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Diffusion of a massive particle in a periodic potential: Application to adiabatic ratchets

Viktor M. Rozenbaum,1,2,3,*Yurii A. Makhnovskii,1,4Irina V. Shapochkina,1,2,5Sheh-Yi Sheu,6, Dah-Yen Yang,1,and Sheng Hsien Lin1,2

1Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan

2Department of Applied Chemistry, National Chiao Tung University, 1001 Ta Hsuen Road, Hsinchu, Taiwan

3Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Generala Naumova Street 17, Kiev 03164, Ukraine 4Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Leninsky Prospect 29, 119991 Moscow, Russia

5Department of Physics, Belarusian State University, Prospekt Nezavisimosti 4, 220050 Minsk, Belarus 6Department of Life Sciences and Institute of Genome Sciences, Institute of Biomedical Informatics,

National Yang-Ming University, Taipei 112, Taiwan (Received 2 September 2015; published 18 December 2015)

We generalize a theory of diffusion of a massive particle by the way in which transport characteristics are described by analytical expressions that formally coincide with those for the overdamped massless case but contain a factor comprising the particle mass which can be calculated in terms of Risken’s matrix continued fraction method (MCFM). Using this generalization, we aim to elucidate how large gradients of a periodic potential affect the current in a tilted periodic potential and the average current of adiabatically driven on-off flashing ratchets. For this reason, we perform calculations for a sawtooth potential of the period L with an arbitrary sawtooth length (l < L) instead of the smooth potentials typically considered in MCFM-solvable problems. We find nonanalytic behavior of the transport characteristics calculated for the sharp extremely asymmetric sawtooth potential at l→ 0 which appears due to the inertial effect. Analysis of the temperature dependences of the quantities under study reveals the dominant role of inertia in the high-temperature region. In particular, we show, by the analytical strong-inertia approach developed for this region, that the temperature-dependent contribution to the mobility at zero force and to the related effective diffusion coefficient are proportional to T−3/2and T−1/2, respectively, and have a logarithmic singularity at l→ 0.

DOI:10.1103/PhysRevE.92.062132 PACS number(s): 05.40.−a, 05.60.Cd

I. INTRODUCTION

Diffusion of a massive particle is described by the Klein-Kramers equation [1,2], the analytical solutions of which are restricted to the cases of simple potentials corresponding to a linear coordinate dependence of the force [3]. That is why a number of results of the reaction-rate theory [4–6] as well as the calculated particle velocity in a tilted periodic potential [7] were obtained in the approximation of high potential barriers (compared to the thermal energy) when the vicinities of potential extrema are only important and can be considered for smooth potentials as parabolic. The Klein-Kramers equation for arbitrary potentials is solvable only by various numerical methods [8–15], among which the Risken’s matrix continued fraction method (MCFM) [3,16,17] occupies a particular place [3,18–21]. This method allows the particle current to be formally expressed in terms of concentration so that analytical relations can be found in some particular cases [22].

One of the most exciting applications of diffusion transport is a Brownian motor which models the drift of a Brownian par-ticle in a fluctuating periodic potential. It is usually considered in the overdamped regime when the inertial term is negligible compared to the damping one [23–25]. A deep insight into the motion-inducing mechanism was given by Parrondo [26], who considered, as an elementary act of directed motion, a particle move resulting from a fast transition (a jump) from one

*vik-roz@mail.ru

sysheu@ym.edu.tw

dyyang@pub.iams.sinica.edu.tw

periodic potential profile to another. He also calculated the net fraction of particles crossing some point in a certain direction over a long period of time (the integrated current) as a pivotal characteristic of adiabatic transport (Parrondo’s lemma). Some further development of this overdamped adiabatic approach can be found in Refs. [27,28]. A generalization of Parrondo’s lemma to include small inertial corrections has been proposed recently [29,30]. A notable observation made in these studies is that even small inertial corrections may help to overcome some of the symmetry restrictions inherent to the zero-mass limit and thus to produce otherwise prohibited directed motion. In the present paper we investigate the effect of large gradients in a potential profile on the transport of a massive particle. More specifically, we study Brownian motion in a periodic sawtooth potential for arbitrary inertia with a special emphasis on the limiting cases where jumps in the potential profile make it extremely asymmetric. It is important to note that, at present, a sawtooth potential is not only a theoretical idealization, but can be realized experimentally. We exemplify, following Ref. [31], such a realization by the experimental scheme of a Brownian ratchet that manipulates charged components within supported lipid bilayers. One side of the patterned bilayers was of a sawtooth shape (a planar surface was the opposite side), the asymmetry of which controlled the amount of effect. Particularly, the maximum effect was reached for the extremely asymmetric case of the side of the pattern. The same regularity for inertialess particles in a sawtooth potential was found theoretically in Refs. [32–34]. Dynamics of a Brownian particle differs essentially in a sharp and smooth potential (with and without jumps, respec-tively), the physics behind this being quite transparent. The

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energy conservation law dictates a strict relationship between the particle velocities on two sides of the potential jump and its height. Thus, a discontinuity in the velocity distribution arises in the vicinity of the jump point. We recall that in the overdamped regime the velocity distribution is Maxwellian and the potential jumps are easily taken into account by the jump conditions for the particle position probability density [3]. This is why sharp potentials are acceptable in solving overdamped diffusion problems but make impossible numerical treatment of underdamped Brownian motion. In view of this fact, the approach developed here appears uniquely helpful to study diffusion of a massive particle in a sharp periodic potential.

Another aim of the paper is to describe inertial adiabatic transport in terms of equations resembling those in the over-damped approach. For the stationary current in a tilted periodic potential and for the integrated current arising from a changing periodic potential [26], we derive analytical expressions that formally coincide with those for the overdamped approach. The expressions contain a factor (accounting for inertia effects) which can be calculated by MCFM. Applying this approach to adiabatic ratchets, we obtain explicit formulas for the average velocity of adiabatically driven on-off flashing and rocking ratchets with inertial effects included. Thus we suggest a unified inertia-relevant description of both ratchet types. A clear mathematical structure of the approach proposed makes the problem analytically treatable provided the inertia factor is analytically expressible. This is demonstrated in Sec. VI

by considering the problem at high temperatures when the potential amplitude is small compared to the thermal energy.

The structure of the paper is as follows. In Secs.IIandIII

we define the transport characteristics under study and derive equations containing the above-mentioned inertia-dependent factor. The scheme for calculating this factor, both in general operator notation and in terms of matrix Fourier-transformed equations, is given in Sec.IV. Findings concerning diffusion in a sawtooth potential are presented and discussed in Sec.Vwith the observation that the quantities concerned lose analyticity in the vicinity of the extremely asymmetric limit of the potential. The latter fact is corroborated in Sec. VI using the high-temperature approximation. In addition, the high-high-temperature asymptotics for the effective mobility and diffusivity are ob-tained in this section. The results are summarized in Sec.VII.

II. THE INTEGRATED CURRENT

Consider a Brownian particle moving along x in the spatially periodic potential V (x) with the period L during the large time interval T sufficient for the equilibrium state in this potential to be established. The initial state is assumed equilibrium in some other potential V0(x). Following Parrondo [26], we are interested in finding the net fraction of particles (x) crossing point x to the right for time T which determines the main characteristics of adiabatic transport. This quantity is defined as the reduced current J (x,t) integrated over the large time interval T ,

(x)=  T

0

J(x,t)dt. (1)

The continuity equation connecting the reduced probability density ρ(x,t) and the corresponding current J (x,t),

∂ρ(x,t)

∂t +

∂J(x,t)

∂x = 0, (2)

gives the interrelation between values (x) at points x and x0,

(x)= (x0)−  x

x0

dy[ρ(y)− ρ0(y)], (3) where ρ(x)= ρ(x,T ) and ρ0(x)= ρ(x,0) are the equilibrium probability densities in the states with potentials V (x) and V0(x) [

L

0 dx ρ(x)= 1 and the same for ρ0(x)]. Thus the difference of (x) values at two points is simply the difference of the probabilities to find the particle between these points in the initial and final distributions.

Note that Eqs. (1)–(3) are valid for both the overdamped (massless) description and the inertial one. In the first case, the integration constant (x0) is easily found since

J(x,t) is directly expressible in terms of ρ(x,t), J (x,t)= (βζ )−1exp[−βV (x)](∂/∂x) exp[βV (x)]ρ(x,t), where ζ and β = (kBT)−1, respectively, denote the friction coefficient and

the inverse thermal energy (kBis the Boltzmann constant and T

is the absolute temperature). This additional equation, together with Eqs. (1) and (3), leads to Parrondo’s result [26]. In the second inertial case, the determination of (x0) requires the probability density ρ(x,v,t) to find an inertial particle at point x with the velocity v at time t to be known. This probability density obeys the Klein-Kramers equation [1,2],

∂tρ(x,v,t)=  − ∂xv+ 1 m ∂v  ζ v+ V(x)+ ζ ∂v  ×ρ(x,v,t), (4)

where m is the particle mass and V(x) is the first spatial derivative of the potential V (x). Then the reduced functions ρ(x,t) and J (x,t) are the zero and first moments of ρ(x,v,t),

ρ(x,t)=  −∞ dv ρ(x,v,t ), J(x,t)=  −∞ dv vρ(x,v,t). (5) These functions, respectively, coincide (up to constant fac-tors) with the two first coefficients cn(x,t) [ρ(x,t)= c0(x,t),

J(x,t)= vthc1(x,t), where vth = (mβ)−1/2 is the thermal velocity] of the expansion ρ(x,v,t) over orthogonal Hermitian polynomials [3]. Equations for cn(x,t) follow from Eq. (4) and

can be written in the form [30]  n+ τv ∂t  cn(x,t)= εn ˆJ(x)cn−1(x,t) − εn+ 1 ∂xcn+1(x,t), (6) where ˆ J(x)= −e−βV (x)∂xeβV(x)= −∂x− βLV(x) (7)

is the dimensionless current operator in the overdamped limit, ∂x = L ∂/∂x is the short notation for the dimensionless

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and the dimensionless parameter ε=√τv/τD characterizes

the strength of inertia (τD = βζ L2 is the diffusion time).

Equation (6) can be easily rewritten for the time-independent quantities ϕn(x)=

T

0 dt cn(x,t), that gives Eq. (3) at n= 0

[taking into account that (x)= vthϕ1(x)] and returns us to Eq. (6) with cn(x,t) substituted by ϕn(x) and without the term

containing ∂/∂t at n 1.

The MCFM developed in Refs. [3,16,17] (and discussed in Sec. IV of this paper) allows one to write down the following recurrent relation: ϕn+1(x)= ˆSn(x)ϕn(x), where the

ˆ

Sn(x) matrix is defined at n= 0 as ˆS0(x)= ε ˆG0(x) ˆJ(x) with ˆ

G0(x) as determined by the recursive procedure which can be properly implemented in the Fourier representation scheme [see Eqs. (28) and (35) below]. Thus ϕ0(x) and ϕ1(x) are related by the equation

eβV(x)Gˆ−10 (x)ϕ1(x)= −ε ∂xeβV(x)ϕ0(x). (8) In view of the periodicity of V (x) and ϕn(x), integration of

Eq. (8) over the spatial period gives  L

0

dx eβV(x)Gˆ−10 (x)(x)= 0. (9) Then, substitution of Eq. (3) into Eq. (9) leads to the equality

(x0)= L 0 dx e βV(x)Gˆ−1 0 (x) x x0dy [ρ(y)− ρ0(y)] L 0 dx eβV(x)Gˆ−10 (x) . (10) Let us define the operator ˆq(x) generalizing the quan-tity q(x)= eβV(x)/0Ldx eβV(x) commonly used in the over-damped case, ˆ q(x)= Zin−1q(x) ˆG−10 (x), Zin=  L 0 dx q(x) ˆG−10 (x). (11)

Then Eq. (10) takes the final form (x0)=  L 0 dxqˆ(x)  x x0 dy[ρ(y)− ρ0(y)]. (12) It is worthy to note that this expression is exact for an arbitrary particle mass. In the overdamped limit ˆG0(x) is the unit operator, so ˆq(x) coincides with q(x) and Eq. (12) reduces to the known Parrondo result [26] [denoted below as m=0(x0)] as it should be. It is expedient to introduce an inertial correction, = (x0)− m=0(x0) =  L 0 dx[ ˆq(x)− q(x)]  x x0 dy[ρ(y)− ρ0(y)], (13) since it is position independent due to the identity L

0 dx[ ˆq(x)− q(x)] = 0 (in accordance with the conclusions of Ref. [30]). Therefore the position-dependent contribution to (x0) is determined solely by the known inertialess expression

m=0(x0). For further consideration, we can set x0= 0 without loss of generality and introduce the average value

of (x), ¯ = L−1  L 0 (x)dx = ¯m=0+ , (14) ¯ m=0= m=0(0)+ ¯x/L, where ¯x= ¯x − ¯x0= L 0 dx x[ρ(x)− ρ0(x)] is the distance between the average particle positions in the equilibrium states with potentials V (x) and V0(x). If V0(x)= 0 and hence

ρ0(x)= L−1, the expression for ¯x/Lcoincides with the net fraction of particles (0) crossing point x= 0 after switching off the potential V (x). That is why the average velocityv of particle motion arising due to a dichotomic process (with period τ ) of switching on and switching off the potential V(x) (so-called on-off ratchet) can be found by the formula v = (L/τ) ¯, where ¯, introduced in Eq. (14), takes the form

¯ =  L 0 dx[ ˆq(x)− L−1]  x 0 dy[ρ(y)− L−1]. (15) III. STATIONARY CURRENT IN A TILTED

PERIODIC POTENTIAL

Consider a massive Brownian particle in the periodic potential V (x) that is driven away from equilibrium by a static force F. At long times, one of the main characteristics of the particle motion in the tilted periodic potential U(x)= V (x) − F x is the stationary current J (F ). The position independence of this quantity follows from the time-independent version of Eq. (2). In order to handle contribution of inertia to J (F ) one typically uses MCFM (see, e.g., Refs. [3,18–21]). Using the F-dependent ˆS0(x; F ) matrix and the identities ρ(x,t)= c0(x,t) and J (x,t)= vthc1(x,t), we have the equality c0(x)= ˆS0−1(x; F )c1 which, with the normalization condition0Lc0(x)dx = 1 taken into account, immediately gives the result [3],

J(F )= vth  L 0 dx ˆS0−1(x; F ) −1 . (16)

The aim of this section is to write the current in the form similar to that known from the studies of the overdamped case, which allows to calculate the mobility μ at zero external force using the same matrix ˆG0(x) as in Sec.II. We start from Eq. (8) in which ϕ0(x), ϕ1(x), and V (x) should be replaced by c0(x),

c1(x), and U (x), respectively, and ˆG−10 (x)= ˆG−10 (x; F ) is F dependent. Integration over x of the thus modified Eq. (8) gives

c0(x)= e−βU(x) 

A− (εL)−1  x

0

dy eβU(y)Gˆ−10 (y; F )c1 

. (17) The two constants A and c1are determined by the normaliza-tion condinormaliza-tion0Lc0(x)dx = 1 and the periodicity condition

c0(0)= c0(L). Therefore the resulting expression for J (F ) takes the form

J(F )= (βζ )−1 1− e −βF L L 0 dx e−βU(x) L 0 dx eβU(x)Gˆ−10 (x; F )− (1 − e−βF L) L 0 dx e−βU(x) x

0 dy eβU(y)Gˆ−10 (y; F )

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This expression, just like Eq. (12), is exact for an arbi-trary particle mass and turns into the known Stratonovich formula [35] in the overdamped case. The success in obtaining this generalized expression is provided by introducing the formal relation between the particle concentration and the corresponding current which is, together with the continuity Eq. (2), equivalent to the description in terms of the Klein-Kramers Eq. (4) [22].

The expansion coefficients of the average velocityv(F ) = J(F )L over F are easily obtainable from Eq. (18), namely,

v(F ) = μF + χF2+ · · · , (19) where μ= ζ−1L2  L 0 dx e−βV (x)  L 0 dx eβV(x) −1 Z−1in (20)

is the mobility μ (at zero external force) and χ = βLμ  L 0 dx[ρ(x)− L−1]  x 0 dy[ ˆq(y)− L−1] + (βLZin)−1  L 0 dxq (x) ∂FG−10 (x; F ) F=0 (21) is the coefficient (the nonlinear response) which defines the average velocity of rocking ratchets adiabatically driven by a small force with zero average value. Note that Eq. (20) is the generalization to the inertial case of the Lifson-Jackson formula for the effective diffusion coefficient Deff of an overdamped Brownian particle in the periodic potential (related to μ by the Einstein relation Deff = kBT μ) [36].

Using the persymmetry of the matrix corresponding to the operator ˆq(x) in Fourier representation [see Eq. (38) below], it is possible to change the order of integration,

 L 0 dx[ρ(x)− L−1]  x 0 dy[ ˆq(y)− L−1] = −  L 0 dx[ ˆq(x)− L−1]  x 0 dy[ρ(y)− L−1] (22) (where the identity0Ldx[ ˆq(x)− L−1]= 0 has been used), and we arrive at the following expression:

χ=−βLμ  ¯ − (βLZin)−1  L 0 dx q(x) ∂FGˆ−10 (x; F ) F=0 , (23) where ¯is given by Eq. (15) for the on-off ratchet. The quantity [∂FGˆ−10 (x; F )]F=0 requires additional MCFM calculations

[see Eqs. (30) and (31) below] from which it follows that this quantity does not contribute to the inertia correction linear in the particle mass. Thus, for small inertia, the average velocities of rocking and on-off flashing ratchets are determined by the same quantity ¯, which is in full agreement with the conclusions of Refs. [29,30].

IV. APPLICATION OF THE MATRIX CONTINUED FRACTION METHOD

The MCFM was developed by Risken [3] and Risken and Vollmer [16,17] on the basis of the theory of continued

fractions considered in Ref. [37] (see also Refs. [38,39]). Application of this approach to different problems is discussed in the literature, e.g., in Refs. [18–21,40–43]. The MCFM is based on the formal solution of a system of equations

− ˆPn(x)cn−1(x)+ cn(x)+ ˆPn+(x)cn+1(x)= 0,

n= 1,2, . . . , (24)

where ˆPn±(x)’s are known differential operators and cn(x)’s

are unknown functions to be found. The solution of Eq. (24) is looked for in the form

cn+1(x)= ˆSn(x)cn(x). (25)

Substitution of Eq. (25) into (24) gives the recurrence relation, ˆ

Sn(x)= [ˆ1 + ˆPn++1(x) ˆSn+1(x)]−1Pˆn−+1(x), (26)

which enables expressing ˆSn(x) through ˆSn+1(x). It is assumed

that one can set ˆSN+1(x)= 0 at a sufficiently large value of

n= N so that all ˆSn(x)’s with n N can be found from

Eq. (26). This allows finding the functions cn(x) from Eq. (25)

if one of them is determined by some additional condition. By comparison of Eq. (24) with the time-independent Eq. (6) for cn(x,t)= cn(x) or its analog for ϕn(x) at n 1, the

explicit forms for ˆPn±(x) can be written as

ˆ Pn(x)= √1 nε ˆJ(x), Pˆ + n(x)= √ n+ 1 n ε∂x. (27) The insertion of Eq. (27) into (26) and introducing the new operator ˆGn(x) by the relation ˆSn(x)= ˆGn(x) ˆPn−+1(x) give the

recurrence relation connecting operators ˆGn−1(x) and ˆGn(x),

ˆ

Gn−1(x)= [ˆ1 + ε2n−1∂xGˆn(x) ˆJ(x)]−1. (28)

In the presence of the biasing force F , V (x) should be replaced by U (x)= V (x) − F x. Then, the dimensionless current operator and the operator ˆGn(x) become F dependent

and equal to ˆJ(x)+ βF L [where ˆJ(x) is given by Eq. (7)] and ˆGn(x; F ), respectively. Thus, the recurrence relation for

the operator ˆGn(x; F ) takes the form

ˆ

Gn−1(x; F )= [ˆ1 + ε2n−1∂xGˆn(x; F ) ˆJ(x)

+ ε2n−1βF L ∂

xGˆn(x; F )]−1. (29)

This relation is needed for the calculation of ˆG−10 (x; F ) that is contained in Eq. (18) for J (F ).

It follows from Eq. (29) that: ∂FGˆ−1n−1(x; F ) F=0= ε 2n−1{∂ x[∂FGˆn(x; F )]F=0Jˆ(x) + βL ∂xGˆn(x)}, (30)

where ˆGn(x)= ˆGn(x; 0) and [∂FGˆn(x; F )]F=0 can be found

from the following recurrence relation:

[∂FGˆn−1(x; F )]F=0 = −ε2n−1Gˆn−1(x){∂x[∂FGˆn(x; F )]F=0

× ˆJ(x) + βL ∂xGˆn(x)} ˆGn−1(x). (31)

Equations (30) and (31) allow the calculation of the quan-tity [∂FGˆ−10 (x; F )]F=0 needed for determining the nonlinear

response [see Eq. (23)].

Note that the above relations give small-inertia corrections [taking into account the contributions linear in ε2and setting

ˆ

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For arbitrary inertia (characterized by the parameter ε), the recurrence procedure [presented by Eq. (28)] should be limited to a definite value of n= N at which ˆGN(x)= ˆ1. The more

inertia, the larger value of N should be taken. At the same time, the formal solutions obtained in Secs.IIandIIIbecome computationally treatable by using the Fourier representation in which operator equations become matrix ones. Due to the periodicity of the potential V (x) and, as consequence, all functions of x in Eqs. (1)–(15) and (20)–(23), we define the matrix elements of an arbitrary operator ˆA(x) composed by the product of the differential operators ∂x and x-dependent

periodic functions as App = L−1  L 0 dx e−ikpxAˆ(x)eikpx, (32) p,p= 0, ± 1, ± 2, . . . ; kp = 2πL−1p.

With this notation, the matrix elements of the operators ∂x= L ∂/∂x and ˆJ(x) [defined by Eq. (7)] take the form

(∂x)pp = iLkpδpp, Jpp = −iL p kp(E−1)p−pEp−p = −iL(kpδpp+ βkp−pVp−p), (33) where Ep = L−1  L 0 dx eβV(x)−ikpx, (34) (E−1)p = L−1  L 0 dx e−βV (x)−ikpx.

Thus each recurrence step in Eq. (28) corresponds to solving the system of linear equations,

p  δpp+ iLε2n−1kp p ( ˆGn)ppJpp  ( ˆGn−1)pp = δpp. (35) Of course, for computational procedure, the allowed values of pshould be limited by a definite value pm so that p= 0, ±

1, . . . ,± pmand the number of equations in Eq. (35) equals

2pm+ 1. Parameters N and pmshould be chosen properly to

provide the required accuracy of the results (see Sec.V). The expression for the main quantity of interest ¯, given by Eq. (15), contains the integral r(x)=0xdy[ρ(y)− L−1], which can be expanded into Fourier series due to the periodic-ity condition r(L)= r(0) = 0. Since the Fourier components of ρ(x) are given by ρp = (E−1)p/L(E−1)0, the Fourier components of r(x) equal rp = p=0−i[L(E −1) 0]−1kp−1(E−1)p, (36) r0= i[L(E−1)0]−1 p=0 kp−1(E−1)p.

Thus Eq. (15) takes the form ¯ = L p=0 q0prp = −i(E−1)−10 p=0 q0pk−1p (E−1)p, (37) where qpp = (ZinE0L)−1 p Ep−p Gˆ−10 pp, (38) Zin= (E0)−1 p E−p Gˆ−10 p 0,

and we have used the equality qp = Ep/LE0for the Fourier components of q(x).

Let us now prove the identity (22). The left side of Eq. (22) contains the integral Q(x)=0xdy[ ˆq(y)− L−1], the Fourier components of which are equal to Qp= −iqp0/kpso that

 L 0 dx[ρ(x)− L−1]  x 0 dy[ ˆq(y)− L−1] = i(E−1)−1 0 p=0 q−p0k−1p (E−1)p. (39)

Comparing Eqs. (37) and (39), we conclude that the iden-tity (22) is true if q−p0= q0p. The more general equality qpp =

q−p,−p defines a property of persymmetry (the symmetry

with respect to the reverse diagonal) [44]. Thus we need to prove that the matrix qpp really possesses this property. It is

convenient to introduce the operator ˆqn(x)= q(x) ˆG−1n (x) so

that ˆq(x)= Zin−1qˆ0(x). Using Eq. (28), it is easy to show that ˆ

qn(x) satisfies the following recurrence relation:

ˆ

qn−1(x)= q(x) − ε2n−1q(x)∂x[ ˆqn(x)]−1∂xq(x), (40)

which corresponds to the following relation for matrix ele-ments: ( ˆqn−1)pp = (LE0)−1Ep−p− ε2n−1(LE0)−2 × p,p Ep−pkp ˆ qn−1ppkpEp−p. (41)

Since any matrix App describing multiplication by periodic

function A(x) is expressed through its Fourier component Ap−p and belongs to persymmetric matrices, the matrix

( ˆqN)pp = qp−p is a persymmetric one. Let us assume that

the matrix ( ˆqn)pp is also persymmetric, and therefore the

same property is true for the matrix ( ˆqn−1)pp of the reverse

operator ˆqn−1 [44]. Substitution of ( ˆqn−1)pp = ( ˆqn−1)−p,−p

to Eq. (41) and changing the summation indices pto−pand pto−plead to the equality ( ˆqn−1)pp = ( ˆqn−1)−p,−pwhich

completes the proof of persymmetry of the matrix ( ˆqn)pp by

the mathematical induction method. Since qpp = Z−1in ( ˆq0)pp,

the equality qp0= q0,−pfollows from ( ˆqn)pp= ( ˆqn)−p,−pat

n= 0 and p= 0. Thus, the identity (22) has been proved. V. APPLICATION TO A SAWTOOTH POTENTIAL A sawtooth potential is commonly used in theoretical stud-ies due to its piecewise-linear structure that allows analytical calculations. For our purposes, using this potential gives a possibility to elucidate how a large gradient of a periodic potential affects the characteristics under study (Secs. II

and III). Besides, this allows us to operate with analytical expressions for matrix elements entering into the numerical scheme discussed above. On the other hand, the presence of large gradients requires the summation over many harmonics of the potential and a large value of the limiting parameter pm

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(for prescribed accuracy to be reached) unlike, for example, the simple case of cosine potential where pm can be taken

small enough [3].

It is convenient to choose a coordinate system in which the Fourier components of the quantities under study have the simplest form. Of course, such a choice has no influence on the final results. If Ap is a Fourier component in the initial

coordinate system, the Fourier component ˜Ap in the new

coordinate system shifted along the x axis by a can be found by the formula ˜Ap = exp(ikpa)Ap. One can check that the

factor exp(ikpa) does not contribute to the quantities under

study. We introduce dimensionless units for length and energy so that the length is measured in units of the potential period L and the energy in units of the potential barrier V . A sawtooth potential can be parametrized in the main period by the following odd function: V (x)= x/l for −l/2 < x < l/2 and V(x)= (±1/2 − x)/(1 − l) for the intervals l/2 < x < 1/2 and−1/2 < x < −l/2, respectively, where l and 1 − l are the sawtooth lengths (0 < l < 1). The Fourier components of the potential gradient f (x)= V(x) are written as fp =

sin(πpl)/[πpl(1− l)]. Figure 1(a) presents the dependence fp on p for several values of l. The smaller l (or 1− l), the

more harmonics should be taken into account in the calculation scheme. The number of sum terms pmrequired for calculations

can be estimated as pm∼ l−1 at l→ 0. On the other hand,

fp→ 1 (for arbitrary p) at l → 0 (or 1 − l → 0) so that we

can expect the singular behavior of quantities of interest in the case of an extremely asymmetric sawtooth potential.

As pointed out above, the advantage of the sawtooth potential is that it provides analytical expressions, including the expression for Epdefined by Eq. (34),

Ep = 2u sinh[(u− ikpl)/2] u2+ i(1 − 2l)uk p+ l(1 − l)k2p , (42) u= βV, kp = 2πp

The expression for (E−1)pis obtained by the replacement of u

by−u in Eq. (42). The dimensionless parameter u specifies the ratio of the potential barrier V to the thermal energy kBT and

serves as the dimensionless inverse temperature. In addition to Epand (E−1)p, we need the matrix ( ˆG−10 )ppfor the quantities

¯

, Zin, and μ to be calculated from Eqs. (37), (38), and (20). It can be found from the recurrence procedure,

pm p=−pm ⎧ ⎨ ⎩δpp+ ε2n−1kp⎣kp( ˆGn)pp − iu pm p=−pm ( ˆGn)ppfp−p ⎤ ⎦ ⎫ ⎬ ⎭( ˆGn−1)pp = δpp. (43)

Note that in the overdamped case the explicit analytical expressions for ¯and μ (marked with subscript m= 0 below) are known for a sawtooth potential [29,30],

¯

m=0 = (1/2 − l)[coth(u/2) + (u/2)/sinh2(u/2)− 4/u],

(44) μm=0 = ζ−1(E−1)−10 E0−1= ζ−1(u/2)2/sinh2(u/2).

(a) -0.4 0 0.4 0.8 1.2 0 50 100 150 200 p fp l = 0.025 l = 0.1 l = 0.01 (b) 0.066 0.067 0.068 0.069 0.07 20 40 60 80 100 120

p

m ΔΔΦ N = 50 N = 100 N = 150 ξ = 4, u = 1, l = 0.1

FIG. 1. (Color online) (a) The Fourier components fp of the

gradient of the sawtooth potential V (x) (depicted in the top right frame) for several sawtooth lengths l. (b) The dependence of the inertial correction  calculated by MCFM on the limiting parameter pmfor several values N .

Thus the mobility μ (at F = 0) can be calculated as μ = μm=0/Zinand the correction  as ¯− ¯m=0from Eq. (14).

It is convenient to characterize inertia by the temperature-independent parameter,

ξ = ε2u= τv/τL= mV /ζ2L2, (45)

which is the ratio of the velocity relaxation time τvto the sliding

time τL= ζ L2/Von the distance L in the overdamped regime.

Figure 1(b) provides insight into the values of the limiting parameters N and pm which ensure the required calculation

accuracy. In a wide range of values of ξ , u, and 0.01 < l < 0.99, an accuracy of up to three significant figures can be reached at N= 100 and pm= 100. It is worthy to mention that

MCFM calculations with the cosine potentials require much lower values of N and pm(namely, N ≈ 20 and pm≈ 12 for

the same accuracy) [3].

Figures2(a)and2(b)demonstrate the l dependences of ¯ and μ for several values of ξ calculated at u= 1, N = 100, and pm= 100. The point l = 0.5 is the center of symmetry

for ¯(l) in the interval 0 < l < 1 [so that ¯(l) < 0 at 0.5 < l <1] and determines the position of the symmetry axis for

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(a) 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 l 0.026 0.028 0.03 0.032 0.034 0 0.005 0.01 l

Φ

Φ (b) -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0 0.1 0.2 0.3 0.4 0.5 l μ (l ) - μ (0.5) 0.72 0.732 0.744 0 0.005 0.01 l μ 0.5 0.75 1 0 1 ξ 2 μ (0 .5 )

FIG. 2. (Color online) (a) The l dependences of the average integrated current after switching on the sawtooth potential and (b) the shift of the particle mobility μ(l) (at zero external force) relative to its value μ(0.5) at the central point l= 0.5. The calculations have been performed by MCFM at u= 1, N = 100, and pm= 100. The

detailed l dependences near the point l= 0 for different pm’s are given

in the corresponding frames. The curves in panel (b) are marked as in panel (a). The dependence of μ(0.5) on ξ is presented in the bottom left frame in panel (b).

μ(l) in the same interval. In other words, ¯and μ are odd and even functions of the asymmetry parameter κ= 1 − 2l. These dependences exhibit a jumplike behavior near the point l= 0 (or l = 1). The more detailed behavior near this point is depicted in the insets of Fig.2. The greater the value of pm,

the closer the smooth portion of the curve approaches the y axis. One can say that there is such a value of pmat which the

calculated value of ¯or μ belongs to the smooth portion of the corresponding curve. This is true for an arbitrarily small but nonzero l. On the other hand, the calculation of ¯and μ at l= 0 (when fp= 1 for all p) does not require large values of

pm, and the result can differ from that of l→ 0. This means that

the function A(l) ( ¯or μ in our case) satisfies the inequality: liml→0A(l)= A(0). An attempt to explain the physical cause

of such a nonanalytic behavior is given in Sec.VIwhere we develop an analytical approach valid in the high-temperature region.

The mobility μm=0 (at F = 0) of an inertialess particle

in a sawtooth potential does not depend on the sawtooth lengths l and 1− l [see Eq. (44)]. It is impressive that such

0 0.1 0.2 0.3 0.4 12 8 4 0 u 0 0.1 0.2 0 4 8 12 u ΔΔΦ 0 0.5 1 0 5 10 μ

Φ

l = 0.1

FIG. 3. (Color online) The inverse temperature dependences of the average integrated current after switching on the sawtooth potential. The corresponding dependences of the inertial correction are depicted in the top left frame. The curves, from bottom to top, are in the order of increasing the inertial parameter ξ from ξ= 0 (the inertialess case, the dotted curve) to ξ = 0.01,0.05,0.25,1,4, and 25 (the solid curves). The inverse temperature dependences of the particle mobility μ (at zero external force) are presented in the bottom right frame. The curves, from top to bottom, correspond to the same increasing order of ξ values. The calculations have been performed by MCFM at l= 0.1, N = 100, and pm= 100.

a dependence arises in the inertial case. The dependences presented in Fig. 2(b) show that the mobility decreases monotonically with increasing the asymmetry parameter κ for ξ <4 with maximum deviation at ξ ∼ 0.1. For strong inertia (the underdamped case) at ξ > 4, the dependences become nonmonotonic.

It follows from the inverse temperature dependences pre-sented in Fig. 3 that inertia has the opposite effect on the average integrated current ¯ and the particle mobility μ: The former increases, and the latter decreases with increasing inertia. In Ref. [29] we have proved such a behavior of μ for arbitrary periodic potentials but only by consideration of small inertia corrections linear in ξ . In this approximation,

¯

 increases with the increase in inertia at all temperatures for a sawtooth potential and, at high temperatures, for simple periodic potentials described by the first two harmonics. The high-temperature asymptotics of ¯ is proportional to T−3 for the overdamped case ( ¯= ¯m=0) [34,45–47] and to T−2

when the inertia is small [29,30]. For strong inertia, the linearlike dependence of ¯is observed in Fig.3at small u (but not belonging to the very narrow area near u= 0 considered in Sec.VI) so that ¯∝ T−1. Although ¯dominates ¯m=0at

high temperatures, the deviation ¯from ¯m=0is maximal at

u∼ 4.

VI. HIGH-TEMPERATURE BEHAVIOR

The high-temperature approximation is a fruitful analytical approach which allowed us to obtain a number of useful regularities both for inertialess ratchets [34,45–47] and for the ratchets with the small inertia [29,30]. That is why it is reasonable to apply this approximation to the strong-inertia case considered in this paper. The small parameter is u= βV , where V is the maximum potential change on its spatial

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period. At u 1, the inertia parameter ε can be arbitrarily chosen (more precise inequalities are given at the end of this section after the final expressions are derived). With this approximation, it is convenient to rewrite Eqs. (37) and (38) for the quantities of interest as follows:

Zin= 1 + uε2(E0)−1 p,p=0 kpkpE−p( ˆG1)ppVp, (46) ¯ = Zin−1¯m=0− iε2Zin−1(E0)−1(E−1)−10 × p,p=0 kpE−p(E−1)p( ˆG1)pp + u p,p=0,p(=p) kp−1kpkp−pE−p(E−1)p( ˆG1)ppVp−p  , (47) where ¯ m=0 = −i(E0)−1(E−1)−10 p=0 kp−1E−p(E−1)p (48)

is the inertialess contribution and kp = 2πp. Since Ep and

(E−1)p are of order u at p= 0, the lowest approximation in

urequires the matrix elements ( ˆG1)ppto be independent of u

for Zinand to contain the linear correction in u for ¯. Since the matrices ( ˆGn)ppare diagonal at u= 0, the matrix ( ˆG1)pp

can be written in the approximate form

( ˆG1)pp ≈ gp(1)δpp+ ugpp(1). (49)

Here g(1)p and g

(1)

pp define diagonal and nondiagonal [see

Eq. (52) below] contributions, respectively.

As follows from Eqs. (28) and (35), the continued fraction g(1)

p is determined by the recurrence relation,

gp(n−1)=  1+ε 2k2 p n g (n) p −1 , n= 2,3, . . . ,N, (50) gp(N )= 1, N 1,

and, in accordance with Ref. [3], is analytically representable, ξp≡ g(1)p = p=0  l 0 dz exp −ε2kp2(z− 1 − ln z)ε>1  π 2 1 ε|kp| , g0(1)= 1. (51)

Then, the nondiagonal contribution gpp(1) can be found by the

following procedure: gpp(1) = k−1p kp−pVp−pξ−p,p, ξpp = N n=2 g(n)p A(n)pp, (52) A(n)pp = n  n=2 a(npp), app(n)= ε2 nkpg (n−1) p kpgp(n−1) .

Expanding Epand (E−1)pin powers of u in Eqs. (46)–(48)

and using Eqs. (49), (51), and (52), we arrive at the following

general relations: Zin= 1 + u2ε2 p=0 k2pξp|Vp|2+ O(u3) ≈ ε>11+ √ 2π u2εp=1 kp|Vp|2+ O(u3), (53) ¯ = iu3 p,p=0 (p+p=0) kp−1[1+ ε2kp(kp+pξpp+ kp+2pξp)] × VpVpV−p−p+ O(u4). (54)

These relations in the case of small inertial corrections (ε 1, ξp = 1, and ξpp = 0) are reduced to those obtained

in Ref. [30]. Since u∝ T−1 and ε∝ T1/2, we can see from Eq. (53) that (Zin− 1) ∝ T−1 for a small inertia and ∝T−3/2 for a strong one. We cannot exactly determine the high-temperature asymptotics of ¯since the temperature de-pendence of ξppis unknown in the general case. Nevertheless,

one can state that ¯∝ T−α where α⊂ (2,5/2).

The comparison between the exact and the approximate l dependences of (Zin− 1)/u2 and ¯/u3 for the sawtooth potential [when Vp= −ifp/kpin Eqs. (53) and (54)] is given

in Figs.4(a) and4(b). The agreement is good when l is not too small. The cause of the discrepancy lies in the fact that the expansion of Epin u depends on whether there is a region

of rapidly changing potential or not. Indeed, the expansion of Eq. (42) for the sawtooth potential in powers of u (u 1) is different for u |kp|l and u |kp|l. Since |kp| = 2π|p| 

2π , the inequality u |kp|l is valid in the wide region of l, l

u/2π where the expansions (53) and (54) are justified, whereas the validity of the inequality u |kp|l at u 1 depends on

kp. The numerical procedure limits the maximal value of|kp|

(the parameter pm) so that, strictly speaking, the region of

small l values with u |kp|l and u 1 is narrowed to the

point l= 0. This is an explanation of the nonanalytic behavior which we observe at small l.

In the case of the sawtooth potential and strong inertia (ε > 1), the summation in Eq. (53) can be carried out analytically [22], which leads to the following expressions for the quantity Zin and the high-temperature mobility μ (at zero external force):

Zin ≈ ε>11− 2u2ε2π l2(1− l)2  l 0 dx(l− x) ln(2 sin πx)l→01− u2ε  3 2+ ln(2πl)  , ζ μ≈ (1 − u2/12)Zin−1 → l→0(ε>1)1+ u2ε  3 2 + ln(2πl)  . (55) As we mentioned above, the approximation used is valid for l u/2π so that the logarithmic divergence at l → 0 must be eliminated by setting 2π l∼ u in the logarithmic argument. Thus we obtain the following nonanalytic behavior at the point l= 0 and at u → 0: Zin → 1 − (2π)−1/2εu2ln(u) (ε > 1), which is in qualitative agreement with the exact results

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(a) 0 2 4 6 l (Z in -1)/ u 2 (b) 0 2 4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 l 3 / u Φ

FIG. 4. (Color online) The comparison of the exact (markers) and high-temperature approximate (lines) l dependences of the normalized quantities (a) (Zin− 1)/u2 and (b) ¯/u3 calculated for the sawtooth potential. The dashed and solid curves as well as the open and filled markers correspond to ε= 2 and ε = 4, respectively. The square and triangle markers correspond to u= 0.05 and u = 0.1.

presented in Fig.4(a): The smaller u, the larger (Zin− 1)/u2 at l= 0. The corresponding high-temperature dimensionless mobility ζ μ decreases from the value of ζ μ≈ 1 − 0.68εu2at

l= 1/2 to the asymptotic value of ζ μ → 1 + 0.40εu2ln(u) at

l= 0. Since the high-temperature contribution is assumed to be small, the inertial parameter ε is majorized by the inequality ε u−1.

The high-temperature analysis allows us to conclude that the presence of large gradients in a periodic potential (in the region of the width l) leads to the nonanalytic behavior of the quantities of interest at l→ 0. For small inertia, the quantities diverge as l−1[29,30], whereas for strong inertia they diverge as − ln l [see Eq. (55)]. One can say that the nonanalytic area contracts to a point with increasing inertia. In fact, the smallness of l is limited by the smallness of u so that we can expect that the quantities of interest have the logarithmic singularity− ln u at small u. This is true, in particular, for the effective diffusion coefficient since it is related to the effective mobility μ (at zero external force) by the Einstein relation. In addition, we have actually shown that the high-temperature asymptotics of the inertial contribution to the mobility and the

effective diffusion coefficient are proportional to T−3/2and T−1/2, respectively, for strong inertia.

VII. CONCLUSIONS

We have developed a theory of inertial adiabatic transport in which the main transport characteristics are described by analytical expressions formally coinciding with known ones for the overdamped massless case but containing a factor which just includes all information about inertia and can be calculated in terms of Risken’s MCFM. One of the important transport characteristics is the average integrated current ¯ arising after switching on a periodic potential which determines the average velocity of an adiabatic on-off flashing ratchet. The second characteristic is the stationary current arising in a tilted periodic potential. In contrast to the traditional computational MCFM scheme [3], our expression for this current formally coincides with the Stratonovich formula [35] which is generalized to the case of inertial particles. Such representation has allowed us to obtain the coefficients of expansion over the small force, namely, the mobility μ (at zero external force) and the effective diffusion coefficient as well as the nonlinear response which deter-mines the average velocity of adiabatically driven rocking ratchets.

Unlike the commonly considered in MCFM cosinelike potentials, the sawtooth potential has been chosen as an example. This choice has allowed us to elucidate how a large gradient of a periodic potential affects the characteristics under study. We have revealed nonanalytic (jump) behavior of the transport characteristics for a stepwise potential when the width l of the steps tends to zero. In contrast to the overdamped case in which the mobility μ (at zero external force) and the effective diffusion coefficient in a sawtooth potential do not depend on l, they decrease with increasing the asymmetry parameter κ= 1 − 2l/L when inertia is not very strong and have nonmonotonic behavior for strong inertia. Analysis of the temperature dependencies μ(T ) and ¯(T ) (determining the average particle velocity of the on-off ratchet) revealed the dominant role of inertia in the high-temperature region. In this region, we have developed an analytical approach which allows us to conclude that the temperature-dependent contribution to the mobility μ is proportional to T−3/2 for strong inertia and has the logarithmic singularity − ln l at u < l → 0 changing to the singularity − ln u at l u → 0.

ACKNOWLEDGMENTS

This work was supported by National Chiao Tung Uni-versity and Academia Sinica. V.M.R., I.V.S., and S.H.L. thank the Ministry of Education, Taiwan (“Aim for the Top University Plan” of National Chiao-Tung University). S.H.L. thanks the Taiwan Ministry of Science and Technology for partial support (Grant No. MOST104-2923-M-009-001-). Y.A.M. thanks the Russian Foundation for Basic Research for partial support (Grants No. 14-03-00343 and No. 15-59-32405 RT-omi). V.M.R., Y.A.M., and I.V.S. gratefully acknowledge the kind hospitality received from the Institute of Atomic and Molecular Sciences.

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數據

FIG. 1. (Color online) (a) The Fourier components f p of the
FIG. 3. (Color online) The inverse temperature dependences of the average integrated current after switching on the sawtooth potential
FIG. 4. (Color online) The comparison of the exact (markers) and high-temperature approximate (lines) l dependences of the normalized quantities (a) (Z in − 1)/u 2 and (b) ¯ /u 3 calculated for the sawtooth potential

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