- Email: [email protected]
Relaxation high-temperature ratchets
Accepted Manuscript Relaxation high-temperature ratchets I.V. Shapochkina, V.M. Rozenbaum, S.-Y. Sheu, D.-Y. Yang, S.H. Lin, L.I. Trakhtenberg
PII: DOI: Reference:
S0378-4371(18)31168-3 https://doi.org/10.1016/j.physa.2018.09.039 PHYSA 20100
To appear in:
Physica A
Received date : 1 May 2017 Revised date : 6 July 2018 Please cite this article as: I.V. Shapochkina, et al., Relaxation high-temperature ratchets, Physica A (2018), https://doi.org/10.1016/j.physa.2018.09.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
RELAXATION HIGH-TEMPERATURE RATCHETS
I. V. Shapochkina,1-3* V. M. Rozenbaum,1,2,4 S.-Y. Sheu,5† D.-Y. Yang,1‡, S. H. Lin,1,2 and L. I. Trakhtenberg6 1
Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan Department of Applied Chemistry, National Chiao Tung University, 1001 Ta Hsuen Road, Hsinchu, Taiwan 3 Department of Physics, Belarusian State University, Prospekt Nezavisimosti 4, 220050 Minsk, Belarus 4 Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Generala Naumova str. 17, Kiev, 03164, Ukraine 5 Department of Life Sciences and Institute of Genome Sciences, Institute of Biomedical Informatics, National Yang-Ming University, Taipei 112, Taiwan 6 Semenov Institute of Chemical Physics, Russian Academy of Sciences, Kosygin Street 4, Moscow 119991, Russia; State Scientific Center of Russian Federation, Karpov Institute of Physical Chemistry, Moscow, 105064 Russia 2
HIGHLIGHTS Periodic relaxation processes are included in ratchet operation. Jump-like spatial changes of potential profiles are considered. Different asymptotics of the ratchet velocity vs the relaxation time are revealed. ABSTRACT
We consider the overdamped motion of a Brownian particle in an asymmetric spatially periodic potential which fluctuates periodically in time, under assumption of finite duration of the relaxation response of the system on deterministic dichotomous fluctuations. It is assumed that the period of these fluctuations is much larger than the characteristic diffusion time and the potential barrier height is small as compared to the thermal energy (an adiabatic hightemperature flashing ratchet). We derive an analytical expression for the average particle velocity, which is concretized for a saw-tooth potential profile. It is revealed the different, linear and quadratic, asymptotic behavior of the average velocity as a function of the relaxation time for extremely and not extremely asymmetric potential profiles, respectively. The result is interpreted in terms of the self-similar representation.
Keywords: Driven diffusive systems; Brownian motor; Adiabatic flashing ratchet __________________________
* [email protected] ; † [email protected] ; ‡ [email protected]
1. Introduction
Driven diffusive systems, which function far from equilibrium in spatially periodic structures [1,2], can serve as a good example of those with many properties following from the competition between their characteristic times [3-5]. For example, the spatiotemporal synchronization between the spatial period of a potential and its time modulation leads to the acceleration of diffusion (in comparison with the cases of free diffusion) and to a finite net diffusion in the absence of thermal noise [6,7]. There are many papers on diffusion and dynamics in ratchet structures in which peculiarities of the observed phenomena are the consequence of the competition between parameters of the nonequilibrium noise and the system considered [8-10]. Artificial nano-devices that can rectify non-equilibrium fluctuations of different nature to the directional motion of particles are usually described by the so-called Brownian motors (ratchets) [11,12], theoretical models based on diffusion dynamics, in which particle potential energy itself U ( x, t ) or its first derivative U x ( x, t ) is considered as a periodic function of coordinate x and
time t [1]. The spatial period L , together with the diffusion coefficient D , determines the main characteristic time: the diffusion time L L2 / D . For potential profiles with large gradients, one more system characteristic time arises, l l 2 / D L , where l L is the width of a sharp part of the profile. These two characteristic times can compete with those resulting from the certain temporal peculiarities of potential profiles, for example, with the period of the potential energy changes or with a much smaller time rel determined by the portion of a sharp temporal change, if any, of U ( x, t ) . This competition typically leads to the nontrivial behavior of Brownian motors [5]; here the jump-like behavior of motor characteristics is a prominent example, specific for both analytical and numerical treatment. One more time which can “come into play” is the velocity relaxation time [13] (the time that is required for the system to establish a Maxwell distribution) in the theory of inertial Brownian motors. For quantum [4] and rotary [14,15] ratchets, there also exist competition-induced effects. The interplay of characteristic time scales should be analyzed analytically (especially when in competition of limits) rather than numerically: Numerical analysis is hardly successful for infinitely large or vanishingly small quantities. So, the analytical expressions for quantities under study are very preferable. There is a possibility to have them provided by the theory of high-temperature ratchets, initially developed in Ref. [16] and then adapted to so-called Brownian photomotors [17]. These objects are one of the bright examples of ratchet systems with competing characteristic times because in photomotors there always exist relaxation processes of finite duration. They are characterized by the delay between laser impulses and the 2
response of the electron subsystem of photomotors, so that one can expect new effects as a result of what place relaxation times occupy in the hierarchy of the system characteristic times. In the present paper, we consider an adiabatic “version” of a Brownian photomotor (an adiabatic flashing ratchet) with the following characteristic times: the relaxation time, rel , and the diffusion time over the small characteristic length, l . As a comparatively simple model potential, implying sharp parts in its profile, we analyse a saw-tooth potential, which plays a key role in ratchet theory due to its piecewise-linear structure (leading to analytical results) and potentiality of its experimental realization [1,5-7,9,18-27]. The structure of the paper is as follows. In Sec. II, using the symmetrized representation for the average particle velocity suggested in Ref. [17] for the additive-multiplicative form of the spatial-temporal dependence of the particle potential energy, we derive, for large values of time period of energy fluctuations ( L , rel ), the expression for the average ratchet velocity, containing a separate summand responsible just for the relaxation-induced effect. The effective fluctuating potential technique [5] turns the resulting expressions into the integral relations (the Green’s function method) which are the core result as they allow analysis of the influence of relaxation times on the ratchet behavior. In Sec. III a model saw-tooth potential characterized by sharp links in its profile is used to get analytical expressions for the quantities of interest, and the mentioned analysis is carried out both analytically and graphically. The subtle interplay of time and spatial limits are analyzed in terms of a universal self-similar representation, which describes continuous transition from quadratic to linear rel -dependence of the average ratchet velocity, also predicted in this Section. Section IV includes discussions and conclusions.
2. Main expressions
It is well-known that the ratchet effect is caused by non-equilibrium fluctuations of parameters of a system, natural or artificial, which result in fluctuations of spatially dependent ratchet potential energy U ( x, t ) in time [1]. The overdamped motion of a Brownian particle under the action of the force F ( x, t ) U ( x, t ) / x is determined by the Langevin equation:
dx / dt F ( x, t ) (t ) ,
3
(1)
where is the friction coefficient, and (t ) is the random force (zero-mean Gaussian deltacorrelated white noise with
(t ) (s) 2 kBT (t s) , where k B is the
(t ) 0 and
Boltzmann constant, T is the absolute temperature, (t ) is the delta function, and ... denotes here the ensemble average). We begin here with the consideration of a so-called hightemperature ratchet in which spatiotemporal changes of the potential energy are small compared to the thermal energy. For ratchets of this type, analytical representation of their characteristics can be obtained, which allows, among others, analyzing the important dependence of the average ratchet velocity on the frequency of potential energy fluctuations of different nature. Limitations of the high-temperature approach are that it is unsuitable for the description of processes of suppressing the backward motion (occurring only at low temperatures) which are important for high-efficient ratchets [28]. We will consider the potential energy of the additive-multiplicative form U ( x, t ) u ( x) (t ) w( x) ,
(2)
with spatially L -periodic functions u ( x) and w( x ) , and -periodic function (t ) which describes temporal variations of the potential energy and satisfies supersymmetric condition
(t ) (t / 2) . Note that the form (2) is a legitimate model for a number of theoretically and practically interesting systems [1]. Using the Fourier-transformation for all the functions involved f ( x, t ) f qj exp(ikq x i j t ), kq 2 L q, j 2 j , q, j 0, 1, 2, ... ,
(3)
qj
we can express the ratchet velocity averaged over the period (from here and after
... 1 dt... ) as a sum of Fourier harmonics of the potential energy [17]: 0
v 2i 3 D3
qq ( 0) ( q q 0)
kq kq kq q kq2 kq2 wq wqu q q Dkq2 , Dkq2 ,
2j j
a, b j 1
2 j
2
a 2 2j b2
4
(4)
,
(5)
(k BT ) 1 is the inverse thermal energy. These expressions are valid for high-temperature expansion up to the third order in the ratio of potential energy changes to the thermal energy [16]. Next, let us assume that the function (t ) describes a periodic relaxation process and hence obeys the relaxation equation d (t ) / dt 0 (t ) (t ) / rel with the relaxation time rel . The
function
0 (t )
/ 2 m t (m 1)
takes
the
1
values
and
1
at
m t / 2 m
and
( m is an integer), respectively, and describes a relaxation-free
deterministic dichotomous process (a stochastic analogue of the dichotomous process is successfully applied to describe ratchet systems, in particular, biological ones [1]; deterministic processes, although not so extensively studied in the context of Brownian motors, are of great importance in the development of nano-devices [11,12]). We can write a periodic solution of the relaxation equation, with the boundary condition ( / 2) (0) on a half-period, and its nonzero frequency Fourier components as follows:
(t ) 1 2 1 e /(2
Fig. 1
rel )
1
et / rel , 0 t / 2, 2 j 1
4i . 2 j 1 (1 i2 j 1 rel )
(6)
Functions (t ) and 0 (t ) are depicted in Fig. 1. Substituting the quantities 2 j 1 from Eq. (6) into Eq. (5) and analytical summation over j gives the result [17]:
3 tanh( / 4 rel ) 2 rel tanh( a / 4) tanh( b / 4) (a, b) . (7) [1 (a rel )2 ][1 (b rel )2 ] [(a rel ) 2 1](a 2 b2 )a [(b rel ) 2 1](b2 a 2 )b
Under the adiabatic assumption, rel and L ( L L2 / D ), Eq. (7) is simplified and takes the form:
Dkq2 , Dkq2
1 q q
3 2 rel
qq (q q )(1 q )(1 q )
, q Dkq2 rel .
(8)
Substituting Eq. (8) into Eq. (4) gives the expression for the desired average particle velocity
v
2i L
3
qq ( 0) ( q q 0)
1 q q
q q wq wqu q q . (1 q )(1 q ) qq
5
(9)
In the absence of the relaxation process ( rel 0, q 0 ), the formula (9) is reduced to the known one [16]:
v
0
4i L
wq wqu q q
3
.
q
qq ( 0) ( q q 0)
(10)
Then the average ratchet velocity can be written in the form
v v 0 v 1,
(11)
that is as a sum of the above relaxation-free part and the relaxation-induced part
v 1
2i L
3
qq ( 0) ( q q 0)
2 4 3 D 2 rel L
qq q q wq wqu q q (1 q )(1 q ) qq
qq ( 0) ( q q 0)
wq
wq
(1 Dkq2 rel ) (1 Dkq2 rel )
(12) u q q
to be found. The primed quantities wq ikq wq and uq ikq wq are the Fourier components of the derivatives w( x) dw( x) / dx and u( x) du ( x) / dx , respectively. Using the concept of the effective fluctuating potential suggested in Ref. [5], we can reduce the double sum in Eq. (12) to the following integral relation: 2 4 3 D 2 rel 2 0 dxu( x)w ( x) , L L
v 1
(13)
which contains the first derivative of the effective potential
w( x) q
wq 1 Dkq2 rel
exp(ikq x)
the function
6
1 L rel
L
dyG( x y)w( y) ; 0
(14)
G( x) rel q
exp(ikq x) 1 Dkq2 rel
rel
z cosh z 1 2 x L 1 L , , z sinh z 2 rel
(15)
is the Laplace representation G( x y, ) dtG( x y, t ) exp(t ) of the Green’s function for 0
1 free diffusion in the interval [0, L] , with the periodic boundary conditions, at rel .
The general expressions for the average velocity of the relaxation high-temperature ratchet represented by Eqs. (11), (13)-(15) are the main result of this paper. The relaxation processes influence the system behavior through the ratio of the system characteristic times, of relaxation rel and of diffusion L , entering the Green’s function (15). At small rel , some additional features in ratchet behavior are attributed to the appearance of the characteristic time
l , which grows from the presence of sharp portions of the small width l in the potential profile, that can be comparable with the time rel . For a sawtooth potential, such a case admits analytical treatment.
3. Sawtooth potential We consider, as the functions u ( x) and w( x ) in Eq. (2), two spatially periodic potentials, both having linear portions proportional to x / l at x [0, l ] and ( L x) /( L l ) at x [l , L] , and Fig. 2
characterizing by energetic barriers, respectively, u0 and w0 (Fig. 2). In the case of an adiabatic relaxation-free flashing ratchet with a saw-tooth potential, the expression for its average velocity is known for arbitrary temperatures (see Eqs. (2) and (3) in Ref. [23]). To apply the hightemperature approximation, this expression must be expanded over the small quantities u0 / kBT and w0 / kBT , or, equivalently, from the analytical summation in Eq. (10), the same result follows
v
0
1 2 l L v0 , , v0 3u0 w02 . 45 L
(16)
The integrals in Eqs. (13) and (14) in their turn give the desired relaxation contribution v 1 :
v 1
1 2 v0 6 f1 ( z , ) 3 f 2 ( z , ) f1 ( z , ) f 2 ( z, ) 16 ( z 2 ) 2
7
(17)
where
f1 ( z, ) 1
sinh z sinh z , z sinh z
f 2 ( z, ) 1
sinh z ( ) l , , ( )sinh z L
1 .
(18)
Thus, we have obtained the average particle velocity in the model of the adiabatic relaxation flashing ratchet with arbitrary values of the relaxation parameter z and with the potential profile shape implying the presence of jumps (parts with large gradient) in it. The temperature dependence is introduced to the average velocity
v
by the dimensionless variable z
(depending on rel and L ) and the scaling velocity factor v0 (see Eqs. (15) and (16), where
L L2 / D L2 is the temperature-dependent diffusion time). The main time parameter of the dichotomous deterministic driving, its period , is also entered into v0 . The notable aspect is that the similar expressions for the average velocity were obtained in Ref. [5] in case of particle motion caused by the stochastic dichotomous (instantaneous) changes of the ratchet potential profile. Such a similarity can be interpreted by the presence of “exponential processes” in the two systems: on the one hand, the relaxation contribution v 1 , considered in the present paper, results from the presence of the exponential relaxation processes in the periodic time dependence (t ) of the potential energy [see Eq. (6)]; on the other hand, the stochastic dichotomous processes, in Ref. [5], are characterized just by the exponential correlation function. Thus, the exponential behavior in the “decay” of a potential profile seems to be the cause of the substantial similarity discussed. The character of rel / L -dependence of the average velocity v (see Fig. 3 for different Fig. 3
values) essentially changes when the extremely asymmetric potential profile ( 0 ) loses this quality ( 0 ). One can see that the more the departure of the profile from the extremely asymmetric one, the less the ratchet effect can be reached. Note that this conclusion agrees with the experimental result described in Ref. [24] that the maximal ratchet effect is observed for the extremely asymmetric shape of the sawtooth side of the pattern. At rel L ( z 0 ), as it is easy to check, the relaxation contribution (17) is compensated by the nonrelaxation one (16) so that the average particle velocity (11) tends to zero in this limit. The opposite limit, rel L ( z ), deserves special consideration because, in this case, the result depends on whether there are large gradients in the potential relief (the potential is sharp) or not. Indeed, in the case of the potential with jump-like spatial changes ( 0 ), the relaxation-induced velocity (17) becomes 8
v 1
v0 cosh 2 z 5 1 coth z 2 v0 rel , 2 2 2 z 4sinh z 4 z z z L
(19)
while at 0 and z it tends to
2
1 2 rel v 1 4 v0 . z ( ) 2 L
(20)
The quadratic asymptotics (20), obtained for the potential profile different from the extremely asymmetric one, is in accordance with the result of Ref. [29] in which the quadratic dependence of the average ratchet velocity on the duration of transient processes has been predicted, but based on the assumption that they are described by linear functions of time. The difference in the asymptotic behavior of the relaxation contribution to the velocity at
0 and 0 means that the result depends on the sequence of the limits 0 and z . If so, a reasonable strategy is to analyse these limits keeping the value of the product z fixed (finite and arbitrary). With this problem definition, we can obtain the both limits from a single continuous description. Indeed, the velocity (17) as a function of two variables, z and , can be rewritten as a function of one self-similar variable:
1 z rel 2 l
1/ 2
,
(21)
which is determined by the ratio of the relaxation time rel to the much smaller diffusion time
l l 2 / D tending to zero at l 0 . Thus, for the desired relaxation contribution to the ratchet velocity, we have the self-similar representation (the scaling law [30])
v
1 0, z z finite
v0 f
rel , L
(22)
where f
1
8 7 4( 2)e 8 3
9
2
e
4
1 , rel 2 l
1/ 2
.
(23)
It is easy to check that the limiting cases (19) and (20) follow from Eqs. (22), (23) at
0 and , respectively. Continuous transition from quadratic to linear rel -dependence of v
1
at rel 0 is depicted in the insert of Fig. 3. Note that the quadratic dependence occurs
in the narrow region of small rel -values corresponding to fast relaxation processes. Thus, at small rel -values, the presence of the relaxation behavior in time dependence of a potential profile suppresses the ratchet effect the more the less the asymmetry of the profile.
4. Discussions and conclusions In a vast majority of ratchet models, a simplified assumption of instantaneous switching of the ratchet potential is used [1,5-7,9,16,18,19,28,31]. This means that there is no energy exchange between the ratchet and the thermal environment during the switching. If, additionally, the period of potential profile fluctuations in time is assumed to be much larger than the characteristic diffusion time L , one deals with the so-called adiabatically fast driven ratchet; just adiabatically driven ratchet systems demonstrate the best (most efficient) regime of operation (maximum values of the average velocity and energy characteristics, in comparison with the nonadiabatic ones) [31-38]. In reality, there always exists a nonzero time interval during which the ratchet potential changes; it can be the duration of a transient process [29] or the relaxation time rel . Corrections to ratchet characteristics resulting from the presence of relaxation processes can be interpreted as nonadiabatic contributions. Brownian photomotors [17] can serve as an example of systems with relaxation processes of finite duration, which is characterized by the delay between rectangular-shaped laser pulses and the response of the electron subsystem of a photomotor. Thus, understanding the effect of relaxation processes on ratchet characteristics is of theoretical and practical relevance, especially in the design of molecular nanomachines. It is known that nontrivial effects in ratchet behavior result from the interplay of the system characteristic times. For example, in Ref. [5], it has been shown that just the competition of the average period of stochastic fluctuations of a sharp periodic ratchet potential and the characteristic diffusion time l over its small characteristic length l leads to the jump-like changes in ratchet characteristics. A nonanalytical behavior in characteristics of inertial diffusion transport can be also expected as a result of the interplay of times: the sliding time on a sharp slope of the potential and the characteristic velocity relaxation time [13]. In the present paper, the diffusion time l competes with the relaxation time rel , and the analysis of ratchet systems under this competition revealed the similarity (heretofore nonobvious) of stochastic and 10
deterministic ratchets, in their operation and description, resulting from the presence of exponential behavior in the “decay” of the potential profile. Using the derived analytical expression for the average velocity of the relaxation hightemperature ratchet, which has been concretized for a saw-tooth potential profile of arbitrary asymmetry, we have revealed the distinction in asymptotic behavior, at small rel / L values, of the velocity as a function of the relaxation time, namely, linear and quadratic asymptotics, for potential profiles with and without jumps, respectively. The effect has been interpreted in a uniform manner with the self-similar representation, and the both asymptotics appeared at different limiting values of the self-similar variable. Thus, the analysis of the position of relaxation times in the hierarchy of system characteristic times allows a step forward in understanding Brownian ratchets. We considered the effect of presence of sharp parts in a potential profile on characteristics of high-temperature deterministic ratchets with potential energy fluctuations described by relaxation-type periodic processes. As a simplest example accounting for existence of a sharp slope in a potential profile, a sawtooth potential with relaxation changes of its amplitude has been chosen. Sawtooth shapes (including sawtooth potential profiles) are widely used in both experimental set-ups and theoretical description of ratchet systems. Sawtooth ratchet elements are suggested to be used, e.g., in molecular pumps [39,40], selective sorting of charged membrane components [24,25], organic electronic ratchets [26,27], etc. One should take into account that such potential reliefs contain cusp points resulting from the piecewise-linear shape of approximating functions which simplify mathematical treatment. Real potential profiles can be arbitrary close to those of a sawtooth shape, but cusp points of those are somehow or other rounded with a small rounding radius. This rounding can change ratchet characteristics only at low temperatures. The reason is that, at low temperatures, the main contribution to distribution functions and barrier factors give just the regions near cusp points (see the quantitative analysis of the influence of cusp point rounding on ratchet characteristics on page 5 in Ref. [41]). Within the high-temperature approximation which we use in this paper, the results weakly depend on these features in a potential shape. Therefore using a sawtooth potential with a sharp link l is not only very convenient for deriving analytical dependencies of desired quantities on a single parameter l (which characterizes applied forces on narrow links of the potential profile) but well grounded.
11
Acknowledgments This work was supported by National Chiao Tung University and Academia Sinica. I.V.S., V.M.R., and S.H.L. thank the Ministry of Education, Taiwan (“Aim for the Top University Plan” of National Chiao-Tung University). S.Y.S. thanks the Taiwan Ministry of Science and Technology for partial support (Grant No. MOST105-2113-M-010-003). D.Y.Y. thanks the Taiwan Ministry of Science and Technology for partial support (Grant No. MOST105-2119-M-001-022). S.H.L. thanks the Taiwan Ministry of Science and Technology for partial support (Grant No. MOST104-2923-M-009-001-). L.I.T. thanks the Russian Foundation for Basic Research (Grants No 18-57-00003 and 18-29-02012) and Ministry of Science of Russia in the frame of state assignment 0082-2018-0003 (the registration number АААА-А18118012390045-2) for financial support. I.V.S. thanks Belarusian Republican Foundation for Fundamental Research for partial support (Grant No Ф18P-022). I.V.S. and V.M.R. gratefully acknowledge the kind hospitality received from the Institute of Atomic and Molecular Sciences.
12
References
[1] P. Reimann, Brownian motors: noisy transport far from equilibrium, Phys. Rep. 361 (2002) 57. [2] A. Schadschneider, D. Chowdhury, K. Nishinari, Stochastic Transport in Complex Systems: From Molecules to Vehicles, Elsevier, 2010. [3] Delvenne, J.-C. et al. Diffusion on networked systems is a question of time or structure. Nat. Commun. 6:7366 doi: 10.1038/ncomms8366 (2015). [4] B. Lau, O. Kedem, M. A. Ratner, and E. A. Weiss, Identification of two mechanisms for current production in a biharmonic flashing electron ratchet, Phys. Rev. E 93 (2016) 062128. [5] V. M. Rozenbaum, I. V. Shapochkina, S.-Y. Sheu, D.-Y.Yang, and S. H. Lin, Hightemperature ratchets with sawtooth potentials, Phys. Rev. E 94 (2016) 052140. [6] A. A. Dubkov and B. Spagnolo, Acceleration of diffusion in randomly switching potential with supersymmetry, Phys. Rev. E 72, (2005) 041104. [7] B. Spagnolo and A. A. Dubkov, Diffusion in flashing periodic potentials, Eur. Phys. J. B 50 (2006) 299-303. [8] D. Valenti, C. Guarcello, B. Spagnolo, Switching times in long-overlap Josephson junctions subject to thermal fluctuations and non-Gaussian noise sources, Phys. Rev. B 89 (2014) 214510. [9] A. Fiasconaro and B. Spagnolo, Resonant activation in piecewise linear asymmetric potentials, Phys. Rev. E 83 (2011) 041122. [10] B. Lisowski, D. Valenti, B. Spagnolo, M. Bier, E. Gudowska-Nowak, Stepping molecular motor amid Lévy white noise, Physical Review E 91 (2015) 042713. [11] P. Hänggi and F. Marchesoni, Artificial Brownian motors: Controlling transport on the nanoscale, Rev. Mod. Phys. 81 (2009) 387. [12] I. Goychuk, Molecular machines operating on the nanoscale: from classical to quantum, Beilstein J. Nanotechnol. 7 (3) (2016) 328-350. [13] V. M. Rozenbaum, Y. A. Makhnovskii, I. V. Shapochkina, S.-Y. Sheu, D.-Y.Yang, and S. H. Lin, Diffusion of a massive particle in a periodic potential: Application to adiabatic ratchets, Phys. Rev. E 92 (2015) 062132. [14] V. M. Rozenbaum, O. Ye. Vovchenko, and T. Ye. Korochkova, Brownian dipole rotator in alternating electric field, Phys. Rev. E 77 (2008) 061111. [15] T. Hiroki, T. Horita, K. Ouchi, Performance estimation for two-dimensional Brownian rotary ratchet systems, J. Phys. Soc. Jpn. 84 (4) (2015) 044004. [16] V. M. Rozenbaum, Pis’ma Zh. Eksp. Teor. Fiz. 88, 391 (2008) [JETP Lett. 88, 342 (2008)]. 13
[17] V. M. Rozenbaum, M. L. Dekhtyar, S. H. Lin, and L. I. Trakhtenberg, J. Chem. Phys 145, 064110 (2016). [18] R. D. Astumian and M. Bier, Fluctuation driven ratchets: molecular motors, Phys. Rev. Lett. 72, (1994) 1766. [19] J.-F. Chauwin, A. Ajdari, and J. Prost, Force-free motion in asymmetric structures: a mechanism without diffusive step, Europhys. Lett. 27 (1994) 421. [20] A. Parmeggiani, F. Jülicher, A. Ajdari, and J. Prost, Energy transduction of isothermal ratchets: Generic aspects and specific examples close to and far from equilibrium, Phys. Rev. E 60, (1999) 2127. [21] I. M. Sokolov, Irreversible and reversible modes of operation of deterministic ratchets, Phys. Rev. E 63 (2001) 021107. [22] K. Sumithra and T. Sintes, Effciency optimization in forced ratchets due to thermal fluctuations, Physica A 297 (2001) 1. [23] M. L. Dekhtyar, A. A. Ishchenko, and V. M. Rozenbaum, Photoinduced molecular transport in biological environments based on dipole moment fluctuations J. Phys. Chem. B 110 (2006) 20111. [24] J. S. Roth, Y. Zhang, P. Bao, M. R. Cheetham, X. Han, and S. D. Evans, Optimization of Brownian ratchets for the manipulation of charged components within supported lipid bilayers, Appl. Phys. Lett. 106 (2015) 183703. [25] M. R. Cheetham, J. P. Bramble, D.G.G. McMillan, R. J. Bushby, P. D. Olmsted, L. J. C. Jeuken, and S. D. Evans, Manipulation and sorting of membrane proteins using patterned diffusion-aided ratchets with AC fields in supported bilayers // Soft Matter. 8 (2012) 5459. [26] E. M. Roeling, W. C. Germs, B. Smalbrugge, E. J. Geluk, T. de Vries, R. A. J. Janssen, and M. Kemerink, Organic electronic ratchets doing work, Nature Mater. 10 (2011) 51. [27] P. Hänggi, Organic electronics: Harvesting randomness, Nature Mat. 10 (2011) 6. [28] V. M. Rozenbaum, T. E. Korochkova, K. K. Liang, Conventional and generalized efficiencies of flashing and rocking ratchets: Analytical comparison of high-efficiency limits, Phys. Rev. E 75 (2007) 061115. [29] V. M. Rozenbaum and I. V. Shapochkina, Nonadiabatic corrections to the velocity of a Brownian motor with a complex potential profile, Pis’ma Zh. Eksp. Teor. Fiz. 92 (2010) 124 [JETP Lett. 92 (2010) 120]. [30] G. I. Barenblatt, Scaling, selfsimilarity and intermediate asymptotics (University Press, Cambridge, 1996). [31] J. M. R. Parrondo, Reversible ratchets as Brownian particles in an adiabatically changing periodic potential, Phys. Rev. E 57 (1998) 7297. 14
[32] R. Krishnan, M. C. Mahato, and A. M. Jayannavar, Brownian rectifiers in the presence of temporally asymmetric unbiased forces, Phys. Rev. E 70 (2004) 021102. [33] V. M. Rozenbauma, D.-Y. Yang, S. H. Lin, and T. Y. Tsong, Energy losses in a flashing ratchet with different potential well shapes, Physica A 363 (2006) 211–216. [34] Y. X. Zhang, The efficiency of molecular motors, J. Stat. Phys. 134 (2009) 669-679. [35] J. M. Horowitz and C. Jarzynski, Exact formula for currents in strongly pumped diffusive systemsJ. Stat. Phys. 136 (2009) 917-925. [36] H. J. Chen, J. L. Huang, C. Y. Wang, and H. C. Tseng, Flow density of a reversible ratchet, Phys. Rev. E 82 (2010) 052103. [37] V. M. Rozenbaum and I. V. Shapochkina, Quasiequilibrium directed hopping in a timedependent two-well periodic potential, Phys. Rev. E 84 (2011) 051101. [38] V. M. Rozenbaum, Y. A. Makhnovskii, I. V. Shapochkina, S.-Y. Sheu, D.-Y.Yang, and S. H. Lin, Adiabatically slow and adiabatically fast driven ratchets, Phys. Rev. E 85 (2012) 041116. [39] R. D. Astumian and P. Hänggi, Brownian Motors, Phys. Today 55, No 11 (2002) 33. [40] R. D. Astumian, Thermodynamics and kinetics of a Brownian motor, Science 276 (1997) 917. [41] V. M. Rozenbaum, Yu. A. Makhnovskii, I. V. Shapochkina, S.-Y. Sheu, D.-Y. Yang, and S. H. Lin, Inertial effects in adiabatically driven flashing ratchets, Phys. Rev. E 89 (2014) 052131.
15
Figure caption Fig. 1. Periodic ( is the period) functions 0 (t ) and (t ) describing deterministic relaxationfree (dichotomous) and periodic relaxation (with the relaxation time rel ) processes, respectively. Fig. 2. Two spatially periodic functions u ( x) and w( x ) with energetic barriers u0 and w0 describing the static and the fluctuating parts of the sawtooth ratchet potential U ( x, t ) u ( x) (t ) w( x) .
Fig. 3. The normalized average velocity
v / v0 of the relaxation high-temperature ratchet
[calculated from Eqs. (11), (16)-(19)] vs the normalized relaxation time rel / L at different values of the asymmetry parameter: 0 (solid line) and 0.3 (dashed line); the shapes of the potential profile are shown schematically under the curves. The normalized relaxation contribution v 1 / v0 to the average velocity [calculated from Eqs. (22), (23)] at
0 , 7 104 and 2 103 is depicted in the top right inset with solid, dotted, and dashed lines, respectively.
16
1.5
0(t )
1 0.5
(t )
/ rel = 1 0
/ rel = 5
-0.5 -1 -1.5 -0.2
0
0.2
0.4
t /
0.6
0.8
1
1.2
Fig. 1
17
Fig. 2
18
0.025
0
10-6
0
0.000001
0
1/v 0
/v 0
0.02
0.015
-10-6
rel/ L
-0.000001
0.01
0.005
0 0
0.1
0.2
rel/ L
Fig. 3
19
Figure
1.5
σ 0(t )
1
0.5
σ (t )
τ /τ rel = 1 0
τ /τ rel = 5
-0.5
-1
-1.5 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
t /τ
Fig. 1
Figure
Fig. 2
Figure
0.025
0
10-6
0
0.000001
0
1/v 0
/v 0
0.02
0.015
τ rel/τ L
-10-6
-0.000001
0.01
0.005
0 0
0.1
0.2
τ rel/τ L
Fig. 3
PII: DOI: Reference:
S0378-4371(18)31168-3 https://doi.org/10.1016/j.physa.2018.09.039 PHYSA 20100
To appear in:
Physica A
Received date : 1 May 2017 Revised date : 6 July 2018 Please cite this article as: I.V. Shapochkina, et al., Relaxation high-temperature ratchets, Physica A (2018), https://doi.org/10.1016/j.physa.2018.09.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
RELAXATION HIGH-TEMPERATURE RATCHETS
I. V. Shapochkina,1-3* V. M. Rozenbaum,1,2,4 S.-Y. Sheu,5† D.-Y. Yang,1‡, S. H. Lin,1,2 and L. I. Trakhtenberg6 1
Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan Department of Applied Chemistry, National Chiao Tung University, 1001 Ta Hsuen Road, Hsinchu, Taiwan 3 Department of Physics, Belarusian State University, Prospekt Nezavisimosti 4, 220050 Minsk, Belarus 4 Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Generala Naumova str. 17, Kiev, 03164, Ukraine 5 Department of Life Sciences and Institute of Genome Sciences, Institute of Biomedical Informatics, National Yang-Ming University, Taipei 112, Taiwan 6 Semenov Institute of Chemical Physics, Russian Academy of Sciences, Kosygin Street 4, Moscow 119991, Russia; State Scientific Center of Russian Federation, Karpov Institute of Physical Chemistry, Moscow, 105064 Russia 2
HIGHLIGHTS Periodic relaxation processes are included in ratchet operation. Jump-like spatial changes of potential profiles are considered. Different asymptotics of the ratchet velocity vs the relaxation time are revealed. ABSTRACT
We consider the overdamped motion of a Brownian particle in an asymmetric spatially periodic potential which fluctuates periodically in time, under assumption of finite duration of the relaxation response of the system on deterministic dichotomous fluctuations. It is assumed that the period of these fluctuations is much larger than the characteristic diffusion time and the potential barrier height is small as compared to the thermal energy (an adiabatic hightemperature flashing ratchet). We derive an analytical expression for the average particle velocity, which is concretized for a saw-tooth potential profile. It is revealed the different, linear and quadratic, asymptotic behavior of the average velocity as a function of the relaxation time for extremely and not extremely asymmetric potential profiles, respectively. The result is interpreted in terms of the self-similar representation.
Keywords: Driven diffusive systems; Brownian motor; Adiabatic flashing ratchet __________________________
* [email protected] ; † [email protected] ; ‡ [email protected]
1. Introduction
Driven diffusive systems, which function far from equilibrium in spatially periodic structures [1,2], can serve as a good example of those with many properties following from the competition between their characteristic times [3-5]. For example, the spatiotemporal synchronization between the spatial period of a potential and its time modulation leads to the acceleration of diffusion (in comparison with the cases of free diffusion) and to a finite net diffusion in the absence of thermal noise [6,7]. There are many papers on diffusion and dynamics in ratchet structures in which peculiarities of the observed phenomena are the consequence of the competition between parameters of the nonequilibrium noise and the system considered [8-10]. Artificial nano-devices that can rectify non-equilibrium fluctuations of different nature to the directional motion of particles are usually described by the so-called Brownian motors (ratchets) [11,12], theoretical models based on diffusion dynamics, in which particle potential energy itself U ( x, t ) or its first derivative U x ( x, t ) is considered as a periodic function of coordinate x and
time t [1]. The spatial period L , together with the diffusion coefficient D , determines the main characteristic time: the diffusion time L L2 / D . For potential profiles with large gradients, one more system characteristic time arises, l l 2 / D L , where l L is the width of a sharp part of the profile. These two characteristic times can compete with those resulting from the certain temporal peculiarities of potential profiles, for example, with the period of the potential energy changes or with a much smaller time rel determined by the portion of a sharp temporal change, if any, of U ( x, t ) . This competition typically leads to the nontrivial behavior of Brownian motors [5]; here the jump-like behavior of motor characteristics is a prominent example, specific for both analytical and numerical treatment. One more time which can “come into play” is the velocity relaxation time [13] (the time that is required for the system to establish a Maxwell distribution) in the theory of inertial Brownian motors. For quantum [4] and rotary [14,15] ratchets, there also exist competition-induced effects. The interplay of characteristic time scales should be analyzed analytically (especially when in competition of limits) rather than numerically: Numerical analysis is hardly successful for infinitely large or vanishingly small quantities. So, the analytical expressions for quantities under study are very preferable. There is a possibility to have them provided by the theory of high-temperature ratchets, initially developed in Ref. [16] and then adapted to so-called Brownian photomotors [17]. These objects are one of the bright examples of ratchet systems with competing characteristic times because in photomotors there always exist relaxation processes of finite duration. They are characterized by the delay between laser impulses and the 2
response of the electron subsystem of photomotors, so that one can expect new effects as a result of what place relaxation times occupy in the hierarchy of the system characteristic times. In the present paper, we consider an adiabatic “version” of a Brownian photomotor (an adiabatic flashing ratchet) with the following characteristic times: the relaxation time, rel , and the diffusion time over the small characteristic length, l . As a comparatively simple model potential, implying sharp parts in its profile, we analyse a saw-tooth potential, which plays a key role in ratchet theory due to its piecewise-linear structure (leading to analytical results) and potentiality of its experimental realization [1,5-7,9,18-27]. The structure of the paper is as follows. In Sec. II, using the symmetrized representation for the average particle velocity suggested in Ref. [17] for the additive-multiplicative form of the spatial-temporal dependence of the particle potential energy, we derive, for large values of time period of energy fluctuations ( L , rel ), the expression for the average ratchet velocity, containing a separate summand responsible just for the relaxation-induced effect. The effective fluctuating potential technique [5] turns the resulting expressions into the integral relations (the Green’s function method) which are the core result as they allow analysis of the influence of relaxation times on the ratchet behavior. In Sec. III a model saw-tooth potential characterized by sharp links in its profile is used to get analytical expressions for the quantities of interest, and the mentioned analysis is carried out both analytically and graphically. The subtle interplay of time and spatial limits are analyzed in terms of a universal self-similar representation, which describes continuous transition from quadratic to linear rel -dependence of the average ratchet velocity, also predicted in this Section. Section IV includes discussions and conclusions.
2. Main expressions
It is well-known that the ratchet effect is caused by non-equilibrium fluctuations of parameters of a system, natural or artificial, which result in fluctuations of spatially dependent ratchet potential energy U ( x, t ) in time [1]. The overdamped motion of a Brownian particle under the action of the force F ( x, t ) U ( x, t ) / x is determined by the Langevin equation:
dx / dt F ( x, t ) (t ) ,
3
(1)
where is the friction coefficient, and (t ) is the random force (zero-mean Gaussian deltacorrelated white noise with
(t ) (s) 2 kBT (t s) , where k B is the
(t ) 0 and
Boltzmann constant, T is the absolute temperature, (t ) is the delta function, and ... denotes here the ensemble average). We begin here with the consideration of a so-called hightemperature ratchet in which spatiotemporal changes of the potential energy are small compared to the thermal energy. For ratchets of this type, analytical representation of their characteristics can be obtained, which allows, among others, analyzing the important dependence of the average ratchet velocity on the frequency of potential energy fluctuations of different nature. Limitations of the high-temperature approach are that it is unsuitable for the description of processes of suppressing the backward motion (occurring only at low temperatures) which are important for high-efficient ratchets [28]. We will consider the potential energy of the additive-multiplicative form U ( x, t ) u ( x) (t ) w( x) ,
(2)
with spatially L -periodic functions u ( x) and w( x ) , and -periodic function (t ) which describes temporal variations of the potential energy and satisfies supersymmetric condition
(t ) (t / 2) . Note that the form (2) is a legitimate model for a number of theoretically and practically interesting systems [1]. Using the Fourier-transformation for all the functions involved f ( x, t ) f qj exp(ikq x i j t ), kq 2 L q, j 2 j , q, j 0, 1, 2, ... ,
(3)
qj
we can express the ratchet velocity averaged over the period (from here and after
... 1 dt... ) as a sum of Fourier harmonics of the potential energy [17]: 0
v 2i 3 D3
qq ( 0) ( q q 0)
kq kq kq q kq2 kq2 wq wqu q q Dkq2 , Dkq2 ,
2j j
a, b j 1
2 j
2
a 2 2j b2
4
(4)
,
(5)
(k BT ) 1 is the inverse thermal energy. These expressions are valid for high-temperature expansion up to the third order in the ratio of potential energy changes to the thermal energy [16]. Next, let us assume that the function (t ) describes a periodic relaxation process and hence obeys the relaxation equation d (t ) / dt 0 (t ) (t ) / rel with the relaxation time rel . The
function
0 (t )
/ 2 m t (m 1)
takes
the
1
values
and
1
at
m t / 2 m
and
( m is an integer), respectively, and describes a relaxation-free
deterministic dichotomous process (a stochastic analogue of the dichotomous process is successfully applied to describe ratchet systems, in particular, biological ones [1]; deterministic processes, although not so extensively studied in the context of Brownian motors, are of great importance in the development of nano-devices [11,12]). We can write a periodic solution of the relaxation equation, with the boundary condition ( / 2) (0) on a half-period, and its nonzero frequency Fourier components as follows:
(t ) 1 2 1 e /(2
Fig. 1
rel )
1
et / rel , 0 t / 2, 2 j 1
4i . 2 j 1 (1 i2 j 1 rel )
(6)
Functions (t ) and 0 (t ) are depicted in Fig. 1. Substituting the quantities 2 j 1 from Eq. (6) into Eq. (5) and analytical summation over j gives the result [17]:
3 tanh( / 4 rel ) 2 rel tanh( a / 4) tanh( b / 4) (a, b) . (7) [1 (a rel )2 ][1 (b rel )2 ] [(a rel ) 2 1](a 2 b2 )a [(b rel ) 2 1](b2 a 2 )b
Under the adiabatic assumption, rel and L ( L L2 / D ), Eq. (7) is simplified and takes the form:
Dkq2 , Dkq2
1 q q
3 2 rel
qq (q q )(1 q )(1 q )
, q Dkq2 rel .
(8)
Substituting Eq. (8) into Eq. (4) gives the expression for the desired average particle velocity
v
2i L
3
qq ( 0) ( q q 0)
1 q q
q q wq wqu q q . (1 q )(1 q ) qq
5
(9)
In the absence of the relaxation process ( rel 0, q 0 ), the formula (9) is reduced to the known one [16]:
v
0
4i L
wq wqu q q
3
.
q
qq ( 0) ( q q 0)
(10)
Then the average ratchet velocity can be written in the form
v v 0 v 1,
(11)
that is as a sum of the above relaxation-free part and the relaxation-induced part
v 1
2i L
3
qq ( 0) ( q q 0)
2 4 3 D 2 rel L
qq q q wq wqu q q (1 q )(1 q ) qq
qq ( 0) ( q q 0)
wq
wq
(1 Dkq2 rel ) (1 Dkq2 rel )
(12) u q q
to be found. The primed quantities wq ikq wq and uq ikq wq are the Fourier components of the derivatives w( x) dw( x) / dx and u( x) du ( x) / dx , respectively. Using the concept of the effective fluctuating potential suggested in Ref. [5], we can reduce the double sum in Eq. (12) to the following integral relation: 2 4 3 D 2 rel 2 0 dxu( x)w ( x) , L L
v 1
(13)
which contains the first derivative of the effective potential
w( x) q
wq 1 Dkq2 rel
exp(ikq x)
the function
6
1 L rel
L
dyG( x y)w( y) ; 0
(14)
G( x) rel q
exp(ikq x) 1 Dkq2 rel
rel
z cosh z 1 2 x L 1 L , , z sinh z 2 rel
(15)
is the Laplace representation G( x y, ) dtG( x y, t ) exp(t ) of the Green’s function for 0
1 free diffusion in the interval [0, L] , with the periodic boundary conditions, at rel .
The general expressions for the average velocity of the relaxation high-temperature ratchet represented by Eqs. (11), (13)-(15) are the main result of this paper. The relaxation processes influence the system behavior through the ratio of the system characteristic times, of relaxation rel and of diffusion L , entering the Green’s function (15). At small rel , some additional features in ratchet behavior are attributed to the appearance of the characteristic time
l , which grows from the presence of sharp portions of the small width l in the potential profile, that can be comparable with the time rel . For a sawtooth potential, such a case admits analytical treatment.
3. Sawtooth potential We consider, as the functions u ( x) and w( x ) in Eq. (2), two spatially periodic potentials, both having linear portions proportional to x / l at x [0, l ] and ( L x) /( L l ) at x [l , L] , and Fig. 2
characterizing by energetic barriers, respectively, u0 and w0 (Fig. 2). In the case of an adiabatic relaxation-free flashing ratchet with a saw-tooth potential, the expression for its average velocity is known for arbitrary temperatures (see Eqs. (2) and (3) in Ref. [23]). To apply the hightemperature approximation, this expression must be expanded over the small quantities u0 / kBT and w0 / kBT , or, equivalently, from the analytical summation in Eq. (10), the same result follows
v
0
1 2 l L v0 , , v0 3u0 w02 . 45 L
(16)
The integrals in Eqs. (13) and (14) in their turn give the desired relaxation contribution v 1 :
v 1
1 2 v0 6 f1 ( z , ) 3 f 2 ( z , ) f1 ( z , ) f 2 ( z, ) 16 ( z 2 ) 2
7
(17)
where
f1 ( z, ) 1
sinh z sinh z , z sinh z
f 2 ( z, ) 1
sinh z ( ) l , , ( )sinh z L
1 .
(18)
Thus, we have obtained the average particle velocity in the model of the adiabatic relaxation flashing ratchet with arbitrary values of the relaxation parameter z and with the potential profile shape implying the presence of jumps (parts with large gradient) in it. The temperature dependence is introduced to the average velocity
v
by the dimensionless variable z
(depending on rel and L ) and the scaling velocity factor v0 (see Eqs. (15) and (16), where
L L2 / D L2 is the temperature-dependent diffusion time). The main time parameter of the dichotomous deterministic driving, its period , is also entered into v0 . The notable aspect is that the similar expressions for the average velocity were obtained in Ref. [5] in case of particle motion caused by the stochastic dichotomous (instantaneous) changes of the ratchet potential profile. Such a similarity can be interpreted by the presence of “exponential processes” in the two systems: on the one hand, the relaxation contribution v 1 , considered in the present paper, results from the presence of the exponential relaxation processes in the periodic time dependence (t ) of the potential energy [see Eq. (6)]; on the other hand, the stochastic dichotomous processes, in Ref. [5], are characterized just by the exponential correlation function. Thus, the exponential behavior in the “decay” of a potential profile seems to be the cause of the substantial similarity discussed. The character of rel / L -dependence of the average velocity v (see Fig. 3 for different Fig. 3
values) essentially changes when the extremely asymmetric potential profile ( 0 ) loses this quality ( 0 ). One can see that the more the departure of the profile from the extremely asymmetric one, the less the ratchet effect can be reached. Note that this conclusion agrees with the experimental result described in Ref. [24] that the maximal ratchet effect is observed for the extremely asymmetric shape of the sawtooth side of the pattern. At rel L ( z 0 ), as it is easy to check, the relaxation contribution (17) is compensated by the nonrelaxation one (16) so that the average particle velocity (11) tends to zero in this limit. The opposite limit, rel L ( z ), deserves special consideration because, in this case, the result depends on whether there are large gradients in the potential relief (the potential is sharp) or not. Indeed, in the case of the potential with jump-like spatial changes ( 0 ), the relaxation-induced velocity (17) becomes 8
v 1
v0 cosh 2 z 5 1 coth z 2 v0 rel , 2 2 2 z 4sinh z 4 z z z L
(19)
while at 0 and z it tends to
2
1 2 rel v 1 4 v0 . z ( ) 2 L
(20)
The quadratic asymptotics (20), obtained for the potential profile different from the extremely asymmetric one, is in accordance with the result of Ref. [29] in which the quadratic dependence of the average ratchet velocity on the duration of transient processes has been predicted, but based on the assumption that they are described by linear functions of time. The difference in the asymptotic behavior of the relaxation contribution to the velocity at
0 and 0 means that the result depends on the sequence of the limits 0 and z . If so, a reasonable strategy is to analyse these limits keeping the value of the product z fixed (finite and arbitrary). With this problem definition, we can obtain the both limits from a single continuous description. Indeed, the velocity (17) as a function of two variables, z and , can be rewritten as a function of one self-similar variable:
1 z rel 2 l
1/ 2
,
(21)
which is determined by the ratio of the relaxation time rel to the much smaller diffusion time
l l 2 / D tending to zero at l 0 . Thus, for the desired relaxation contribution to the ratchet velocity, we have the self-similar representation (the scaling law [30])
v
1 0, z z finite
v0 f
rel , L
(22)
where f
1
8 7 4( 2)e 8 3
9
2
e
4
1 , rel 2 l
1/ 2
.
(23)
It is easy to check that the limiting cases (19) and (20) follow from Eqs. (22), (23) at
0 and , respectively. Continuous transition from quadratic to linear rel -dependence of v
1
at rel 0 is depicted in the insert of Fig. 3. Note that the quadratic dependence occurs
in the narrow region of small rel -values corresponding to fast relaxation processes. Thus, at small rel -values, the presence of the relaxation behavior in time dependence of a potential profile suppresses the ratchet effect the more the less the asymmetry of the profile.
4. Discussions and conclusions In a vast majority of ratchet models, a simplified assumption of instantaneous switching of the ratchet potential is used [1,5-7,9,16,18,19,28,31]. This means that there is no energy exchange between the ratchet and the thermal environment during the switching. If, additionally, the period of potential profile fluctuations in time is assumed to be much larger than the characteristic diffusion time L , one deals with the so-called adiabatically fast driven ratchet; just adiabatically driven ratchet systems demonstrate the best (most efficient) regime of operation (maximum values of the average velocity and energy characteristics, in comparison with the nonadiabatic ones) [31-38]. In reality, there always exists a nonzero time interval during which the ratchet potential changes; it can be the duration of a transient process [29] or the relaxation time rel . Corrections to ratchet characteristics resulting from the presence of relaxation processes can be interpreted as nonadiabatic contributions. Brownian photomotors [17] can serve as an example of systems with relaxation processes of finite duration, which is characterized by the delay between rectangular-shaped laser pulses and the response of the electron subsystem of a photomotor. Thus, understanding the effect of relaxation processes on ratchet characteristics is of theoretical and practical relevance, especially in the design of molecular nanomachines. It is known that nontrivial effects in ratchet behavior result from the interplay of the system characteristic times. For example, in Ref. [5], it has been shown that just the competition of the average period of stochastic fluctuations of a sharp periodic ratchet potential and the characteristic diffusion time l over its small characteristic length l leads to the jump-like changes in ratchet characteristics. A nonanalytical behavior in characteristics of inertial diffusion transport can be also expected as a result of the interplay of times: the sliding time on a sharp slope of the potential and the characteristic velocity relaxation time [13]. In the present paper, the diffusion time l competes with the relaxation time rel , and the analysis of ratchet systems under this competition revealed the similarity (heretofore nonobvious) of stochastic and 10
deterministic ratchets, in their operation and description, resulting from the presence of exponential behavior in the “decay” of the potential profile. Using the derived analytical expression for the average velocity of the relaxation hightemperature ratchet, which has been concretized for a saw-tooth potential profile of arbitrary asymmetry, we have revealed the distinction in asymptotic behavior, at small rel / L values, of the velocity as a function of the relaxation time, namely, linear and quadratic asymptotics, for potential profiles with and without jumps, respectively. The effect has been interpreted in a uniform manner with the self-similar representation, and the both asymptotics appeared at different limiting values of the self-similar variable. Thus, the analysis of the position of relaxation times in the hierarchy of system characteristic times allows a step forward in understanding Brownian ratchets. We considered the effect of presence of sharp parts in a potential profile on characteristics of high-temperature deterministic ratchets with potential energy fluctuations described by relaxation-type periodic processes. As a simplest example accounting for existence of a sharp slope in a potential profile, a sawtooth potential with relaxation changes of its amplitude has been chosen. Sawtooth shapes (including sawtooth potential profiles) are widely used in both experimental set-ups and theoretical description of ratchet systems. Sawtooth ratchet elements are suggested to be used, e.g., in molecular pumps [39,40], selective sorting of charged membrane components [24,25], organic electronic ratchets [26,27], etc. One should take into account that such potential reliefs contain cusp points resulting from the piecewise-linear shape of approximating functions which simplify mathematical treatment. Real potential profiles can be arbitrary close to those of a sawtooth shape, but cusp points of those are somehow or other rounded with a small rounding radius. This rounding can change ratchet characteristics only at low temperatures. The reason is that, at low temperatures, the main contribution to distribution functions and barrier factors give just the regions near cusp points (see the quantitative analysis of the influence of cusp point rounding on ratchet characteristics on page 5 in Ref. [41]). Within the high-temperature approximation which we use in this paper, the results weakly depend on these features in a potential shape. Therefore using a sawtooth potential with a sharp link l is not only very convenient for deriving analytical dependencies of desired quantities on a single parameter l (which characterizes applied forces on narrow links of the potential profile) but well grounded.
11
Acknowledgments This work was supported by National Chiao Tung University and Academia Sinica. I.V.S., V.M.R., and S.H.L. thank the Ministry of Education, Taiwan (“Aim for the Top University Plan” of National Chiao-Tung University). S.Y.S. thanks the Taiwan Ministry of Science and Technology for partial support (Grant No. MOST105-2113-M-010-003). D.Y.Y. thanks the Taiwan Ministry of Science and Technology for partial support (Grant No. MOST105-2119-M-001-022). S.H.L. thanks the Taiwan Ministry of Science and Technology for partial support (Grant No. MOST104-2923-M-009-001-). L.I.T. thanks the Russian Foundation for Basic Research (Grants No 18-57-00003 and 18-29-02012) and Ministry of Science of Russia in the frame of state assignment 0082-2018-0003 (the registration number АААА-А18118012390045-2) for financial support. I.V.S. thanks Belarusian Republican Foundation for Fundamental Research for partial support (Grant No Ф18P-022). I.V.S. and V.M.R. gratefully acknowledge the kind hospitality received from the Institute of Atomic and Molecular Sciences.
12
References
[1] P. Reimann, Brownian motors: noisy transport far from equilibrium, Phys. Rep. 361 (2002) 57. [2] A. Schadschneider, D. Chowdhury, K. Nishinari, Stochastic Transport in Complex Systems: From Molecules to Vehicles, Elsevier, 2010. [3] Delvenne, J.-C. et al. Diffusion on networked systems is a question of time or structure. Nat. Commun. 6:7366 doi: 10.1038/ncomms8366 (2015). [4] B. Lau, O. Kedem, M. A. Ratner, and E. A. Weiss, Identification of two mechanisms for current production in a biharmonic flashing electron ratchet, Phys. Rev. E 93 (2016) 062128. [5] V. M. Rozenbaum, I. V. Shapochkina, S.-Y. Sheu, D.-Y.Yang, and S. H. Lin, Hightemperature ratchets with sawtooth potentials, Phys. Rev. E 94 (2016) 052140. [6] A. A. Dubkov and B. Spagnolo, Acceleration of diffusion in randomly switching potential with supersymmetry, Phys. Rev. E 72, (2005) 041104. [7] B. Spagnolo and A. A. Dubkov, Diffusion in flashing periodic potentials, Eur. Phys. J. B 50 (2006) 299-303. [8] D. Valenti, C. Guarcello, B. Spagnolo, Switching times in long-overlap Josephson junctions subject to thermal fluctuations and non-Gaussian noise sources, Phys. Rev. B 89 (2014) 214510. [9] A. Fiasconaro and B. Spagnolo, Resonant activation in piecewise linear asymmetric potentials, Phys. Rev. E 83 (2011) 041122. [10] B. Lisowski, D. Valenti, B. Spagnolo, M. Bier, E. Gudowska-Nowak, Stepping molecular motor amid Lévy white noise, Physical Review E 91 (2015) 042713. [11] P. Hänggi and F. Marchesoni, Artificial Brownian motors: Controlling transport on the nanoscale, Rev. Mod. Phys. 81 (2009) 387. [12] I. Goychuk, Molecular machines operating on the nanoscale: from classical to quantum, Beilstein J. Nanotechnol. 7 (3) (2016) 328-350. [13] V. M. Rozenbaum, Y. A. Makhnovskii, I. V. Shapochkina, S.-Y. Sheu, D.-Y.Yang, and S. H. Lin, Diffusion of a massive particle in a periodic potential: Application to adiabatic ratchets, Phys. Rev. E 92 (2015) 062132. [14] V. M. Rozenbaum, O. Ye. Vovchenko, and T. Ye. Korochkova, Brownian dipole rotator in alternating electric field, Phys. Rev. E 77 (2008) 061111. [15] T. Hiroki, T. Horita, K. Ouchi, Performance estimation for two-dimensional Brownian rotary ratchet systems, J. Phys. Soc. Jpn. 84 (4) (2015) 044004. [16] V. M. Rozenbaum, Pis’ma Zh. Eksp. Teor. Fiz. 88, 391 (2008) [JETP Lett. 88, 342 (2008)]. 13
[17] V. M. Rozenbaum, M. L. Dekhtyar, S. H. Lin, and L. I. Trakhtenberg, J. Chem. Phys 145, 064110 (2016). [18] R. D. Astumian and M. Bier, Fluctuation driven ratchets: molecular motors, Phys. Rev. Lett. 72, (1994) 1766. [19] J.-F. Chauwin, A. Ajdari, and J. Prost, Force-free motion in asymmetric structures: a mechanism without diffusive step, Europhys. Lett. 27 (1994) 421. [20] A. Parmeggiani, F. Jülicher, A. Ajdari, and J. Prost, Energy transduction of isothermal ratchets: Generic aspects and specific examples close to and far from equilibrium, Phys. Rev. E 60, (1999) 2127. [21] I. M. Sokolov, Irreversible and reversible modes of operation of deterministic ratchets, Phys. Rev. E 63 (2001) 021107. [22] K. Sumithra and T. Sintes, Effciency optimization in forced ratchets due to thermal fluctuations, Physica A 297 (2001) 1. [23] M. L. Dekhtyar, A. A. Ishchenko, and V. M. Rozenbaum, Photoinduced molecular transport in biological environments based on dipole moment fluctuations J. Phys. Chem. B 110 (2006) 20111. [24] J. S. Roth, Y. Zhang, P. Bao, M. R. Cheetham, X. Han, and S. D. Evans, Optimization of Brownian ratchets for the manipulation of charged components within supported lipid bilayers, Appl. Phys. Lett. 106 (2015) 183703. [25] M. R. Cheetham, J. P. Bramble, D.G.G. McMillan, R. J. Bushby, P. D. Olmsted, L. J. C. Jeuken, and S. D. Evans, Manipulation and sorting of membrane proteins using patterned diffusion-aided ratchets with AC fields in supported bilayers // Soft Matter. 8 (2012) 5459. [26] E. M. Roeling, W. C. Germs, B. Smalbrugge, E. J. Geluk, T. de Vries, R. A. J. Janssen, and M. Kemerink, Organic electronic ratchets doing work, Nature Mater. 10 (2011) 51. [27] P. Hänggi, Organic electronics: Harvesting randomness, Nature Mat. 10 (2011) 6. [28] V. M. Rozenbaum, T. E. Korochkova, K. K. Liang, Conventional and generalized efficiencies of flashing and rocking ratchets: Analytical comparison of high-efficiency limits, Phys. Rev. E 75 (2007) 061115. [29] V. M. Rozenbaum and I. V. Shapochkina, Nonadiabatic corrections to the velocity of a Brownian motor with a complex potential profile, Pis’ma Zh. Eksp. Teor. Fiz. 92 (2010) 124 [JETP Lett. 92 (2010) 120]. [30] G. I. Barenblatt, Scaling, selfsimilarity and intermediate asymptotics (University Press, Cambridge, 1996). [31] J. M. R. Parrondo, Reversible ratchets as Brownian particles in an adiabatically changing periodic potential, Phys. Rev. E 57 (1998) 7297. 14
[32] R. Krishnan, M. C. Mahato, and A. M. Jayannavar, Brownian rectifiers in the presence of temporally asymmetric unbiased forces, Phys. Rev. E 70 (2004) 021102. [33] V. M. Rozenbauma, D.-Y. Yang, S. H. Lin, and T. Y. Tsong, Energy losses in a flashing ratchet with different potential well shapes, Physica A 363 (2006) 211–216. [34] Y. X. Zhang, The efficiency of molecular motors, J. Stat. Phys. 134 (2009) 669-679. [35] J. M. Horowitz and C. Jarzynski, Exact formula for currents in strongly pumped diffusive systemsJ. Stat. Phys. 136 (2009) 917-925. [36] H. J. Chen, J. L. Huang, C. Y. Wang, and H. C. Tseng, Flow density of a reversible ratchet, Phys. Rev. E 82 (2010) 052103. [37] V. M. Rozenbaum and I. V. Shapochkina, Quasiequilibrium directed hopping in a timedependent two-well periodic potential, Phys. Rev. E 84 (2011) 051101. [38] V. M. Rozenbaum, Y. A. Makhnovskii, I. V. Shapochkina, S.-Y. Sheu, D.-Y.Yang, and S. H. Lin, Adiabatically slow and adiabatically fast driven ratchets, Phys. Rev. E 85 (2012) 041116. [39] R. D. Astumian and P. Hänggi, Brownian Motors, Phys. Today 55, No 11 (2002) 33. [40] R. D. Astumian, Thermodynamics and kinetics of a Brownian motor, Science 276 (1997) 917. [41] V. M. Rozenbaum, Yu. A. Makhnovskii, I. V. Shapochkina, S.-Y. Sheu, D.-Y. Yang, and S. H. Lin, Inertial effects in adiabatically driven flashing ratchets, Phys. Rev. E 89 (2014) 052131.
15
Figure caption Fig. 1. Periodic ( is the period) functions 0 (t ) and (t ) describing deterministic relaxationfree (dichotomous) and periodic relaxation (with the relaxation time rel ) processes, respectively. Fig. 2. Two spatially periodic functions u ( x) and w( x ) with energetic barriers u0 and w0 describing the static and the fluctuating parts of the sawtooth ratchet potential U ( x, t ) u ( x) (t ) w( x) .
Fig. 3. The normalized average velocity
v / v0 of the relaxation high-temperature ratchet
[calculated from Eqs. (11), (16)-(19)] vs the normalized relaxation time rel / L at different values of the asymmetry parameter: 0 (solid line) and 0.3 (dashed line); the shapes of the potential profile are shown schematically under the curves. The normalized relaxation contribution v 1 / v0 to the average velocity [calculated from Eqs. (22), (23)] at
0 , 7 104 and 2 103 is depicted in the top right inset with solid, dotted, and dashed lines, respectively.
16
1.5
0(t )
1 0.5
(t )
/ rel = 1 0
/ rel = 5
-0.5 -1 -1.5 -0.2
0
0.2
0.4
t /
0.6
0.8
1
1.2
Fig. 1
17
Fig. 2
18
0.025
0
10-6
0
0.000001
0
0.02
0.015
-10-6
rel/ L
-0.000001
0.01
0.005
0 0
0.1
0.2
rel/ L
Fig. 3
19
Figure
1.5
σ 0(t )
1
0.5
σ (t )
τ /τ rel = 1 0
τ /τ rel = 5
-0.5
-1
-1.5 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
t /τ
Fig. 1
Figure
Fig. 2
Figure
0.025
0
10-6
0
0.000001
0
0.02
0.015
τ rel/τ L
-10-6
-0.000001
0.01
0.005
0 0
0.1
0.2
τ rel/τ L
Fig. 3