EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT OF A BROWNIAN PARTICLE IN A PERIODIC POTENTIAL

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1 Acta Mathematica Scientia 2,3B6: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT OF A BROWNIAN PARTICLE IN A PERIODIC POTENTIAL Dedicated to Professor Peter D. La on the occasion of his 85th birthday Hongyun Wang Department of Applied Mathematics and Statistics University of California, Santa Cruz, CA 9564, USA hongwang@ams.ucsc.edu Abstract We study the stochastic motion of a Brownian particle driven by a constant force over a static periodic potential. We show that both the effective diffusion and the effective drag coefficient are mathematically well-defined and we derive analytic epressions for these two quantities. We then investigate the asymptotic behaviors of the effective diffusion and the effective drag coefficient, respectively, for small driving force and for large driving force. In the case of small driving force, the effective diffusion is reduced from its Brownian value by a factor that increases eponentially with the amplitude of the potential. The effective drag coefficient is increased by approimately the same factor. As a result, the Einstein relation between the diffusion coefficient and the drag coefficient is approimately valid when the driving force is small. For moderately large driving force, both the effective diffusion and the effective drag coefficient are increased from their Brownian values, and the Einstein relation breaks down. In the it of very large driving force, both the effective diffusion and the effective drag coefficient converge to their Brownian values and the Einstein relation is once again valid. Key words effective diffusion, effective drag coefficient, Einstein relation, Fokker-Planck equation, probability theory, asymptotic analysis 2 MR Subject Classification 35QXX; 65CXX; 6JXX Introduction and Mathematical Equations We consider the one-diemnsional stochastic motion of a Brownian particle driven by a constant force over a static periodic potential. We are interested in the effective diffusion and the effective drag coefficient of the particle over long time. In single molecule eperiments, the average velocity and the effective diffusion are the two important quantities that can be estimated from eperimental data [ 4]. The effective drag coefficient is calculated directly from Received October 9, 2. This work was partially supported by the US National Science Foundation.

2 2324 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B the measured average velocity. The effective diffusion is related to the randomness parameter, which reveals the internal mechanochemical details of a molecular motor [5, 6]. We like to know how the driving force and the static periodic potential affect the effective diffusion and the effective drag coefficient. This knowledge will be useful in deciphering the motor mechanism from eperimental measurements in modeling studies of molecular motors [7 9]. As shown in Figure, a particle is driven by a constant force f over a static periodic potential φ with period L. The combination of a constant force and a periodic potential can be viewed as a simplified model for protein motors [, ]. In addition to the driving force and the static potential, the particle is also subject to the Brownian diffusion caused by the bombardment of surrounding fluid molecules [2]. We consider the one-dimensional motion of the particle. Let Xt denote the stochastic position of the particle at time t. For a small particle in a viscous fluid environment, the inertia is negligible. Consequently, the dynamics of the particle is governed by the Langevin equation without inertia [3], which describes the balance of all forces on the particle: ζ dxt dt }{{} Viscous drag φ X }{{} Force from potential +f }{{} Driving force + dw t 2k B Tζ. }{{ dt } Brownian force In the above, ζ is the Brownian drag coefficient due to collisions between the particle and the surrounding fluid molecules, k B is the Boltzmann constant and T is the absolute temperature [4, 5]. W t is the standard Weiner process modeling the thermal Brownian motion [6]. φ f Fig. A particle driven by a constant force f over a static periodic potential φ and subject to the Brownian diffusion Dividing by ζ and epressing dxt/dt in terms of others, we obtain the stochastic governing equation for Xt: dxt dt φ X+f ζ + 2D dw t, 2 dt where D k BT ζ is the Brownian diffusion coefficient. Equation 3 is called the Einstein relation [7], which relates the Brownian diffusion and the Brownian drag coefficient. The meaning of the Brownian 3

3 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT 2325 diffusion can be seen by eamining the simple case of φ i.e., in the absence of a static periodic potential. In this case, the stochastic position of the particle is given by Xt X + f ζ t + 2DWt. The variance of particle position in the absence of a potential is var {Xt} 2D var {W t} 2Dt. Note that in the absence of a static periodic potential, the variance of particle position is always proportional to the time, for arbitrary time. As we will find out, in the presence of a static periodic potential, the variance of particle position is only asymptotically proportional to the time for long time. In the presence of a static periodic potential, we define the average velocity and the effective diffusion coefficient of the particle as the eistence of the its will be eamined later V avg D eff t + t + Xt, t var {Xt}. 2t We define the effective drag coefficient as the effective resistance to the driving force ζ eff f. V avg In general, all these effective quantities vary with the driving force and the static periodic potential. So the full notation for the effective diffusion coefficient should be D eff f,[φ]. In the absence of a periodic potential, the effective diffusion coefficient is the same as the Brownian diffusion coefficient, which justifies the name of effective diffusion. Since all average quantities can be calculated from the probability density, in this study, we derive the effective diffusion coefficient by following the time evolution of the probability density. Let ρ, t be the probability density that the particle is at position at time t. Mathematically, ρ, t is defined as ρ, t Δ + Pr{ Xt <+Δ}. Δ The time evolution of the probability density is governed by the Fokker-Planck equation [8] corresponding to Langevin equation 2: ρ t D φ f ρ + ρ. 4 k B T The probability density also satisfies the normalizing condition + ρ, td. To facilitate the analysis below, we non-dimensionalize all quantities. We introduce L, X X L, t td L 2, φ φ k B T, fl f k B T, ρ, t Lρ, t.

4 2326 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B It is straightforward to verify that φ is a periodic function with period and that ρ, t satisfies the constraint + ρ, td. In the derivation of the effective diffusion coefficient below, we will work only with the non-dimensionalized quantities. For the simplicity of presentation and without confusion, we drop the tildes in the rest of the paper and denote the non-dimensionalized quantities simply by {X,, t, φ, f, ρ, V avg,d eff,ζ eff }. To distinguish dimensional and non-dimensional quantities, we denote the physical dimensional quantities by {X p, p,t p,φ p,f p,ρ p,v avg p,d p }. For the non-dimensionalized variables, equation 4 becomes eff,ζp eff ρ t φ f ρ + ρ which, mathematically, corresponds to the special case of equation 4 with L,D,and k B T. The dimensional and the non-dimensional V avg,d eff,ζ eff are related by V p avg D L V avg, ζ p eff k BT D ζ eff, The non-dimensional V avg,d eff,ζ eff has the epressions V avg ζ eff D eff 5 D p eff D D eff. 6 t + f V avg, t + Xt, t where the j-th moment of the non-dimensional particle position is X j t X 2 t Xt 2, 7 2t + In the absence of a static periodic potential φ, we have V avg,d eff,ζ eff,,. j ρ, td. 8 The rest of the paper is organized as follows. In Section 2, we derive analytic epressions for the average velocity and the effective diffusion coefficient, based on governing equation 5 and definition 7. In Section 3, asymptotic analysis is carried out to investigate the trend of the effective diffusion and the effective drag coefficient in the regime of small driving force and in the regime of large driving force. The validity of the Einstein relation is also eamined. In Section 4, the integral formulas derived in Section 2 are implemented numerically to compute the effective diffusion and the effective drag coefficient for arbitrary driving force. The numerical simulations will confirm the asymptotic results and will reveal the behaviors of the effective diffusion, the effective drag coefficient and the Einstein relation when the driving force is in the intermediate range. In Section 5, we discuss the results and observations.

5 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT Analytic Epressions for the Average Velocity and the Effective Diffusion Coefficient In 8, the j-th moment of particle position Xt involves an integral over, +. In differential equation 5, the potential φ is periodic but the probability density ρ, t is not periodic in space. We like to eploit the fact that the driving force is constant and the static potential is periodic in space. Specifically, we like to epress the j-th moment X j t in terms of functions in [, ], instead of functions in, +. Note that these moments are used in 7 to calculate the long time its. Thus, we are allowed to use approimations for X j t as long as the effect of errors disappears in the it as t +. We consider a coarse-graining representation of Xt defined as Nt Xt, 9 where z denotes the largest integer that is less than or equal to z. NotethatNt indicates which period the particle is in. Intuition tells us that over long time this coarse-graining representation is adequate for capturing the average velocity and the effective diffusion. The j-th moment of Nt is N j t + n n+ We introduce a sequence of functions: Function ρ, t is periodic: ρ, t andρ 2, t satisfy ρ +,t ρ 2 +,t n ρ j, t + n n j ρ, td + n ρ +,tρ, t. nρn + +,t ρ, t ρ, t, + n n 2 ρn +, t + n n j ρn +, t. n j ρn +, t. + n ρ 2, t 2ρ, t+ρ, t. n ρn +, t With function ρ j, t, the j-th moment of Nt is simply epressed as N j t ρ j, td. To relate the moments of Xt tothoseofnt, we re-write the j-th moment of Xt as X j t + n + n n + j ρn +, td n j ρn +, td + + n n + j n j ρn +, td.

6 2328 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B Thus, the first two moments of Xt are epressed in terms of those of Nt as Xt Nt + X 2 t N 2 t + 2ρ, td + ρ, td, 2 2 ρ, td. 3 In the lemma below we show that Nt is indeed an adequate representation of Xt for the purpose of calculating the average velocity and the effective diffusion. Lemma If var{nt} satisfies var{nt} C t for large t, then the following is true: Xt Nt, 4 t t var{xt} var{nt}. 5 t t Proof From definition, we see that ρ, t satisfies ρ, t > and ρ, td. Consequently, we have < which, when combined with 2, leads directly to conclusion 4. j ρ, td < 6 For the difference between variances, we use relations 2 and 3 to write var{xt} var{nt} as var{xt} var{nt} X 2 t N 2 t Xt 2 Nt 2 2ρ, td 2 Nt ρ, td ρ, td ρ, td To prove 5, we only need to show I + I 2. 7 I + I 2 t t With the help of 6, one can readily show I 2 <. For I, we re-write it in the integralsummation form and then use the Cauchy-Schwarz inequality to derive I 2 2 n. ρ, t Nt ρ, t d n Nt ρn +, td

7 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT n 2 ρn +, td n n Nt 2 ρn +, td 2 2 C t. 2 ρ, td var{nt} It follows that I + I 2 /t and thus, conclusion 5 is true, which completes the proof t of Lemma. Lemma allows us to derive the average velocity and the effective diffusion coefficient using the moments of Nt, which are conveniently epressed in terms of functions ρ j, t in [, ]: Xt Nt ρ, td, t t t t t t var{xt} var{nt} t 2t t 2t t ρ 2, td ρ, td The meaning of the equations above is that if the it on the right hand side eists then the it on the left hand side also eists and is the same. To discuss the properties of ρ j, t, we introduce functions w andw : w w epφ + s φ fsds, 8 w 2 + sepφ + s φ fsds. 9 We start with the governing equation for ρ, t. Since ρ, t satisfies 5, it is straightforward to verify that ρ, t satisfies 5 and the periodic boundary condition ρ t φ f ρ + ρ, 2 ρ +,tρ, t, 2t ρ, td. Properties of ρ, t are described in the lemma below. Lemma 2 As t, function ρ, t converges to the steady state u : where the steady state u satisfies t ρ, t u, 2 φ f u + u J and is given by u J e f w, J e f w d. 23

8 233 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B Proof Below we first derive the steady state u of 2. Then we show that ρ, t converges to the steady state u eponentially as t. At the steady state, the probability flu is a constant J, independent of time and location. The steady state u of 2 satisfies φ f u + u J. 24 Multiplying by the integration factor and integrating from to +,weget + epφs fsu s + J epφs fsds. Applying the periodic condition u +u yields u J ep φ+f e f Making a change of variable s old + s new,weobtain u + J e f w, epφs fsds. where w is defined in 8. The epression for J stated in 23 follows directly from the condition u d. To complete the proof, it remains to show that ρ, t u. We consider function t r, t ρ, t u. 25 u We only need to show r, t. Both ρ, t andu satisfy linear differential t equation 2. As a result, the difference ρ, t u u r, t satisfies equation 2. Substituting u r, t into equation 2 and using 24, we obtain a differential equation for r, t with the periodic boundary condition: r u t r J r + u, 26 r +,tr, t, u r, td. Note that in differential equation 26, J is a constant, independent of, t. We study the time evolution of quantity u r 2, td. Sinceu is positive and is independent of t, we only need to show t u r, 2 td. Using the differential equation and integration by parts, and then applying the periodic boundary condition, we derive d dt u r 2 d 2 r u r t d 2 2 r r J r + u d r 2 2 J 2 r u d 2 r u d. 27

9 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT 233 We like to show that u rd 2 decays eponentially to zero. For that purpose, we only need to show 2 u rd 2 r C u d. Integrating r / from y to z gives us z r r z,tr y, t+ y d. Multiplying by u y, integrating with respect to y from to, and using u ydy and u yr y, tdy,weget r z,t z r u y y ddy. For z [, ], taking the absolute value and using the Cauchy-Schwarz inequality yields r z,t z u y r y d dy u y r 2 d r d. r ddy The inequality above is valid for any z [, ]. u r 2 d is bounded by u r 2 d ma r2, t min Combining 27 and 28, we arrive at d dt u u r 2 d 2min u From this differential inequality, we can derive u r, 2 td C ep 2 r u d d 2 r u d. 28 t 2min u r 2 d. u, which implies r, t. Therefore, we conclude ρ, t u. This completes t t the proof of Lemma 2. Lemma2showsthatρ, t convergestou eponentially as t. Without loss of generality, we assume that ρ, t is already at the steady state at t. Specifically, we set the initial probability density of particle position as u,, ρ,, otherwise.

10 2332 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B With this initial probability density, we have ρ, t u, for all t, ρ, + n nρn +,, for. Net we look at the governing equation for ρ, t. From the definition of ρ, t, we can verify that ρ, t satisfies 5 with the initial and boundary conditions below: ρ t φ f ρ + ρ, 29 ρ +,tρ, t u, ρ,, for [, ]. Before solving for ρ, t, we use the differential equation and the boundary condition of ρ, t to write out the average velocity: d dt Nt d dt ρ, td φ f ρ + ρ φ f ρ + ρ t ρ, td d φ f u + u J, where the steady state flu J is given in 23. The initial condition ρ, gives us N. Thus,wehave Nt J t and the average velocity has the epression Nt d V avg t t t dt Nt J e f w d. 3 When the driving force f is non-zero, Nt ρ, td increases linearly with t. It is clear that we should not epect ρ, t to reach a steady state as t. Instead, we consider function p, t ρ, t J tu. 3 Here the coefficient J t is selected to make p, t so it is possible for p, t toconverge to a steady state. The governing differential equation for p, t can be derived by substituting ρ, t p, t+j tu into equation 29: p t + J u φ f p + p, 32 p +,tp, t u, p, t. The lemma below contains the results for p, t.

11 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT 2333 Lemma 3 As t, function p, t converges to the steady state u : p, t u. 33 t The steady state u satisfies and has the epression φ f u + u J + J u sds, 34 u u J J u sds w 3, 35 J w d where the constant J is given by J w d 3 + J w d Proof Similar to what we did in the proof of Lemma 2, below we first derive the steady state u of 32. Then we show that p, t converges to the steady state u eponentially as t. The steady state u satisfies Integrating both sides yields J u d d φ f u + u. φ f u + u J + J u sds, where J is the integration constant, to be determined in the calculation below. Multiplying by the integration factor and integrating from to +,wehave + s epφs fsu s + epφs fs J + J u ydy ds. 37 Recall that in Lemma 2 when we multiply the differential equation 24 for u by the integration factor, we can derive epφs fs d J ds epφs fsu s. Substituting this result into the right hand side RHS of 37, carrying out integration by parts and using + u ydy, and for the left hand side LHS of 37, applying the boundary condition u +u u, we get LHS epφ f [ e f u e f u ], RHS [ epφ fu e f J + J J + + epφs fsu 2 sds. ] u ydy +e f J

12 2334 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B Using the epression of u that we derived in Lemma 2 and applying change of variable s old + s new, we write the last term of the RHS in terms of function w : + epφs fsu 2 sds + 2 epφs fsw 2 sds w sds epφ f 2 epφ + s φ fsw 2 + sds w sds epφ f e f 3 w, 38 J w sds where w is defined in 9. Setting LHS RHS and solving for u fromitleadsusto the epression of u stated in 35: J u u J u sds w 3. J w sds The epression of J is derived from the constraint u d. Integrating the equation above, and using u d and u u sds 2 u sds 2 2,weobtain which implies 36. J J 2 w d 3, J w sds The second part of the proof is to show that t p, t u. We consider function r, t p, t u. 39 u We only need to show r, t. Both p, t andu satisfy linear differential t equation 32. Consequently, the difference p, t u u r, t satisfies equation 32. Substituting u r, t into equation 32 and using 22, we obtain a differential equation for r, t with the periodic boundary condition: r u t r J r + u, 4 r +,tr, t, u r, td. Notice that system 4 is eactly the same as system 26. Using the same method as we employed in the proof of Lemma 2, we can show that u r, 2 td decays eponentially to zero as t. Thus, we conclude that p, t u, which completes the proof of t Lemma 3.

13 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT 2335 In order to calculate the effective diffusion, we need to know the governing equation for ρ 2, t. From the definition of ρ 2, t, we can verify that ρ 2, t satisfies 5 with the initial and boundary conditions below: ρ 2 t φ f ρ 2 + ρ 2, 4 ρ 2 +,tρ 2, t 2ρ, t+u, ρ 2,, for [, ]. Using this differential equation, we write the derivative of N 2 t as d dt N 2 t d dt ρ 2, td φ f ρ 2 + ρ 2 φ f ρ 2 + ρ 2 t ρ 2, td d After we apply the boundary condition ρ 2 +,tρ 2, t 2ρ, t+u, the relation ρ, t p, t +J tu, and the result φ f u + u J, the derivative of N 2 t takes the form d dt N 2 t 2. φ f p + p +2J t J. For calculating the effective diffusion, we need to study the derivative of var {Nt}. Recall that after Lemma 2 we have derived Nt J t. We write the derivative of var {Nt} as d dt var {Nt} d dt N 2 t d dt Nt 2 2 φ f p + p J. Taking the it as t, using the results which we have proved in Lemma 3 t p, t u, φ f u + u J + J u sds, and using the epression of J given in 36, we finally arrive at var {Nt} D eff t 2t 2 t J J 2 w d d var {Nt} dt w d The main theorem below summarizes the integral epressions we derived for V avg and D eff. Theorem Consider the non-dimensionalized case of a Brownian particle with Brownian diffusion coefficient, driven by a constant force f over a static periodic potential φ

14 2336 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B with period. The average velocity, the effective diffusion and the effective drag coefficient are well defined that is, the its in their definitions eist and they are given by V avg t Nt t e f var{nt} D eff t 2t ζ eff f V avg f e f where functions w andw are defined as w w w sds, 43 w sds 3, 44 w sds w sds, 45 epφ + s φ fsds, 46 w 2 + sepφ + s φ fsds Asymptotic Behaviors of the Effective Diffusion and the Effective Drag Coefficient We study the asymptotic behaviors of the effective diffusion and the effective drag coefficient in two regimes: small driving force and 2 large driving force. Regime f is small. When the driving force f is small, we epand everything with respect to f: [ e f f f 2 + O f 2], f e f +f 2 + O f 2, [ w e φ e φ+s ds f e φ+s s ds + O f 2]. Recall a useful property of periodic functions: if gs is periodic with period, then + gsds gsds. 48 Applying this property in calculating w d, weget [ w d a f a + O f 2 ], a where coefficients a and a are given by a e φs ds e φs ds, 49 a e φ+s φ s ds d. 5

15 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT 2337 To calculate w d, we use property 48 to re-write it as w d w 2 + se φ+s φ fs dds w 2 eφ φ s fs dds [ w 2 eφ e φs ds f e φ s s ds + O f 2] d. Substituting into it the epansion of w and using change of variable s old s new on the term below we obtain w d I 3 a 2 I 3, e φ e φ s s ds d [ 2 e φ e ds φs 2f e φ e φ+s sds d e φ s e φ sds d e φ s e φ ds d I 3 e φs ds [ e φs ds f e φ s s ds + O f 2] d a 2 f 2a a + 2 a2 + O f 2 [ a 2 2a f + + O f 2]. a 2 The effective drag coefficient and the effective diffusion have the epansions: V avg e φ+s s ds + O f 2] e f w d f [ +f 2a a + O f 2 ], 5 a 2a ζ eff f w [ d e f a f 2a a + O f 2 ], 2a 52 D eff d 3 [ +f 2a a + O f 2 ]. w a 2a d 53 The non-dimensional version of the Einstein relation is Drag coefficient Diffusion coefficient.

16 2338 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B For small driving force, the Einstein relation is valid up to the Of term ζ eff D eff +O f 2. When the driving force is small, the leading term of the effective diffusion is D eff /a.the Cauchy-Schwarz inequality guarantees that a : 2 a e φs ds e +φs ds e φs e +φs ds. Coefficient a increases roughly eponentially with the amplitude of the potential. To demonstrate the eponential growth, we consider the special case of φ α2 2. For this potential, coefficient a can be epressed as a e α2s 2 ds e +α2s 2 ds π e α 2 α erf α F Dawson α, α where erfz 2 π z e s2 ds is the Gauss error function and F Dawson z e z2 z e+s2 ds is the Dawson s function. For large z, they have the asymptotic behaviors: erfz, F Dawson z 2z. For large α, the coefficient a is approimately π a 4α 3/2 eα, which increases eponentially with α. Thus, when the amplitude of the potential is moderately large and driving force is small, the effective diffusion is reduced from the Brownian diffusion by a factor that increases eponentially with the potential amplitude. This is not surprising. For small driving force, the particle motion is constrained by the static potential. To move by Brownian diffusion from the minimum energy location in one period to that in the net one, the particle has to jump over a energy barrier of the magnitude of the potential amplitude. It is the energy barriers that reduce the effective diffusion eponentially. When the driving force is large, however, the situation is very different. As we will see later, in the case of large driving force, the effective diffusion is actually larger than the Brownian diffusion. Regime 2 f is large. We use Watson s Lemma the Laplace integral method [9] to epand w as w f ep φ + s φ e fs ds [ +sφ +s 2 φ 2 + φ ] + Os 3 2 [ + f φ + f 2 φ 2 + φ + O f 3 ]. e fs ds Integrating with respect to and using the fact that φ is periodic, we have w d [ + ] f f 2 φ 2 d + O f 3.

17 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT 2339 Substituting the epansion of w into the epression of w d, weotain w d f 3 w 2 e φ φ s fs ds d [ + f φ + [ + f φ + [ f f 2 φ f φ + O f 3 φ f 2 2 φ + O ] φ 2 d + O f 3. f 3 The effective drag coefficient and the effective diffusion have the epansions: V avg [ e f w d f f 2 ζ eff f w d e f + f 2 D eff w d w f d 2 φ 2 d + O φ 2 d + O φ 2 d + O ] d ] 2 ] f 3, 54 f 3, 55 f Note that as f, the effective diffusion decreases to, the value of the non-dimensionalized Brownian diffusion. That means, for a moderately large driving force, the effective diffusion is larger than the Brownian diffusion. In the case of large driving force, the Einstein relation is again valid approimately up to O /f term: ζ eff D eff + 4 f 2 φ 2 d + O f Thus, the Einstein relation is approimately valid in both the case of small driving force and the case of large driving force. When the driving force is in between these two etreme cases, 57 indicates that the product of the effective diffusion and the effective drag coefficient is larger than. 4 Numerical Results We implement the integral formulas 44 and 45 numerically using Romberg integration method to compute the effective diffusion coefficient and the effective drag coefficient. We carry out numerical simulations for the case where the static periodic potential is φ 5sin2π. Figure 2 shows the effective diffusion D eff as a function of the driving force f. As we derived in the asymptotic analysis of the previous section, when the driving force is small, the effective diffusion is reduced eponentially to almost zero by the static periodic potential. When the driving force is very large, the effective diffusion converges from above to the

18 234 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B non-dimensional Brownian diffusion, again confirming the results of the asymptotic analysis. For moderately large driving force, the effective diffusion is significantly above the Brownian diffusion. In Figure 2, the effective diffusion attains a maimum at an intermediate driving force. The location of the maimum of the effective diffusion is approimately the smallest driving force f c that makes φ +f for all, where φ +f is the total active force on the particle the sum of the constant driving force and the force from the periodic potential. For φ 5sin2π, we have f c 5 2π D eff f Fig.2 The effective diffusion as a function of the driving force Figure 3 shows ζ eff D eff, the product of the effective diffusion and the effective drag coefficient, as a function of the driving force f. In the Einstein relation, this product should be if we use the Brownian drag coefficient and Brownian diffusion. Indeed, in Figure 3, the product ζ eff D eff does approach for small f and for large f, consistent with the asymptotic results obtained in the previous section. For driving forces in between, the product ζ eff D eff is always larger than, and it attains a maimum at an intermediate force. However, the maimum of ζ eff D eff is attained at a force appreciably smaller than f c ζ eff * D eff f Fig.3 The product of the effective diffusion and the effective drag coefficient as a function of the driving force

19 No.6 H.Y. Wang: EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT Discussion In this paper, we have studied the stochastic motion of a Brownian particle driven by a constant force over a static periodic potential. We have derived integral epressions for the effective diffusion and the effective drag coefficient of the particle. These analytic epressions provide the theoretical foundation for investigating the behavior of the effective diffusion and the effective drag coefficient. With proper implementation, these integral formulas also serve as efficient numerical tools for computing the effective diffusion and the effective drag coefficient. Note that a direct calculation of the effective diffusion involves solving the Fokker-Planck equation [2 22] in a large numerical spatial domain and for a long time, which is computationally epensive and suffers from numerical discretization error. The integral formulas can be evaluated efficiently and with accuracy close to the machine precision by using Romberg integration method. Based on the analytic epressions, we have studied the asymptotic behaviors of the effective diffusion and the effective drag coefficient, in the regime of small driving force and in the regime of large driving force. It is interesting to compare the asymptotic behaviors in these two cases. As we discussed in the previous sections, for small driving force, the effective diffusion is reduced by a factor that grows eponentially with the amplitude of the potential and the drag coefficient is increased by the same factor. This is primarily caused by that the energy barriers in the static potential decrease the mobility of the particle. In contrast, for large driving force, both the drag coefficient and the effective diffusion are above their Brownian values. It is not very difficult to see why the drag coefficient is above its Brownian value. The static potential barriers slow down the particle motion. In other words, in the absence of the static potential barriers, the particle would move faster. It is, however, not immediately clear how the combination of a static periodic potential and a moderately large driving force increases the effective diffusion. It is important to point that the mechanism of increasing the effective diffusion is definitely a nonlinear one. In the absence of a static periodic potential, the effective diffusion is always the same as the Brownian diffusion after non-dimensionalization. In the absence of a driving force the it of small driving force, the static periodic potential decreases the effective diffusion eponentially. Thus, the mechanism of increasing the effective potential must come from the non-linear interaction between the driving force and the static periodic potential. The physical mechanism of the effective diffusion being significantly above the Brownian diffusion will be investigated in a subsequent study. References [] Shaevitz J W, Block S M, Schnitzer M J. Statistical kinetics of macromolecular dynamics. Biophys J, 25, 89: 2277 [2] Yasuda R, Noji H, Kinosita K, Yoshida M. F-ATPase is a highly efficient molecular motor that rotates with discrete 2 steps. Cell, 998, 93: 7 24 [3] Visscher K, Schnitzer M, Block S. Single kinesin molecules studied with a molecular force clamp. Nature, 999, 4: [4] Schnitzer M J, Block S M. Statistical kinetics of processive enzymes. Cold Spring Harb Symp Quant Biol, 995, 6: 793 [5] Svoboda K, Mitra P P, Block S M. Fluctuation analysis of motor protein movement and single enzyme kinetics. Proc Natl Acad Sci, 994, 9: 782

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