a general picture in two asymptotic cases: for very small correlation length the current vanishes, and for very large correlation length the current approaches to the value, which corresponds with the standard rocking ratchet case. The behavior in the intermediate value of the correlation length much depends on temperature range (see Fig. 4 together with Fig. 5).

T. Zhao et al. / Physica A 312 (2002) 109 – 118

117

Fig. 5. The current J as a function of the correlation length l with Kxed temperature and diEerent strength of the stochastic potential for O–U potential, where i = 1, U0 = 1, F0 = 0:3, kT = 0:5, a = 0:8.

4. Conclusion and remarks To conclude, we summarize the Kndings of our present work. We have presented a modiKed rocking ratchet model for the directed motion of Brownian particle subject to a stochastic potential. As a whole, in the presence of the stochastic potential the maximal value of the current decreases and it emerges in a much high temperature as the intensity of the stochastic potential increases. So it is clear that in the range of low temperature the current decreases with the increase of the stochastic potential intensity, but it increases in the range of high temperature. The 2uctuations of the potential play a particular role in the transport, which could not be referred simply as enhancement or not. Similar situations are also found in the dependences of the net current on various parameters such as the temperature and the intensity of the stochastic potential both for the simple dichotomous potential and the O–U potential.

Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grant No. 10075007 and the Natural Science Foundation of Hebei Province under Grant No. 198027 and B2001113.

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