Arnold Sommerfeld Center for Theoretical Physics Prof. Dr. E. FreyLudwig-Maximilians-Universitat Munchen J. Knebel, M. Weber, C. Weig

Stochastic Processes in Physics and Biology – Exercise sheet 4Winter term 2012/2013

Exercise 8 Fluctuation Driven Ratchets

Download the paper: R.D.Astumian and M.Bier, “Fluctuation Driven Ratchets: Molecular Motors”, Phys. Rev.Lett., 1994, 72 (11), 1766–1769 from http://prl.aps.org/abstract/PRL/v72/i11/p1766_1.

a) Reading:

• Read the abstract and the introduction of the paper up to the end of the first page. Try to get an ideaabout the kind of models studied in the paper.

• Astumian and Bier distinguish between two types of noise: On the one hand, the paper focuses on themotion of a Brownian particle undergoing a continuous, Gaussian random walk. When placed into the gearof a static ratchet like the ones shown in Fig. 1, it turns out that the Brownian particle prefers neitherdirection over the other, and that its time-averaged net flow vanishes (note that the Brownian particle wouldbe the “pawl” in an ordinary ratchet). Due to the different steepness of the ratchet’s teeth this observationmay not be very intuitive, but you will work out the reason below (everything else would violate the SecondLaw of Thermodynamics by the way). On the other hand, Astumian and Bier allow for the possibility thatthe ratchet’s teeth fluctuate over time. In contrast to the Brownian particle’s motion, this additional noisesource is modeled as a discontinuous jump process. You will see in the paper that the interplay betweenthe two noise sources leads to some interesting new effects and causes a non-vanishing net flow along theratchet.

b) Calculation:

• Astumian and Bier focus on the diffusion of Brownian particles in viscous liquids, i.e. at low Reynolds num-ber, for which the Einstein-Stokes relation predicts that the diffusion coefficient D is inversely proportionalto the friction coefficient β: D = k T/β (with Boltzmann constant k and temperature T ). The frictioncoefficient itself is proportional to the liquid’s viscosity η and to the size r of the Brownian particle: β ∝ η r.The trajectory of a single Brownian particle can be described by the Langevin equation:

˙x(t) = − ∂

∂xU(x, t) + ξ(t) .

The derivative of the potential U exerts a downward force into valleys of U , while the Gaussian noise ξdrives the particle’s random walk. The noise is specified by demanding that:

〈ξ(t)〉 = 0 , and

〈ξ(t) ξ(t′)〉 = 2D δ(t− t′) .

Gear Pawl

Base

ToothNotch

Abbildung 1: An ordinary ratchet. We consider infinitely long ratchets and ratchets witch only a singlenotch and periodic boundaries (modified from Wikipedia).

The brackets 〈 and 〉 denote averages over independent realizations of ξ. Rescale time t = D t by thediffusion constant to obtain the Langevin equation used by Astumian and Bier with correlation function:〈ξ(t) ξ(t′)〉 = 2δ(t− t′).

• Proceed to the models discussed under (i) and (ii) on the second page of the paper. Write down the equationsfor the potential U(x, t) for both cases and check your equations against the ratchets shown in Fig. 1. Notethat there are some minor sign errors in part (i). You should interpret the fluctuating terms ∆F (t) and∆E(t) as trajectories of a telegraph process, and compare the model with the switching genetic operatorsdiscussed by Kepler and Elston (see exercise sheet 2).

• In order to derive a Fokker-Planck equation for the Brownian particles’ motion, let us turn to the integralrepresentation of the Langevin equation (with F (x, t) ≡ −∂xU(x, t)):

x(t+ ∆t) = x(t) + F (x, t) ∆t+

∫ t+∆t

t

ds ξ(s) .

Consider an arbitrary (non-anticipatory) function f(x(t)) and use the definition of the noise ξ to show that:

〈f(x(t+ ∆t))〉 = 〈f(x(t))〉+ 〈f ′(x(t))F (x, t) + f ′′(x, t)〉∆t+O(∆t3/2)

(Hint: Apply the Taylor expansion twice). Rewrite averages over the noise ξ in terms of a conditionalprobability distribution p(x, t) ≡ p(x, t|x0, t0):

〈f(x(t))〉 =

∫Rp(x, t)f(x) ,

to derive a Fokker-Planck equation for p(x, t) (Hint: Use partial integrations). Your equation should reduceto (1) in the case of a constant force F (x, t) = F .

• Since you probably did not encounter so many Fokker-Planck equations before (although quite a fewSchrodinger equations), let us consider a simple problem first. As warm-up, assume that the particle isdriven by a constant force F (x, t) = F :

∂tP (x, t|x0, t0) = −F∂xP (x, t|x0, t0) + ∂2xP (x, t|x0, t0) with P (x, t0|x0, t0) = δ(x− x0) . (1)

Use the Fourier transformation

φ(k, t|x0, t0) =

∫Rdx eikxP (x, t|x0, t0)

to show that the Fokker-Planck equation is solved by a Gaussian with mean µ = x0 + F (t− t0) and varianceσ2 = 2(t− t0). Does the Fokker-Planck equation admit a stationary solution?

• Astumian and Bier’s model does not admit such a solution. In order to show that the ratchet’s fluctu-ating teeth may cause a non-vanishing particle flow along the ratchet, they focus on a single notch andassume periodic boundary conditions. If this reduced model admits a stationary solution with non-vanishingprobability flow, such a flow may also assumed for the infinitely long ratchet (since particles crossing theboundary in the flow’s direction get trapped in the next notch). As a first step, show that a static ratcheton the interval x ∈ [0, 1] with periodic boundaries does not admit such a probability flow. For this purpose,rewrite the Fokker-Planck equation as a continuity equation by introducing the probability flux/flow J :

∂tP (x, t|·) + ∂xJ(x, t|·) = 0 ,

with J(x, t|·) = F (x)P (x, t|·)− ∂xP (x, t|·) .

By noting that the force F (x) is a piecewise constant function,

F (x) = −E0

αθ(0 ≤ x < a) +

E0

1− αθ(α ≤ x ≤ 1)

show that the stationary Fokker-Planck equation is solved by:

Pi(x) = eFi(x−α)Pi(α)− F−1i (eFi(x−α) − 1)Ji(x) ,

Ji(x) = Ji(α) ,

where i ∈ {0, 1} denotes the regions x ∈ [0, α), [α, 1), respectively. Use the assumed continuity of P and J atx = α and at the periodic boundary x = 0/1, and, furthermore, the normalization constraint

∫dxP (x) = 1,

to show that the solution reduces to:

P (x) = Z−1(e−E0xa θ(0 ≤ x < a) + e−E0

1−x1−a θ(a ≤ x < 1)

),

Z = E−10 (1− e−E0) ,

J(x) = 0 .

Do you recognize the form of P (x)? What is the name of such a distribution?

• Let us return to the paper. Argue that equation (2) of the paper is obtained by combining your previousresult for the Fokker-Planck equation with a telegraph process governing the motion of the ratchet’s teeth(you may compare the expression with equation (17) in the paper by Kepler and Elston).

• (Bonus) Unfortunately, it is somewhat more complicated to find a stationary distribution for the full model.It helps to ignore Astumian’s and Bier’s ansatz and to introduce the variables:

Pi = P+i + P−i , Ji = J+

i + J−i , Fi =1

2(F+i + F−i ) ,

Pi = P+i − P

−i , Ji = J+

i − J−i , Fi =

1

2(F+i − F

−i ) .

In this way you will easily see that the total flux J is constant over the full ratchet. The remaining equationsfor the variables P , P and J may be reduced to a form that exactly mirrors the equations you encounteredfor the static ratchet, albeit with the forces Fi replaced by matrices A(Fi, γ). Due to these matrices theresulting equations can only be solved numerically, for example with the help of Matlab or Mathematica.

• (Bonus) Combine the Gillespie and Langevin algorithms from the previous exercises to generate sampletrajectories of the Langevin equation. Use the Gillespie algorithm to find the times at which the telegraphprocess switches between different states of the ratchet. You may then use an explicit Euler algorithm tointegrate over the Gaussian noise of the Langevin equation (Ito prescription). Note that your time stepshould be sufficiently small such that the particle cannot jump from one notch into another in a singleintegration step (roughly ∆t = 0.00001 for the parameters used in the paper; you should, however, tunethe amplitudes of the fluctuations to ∆F , ∆E ≈ 7 to get nice trajectories).

c) Reading:

• Read the rest of the paper and try to understand the phenomenological discussions. You should especiallytry to follow Astumian’s and Bier’s phenomenological reasoning why ratchets with fluctuating barriersadmit a non-vanishing particle flow. This ratchet is discussed on the third page, on the one hand by theuse of time scale arguments, and on the other hand in terms of a simplified model.

−8 −6 −4 −2 0 20

5

10

15

20

25

30

35

40

45

50

Position x

Tim

e t

Abbildung 2: Sample trajectory of a particle in a ratchet with fluctuating barriers. The same parametersas in the paper have been used, although with a larger fluctuation amplitude ∆E = 7. The stepsize wasset to ∆t = 0.0000075. Note that the red color indicates time intervals over which the barriers were down,while the green color indicates time intervals over which they were up.

Your solutions should be handed in by dropping them in the markedbox close to lecture hall A348 in Theresienstr. 37 by noon onThursday, 29th of November.