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ARMA manuscript No.(will be inserted by the editor)

Diffusion mediated transport and the

flashing ratchet

David Kinderlehrer�, Micha�l Kowalczyk��

Abstract

Diffusion mediated transport is a phenomenon in which a unidirectionalmotion of particles is achieved as a result of two opposing tendencies: diffu-sion, which spreads the particles uniformly through the medium and trans-port, which concentrates the particles at some special sites. The flashingratchet version of the Brownian motor, a simple model for protein motors,where the switching between transport and diffusion is periodic, illustratesdiffusion mediated transport.

In this paper we show rigorously that the flashing ratchet can be tunedin such a way that the transport of mass against the gradient of the po-tential takes place. the concentration of mass during the transport phaseoccurs at sites located at the wells of an asymmetric potential. This goalis accomplished by comparing the flashing ratchet with an approximatingMarkov chain. A principle achievement of this work is to establish the con-nection between the dynamics of the ratchet and the Markov chain in theweak* topology.

� The first author was supported by ARO DAAH 04960060 and NSF DMS-0072194, DMS-9505078 and DMS-9303054.�� The second author was supported by the Center of Nonlinear Analysis atCarnegie Mellon University and the NSF Mathematical Sciences PostdoctoralFellowship DMS-9705972. This work was completed during the author’s stay atCarnegie Mellon University.

2 David Kinderlehrer, Micha�l Kowalczyk

1. Introduction

1.1. The mathematical framework of diffusion mediated transport

Diffusion mediated transport is a prominent feature of many molecularscale transduction processes. This is especially thought to relate to motorproteins responsible for intracellular transport in eukarya, where the Brow-nian motor serves as a paradigm, [6,25,26]. The explanation of the Janossyeffect, a light actuated dye/nematic liquid crystal interaction also dependson this mechanism [23],[17]. Additional examples are found in the studyof lipid bilayers [20] and dielectrophoresis [10]. They appear to share thisfeature: there is a collaboration between a diffusive process, which tendsto spread density uniformly through the medium, and a transport process,which concentrates density at specific sites, that results in net transportthrough the system. Each process taken separately rapidly approaches equi-librium without net movement of density. It has been increasingly under-stood that these and many other complex systems function away from equi-librium, in a metastable environment. Their energy landscapes are knownto have an important role in structure formation and their pathways of evo-lution, cf. eg. [5]. Here, in a different approach, we study the flashing rachetversion of the Brownian motor, where the energy landscape is quite simple,and focus on its metastable dynamics.

A main attempt of quantitative studies has been the mesoscale descrip-tion and modeling of these systems. A reasonable first step is an approachvia simple distributions, like Fokker-Planck or master equations suggestiveof major active chemical and conformational events, cf. Astumian et al.[1], [2], [3] and Parmeggiani et al. [24]. This leads to a pair of boundaryvalue problems. Our main concern is to verify the phenomenon of diffu-sion mediated transport. Our second aim is to establish a format to predicttransport based on the equations, that is, to establish parameters to tunethe ratchet. The properties of the system are characterized in terms of aMarkov chain. It is of essential importance to understand what it means todetermine a Markov chain in this manner, namely, the sense in which thediscrete problem is an accurate description of the original continuous one.We propose that weak topology dynamics are natural for this purpose andutilize the viewpoint provided by recent work in this area. This perspectiveilluminates the fundamental connection between the problems and presentsan opportunity for future applications.

The flashing ratchet model finds direct application in the interpretationof the processivity of the kinesin superfamily proteins KIF1A, cf. Okadaand Hirokawa [18], [19].

To formulate the question in precise terms, let ψ be a smooth, nonneg-ative function on the interval (0, 1) and set ψx ≡ b. Given Ttr, Tdiff > 0,T = Ttr + Tdiff set

h(t) ={

1, if t ∈ (nT, nT + Ttr], n = 0, 1, . . .0, if t ∈ (nT + Ttr, (n + 1)T ], n = 0, 1, . . .

(1)

Diffusion mediated transport and the flashing ratchet 3

Let ρ be a solution of

ρt = σρxx + h(t)(bρ)x, x ∈ (0, 1), t > 0,σρx + h(t)bρ = 0, x = 0, 1, t > 0,

ρ(x, 0) = ρ0(x) ≥ 0, x ∈ (0, 1) with∫ 1

0ρ0 dx = 1,

(2)

where σ > 0 is a diffusion constant. As we see the above system, known asthe flashing ratchet, fluctuates periodically (”flashes”) between the Fokker-Planck equation with a drift (transport phase) and the diffusion equation(diffusion phase).

The potential ψ which represents the ratchet typically is assumed tobe periodic in (0, 1) with period, X and such that ψ((i − 1)X) = max ψ,i = 1, . . . k+1. In addition we assume that the local minima of ψ are locatedat points ai = (i − 1)X + a, i = 1, . . . , k, a ∈ (0, X) and that ψ(ai) = 0.

The initial condition in (2), and hence the solution, is subject to∫ 1

0

ρ(x, 0) dx = 1 =∫ 1

0

ρ(x, t) dx, t > 0.

In other words, the probability density ρ alternates between solving the twoproblems

ρt = σρxx + (bρ)x, x ∈ (0, 1), t > 0,σρx + bρ = 0, x = 0, 1, t > 0,

(3)

andρt = σρxx, x ∈ (0, 1), t > 0,σρx = 0, x = 0, 1, t > 0.

(4)

The Gibbs distributions for (3) and (4) are

ρtr(x) = 1Z e−ψ(x)/σ, Z =

∫ 1

0e−ψ(x)/σ dx,

ρdiff(x) ≡ 1.(5)

Although, owing to the time dependence of the coefficients, (2) doesnot have a steady state, it may admit a periodic solution of period T . Theintuitive surmise is that this solution will oscillate, for example, as a convexcombination of ρtr and and ρdiff . This does not represent transport sincethe mass of ρ will tend to be equally distributed in all the period intervalsof ψ. Nor need it happen. Indeed we may isolate three principles that leadto the phenomena of diffusion mediated transport:

(i) the potential ψ is asymmetric in its period,(ii) transport tends to accumulate the mass while diffusion dissipates it

(transport is the ”inverse” of diffusion), and(iii) boundary conditions are of natural type.

Referring to Figure 1, we give a brief caricature of how (i), (ii), and(iii) lead to transport. So, for ease of exposition, the domain is temporarilychanged to be the interval (0, 8) and the ratchet potential, the piecewiselinear function, has two equal teeth with minima at x = 1 and x = 5,


0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8x

Fig. 1. Piecewise linear potential with minima at x = 1 and x = 5 and twoGaussians that have diffused from Dirac masses concentrated at the minima

illustrating (ii). If the diffusion coefficient σ is small then, during a transportphase, this potential concentrates all mass in a basin of attraction to theminimum in that basin. Consider now the diffusion of two Dirac measures ofequal mass, one concentrated at x = 1 and the other at x = 5. After passageof some time, they diffuse to distributions not too different from the twoGaussians pictured in the figure. A large fraction of the right Gaussian hasdiffused into the left basin while only a small fraction of the left Gaussianhas diffused to the right. This is exactly because of the asymmetry withinthe period of the starting distribution. When transport begins again, there ismore mass in left basin than in the right. At the end of the ensuing transportphase, the Dirac measures now have unequal mass. Note the significance ofdiffusion; this is the essence of diffusion mediated transport. We shall refinethis picture in what follows.


The flashing ratchet (2) will be approximated by a discrete ratchet,which is a Markov chain that renders the picture above precise. Note thatthe system (2) arises from a pair of stochastic differential equations and thusrepresents a coarse graining of a finer scale problem. Also the discrete ratchetis actually a Markov chain on measures that are combinations of Diracmeasures. For these reasons, a weak topology is appropriate for studyingthe nature of the evolution, and thus we introduce, following [4], [7], [8],[13], [14], [16], [21], [22], Wasserstein metric techniques. This overcomes,in particular, the separation of scales issues prevalent in the analysis ofdiffusion problems. Indeed, as already mentioned, a principle objective ofthis work is to establish the connection between the dynamics of the ratchetproblem and the approximating Markov chain in the weak* topology.

As we will show, in order to establish unidirectional transport betweenthe wells of the potential one needs to analyze the stationary state of a cer-tain Markov chain. In general that can be quite difficult however in section6 we assume that the ratchet has two teeth, i.e., k = 2. We then prove thatif ψ(a) = minψ with a < 1/4 (and a is the unique minimum of ψ in thisinterval), then the unique, periodic solution ρs of (2) satisfies

maxt>0

∫ 1/2

0

ρs(x, t) dx >12.

We shall prove a much deeper result, that ρs is close to a measure µ =ηδa + (1 − η)δ1/2+a, η > 1/2 in the weak* topology.

1.2. Statements of the main results

The potential ψ which represents the ratchet is assumed to be a C4

function on [0, 1] such that ψx(0) = 0 = ψx(1). More generally than in theprevious section we assume that there exist points 0 = x1 < a1 < x2 <. . . < xk < ak < xk+1 = 1 such that ψ is monotone on each interval (xi, ai),(ai, xi+1) and ψx(xi) = 0 = ψx(ai). Observe that if ψ is periodic thenxi = (i − 1)X, where X is the period of ψ. Recall that we have denotedb ≡ ψx. Set

λ = [9|bx|0 + 6|bxx|0 + 3|bxxx|0],where | · |0 denotes C0 norm on (0, 1).

Theorem 1. Under the assumption

2π2σTdiff − λTtr > log 2 (6)

the following statements hold:

(i) The problem (2) has a unique, periodic orbit ρs.(ii) If ρ is any other solution to (2) and the sequence of functions ρn :

(0, 1) × [0, T ) → R+ is defined by ρn(x, t) = ρ(x, nT + t) then

limn→∞

‖ρs(·, t) − ρn(·, t)‖H2 = 0. (7)


(iii) The following estimate holds

‖ρs(0)‖H2 ≤ 21/2 + 121/2 − 1

.

Remark 1. Condition (6) is clearly a sufficient but not a necessary conditionfor the existence of a periodic orbit of (2). However in general periodic orbitsof (2) do not need to be unique and stable. We also expect that the flashingratchet represents diffusion mediated transport in the range of (Ttr, Tdiff)given by a condition similar to (6).

Our next result establishes the relationship between the periodic solutionof the flashing ratchet problem and the stationary state of its approximatingMarkov chain. To explain this connection consider (2) with an initial dataρ(·, 0) and a vector µ∗ = (µ∗

1, . . . , µ∗k) where

µ∗i =

∫ xi+1

xi

ρ(x, 0) dx.

The vector µ∗ represents the initial distribution of mass in the basin ofattraction of each well of ψ.

After time Ttr we can write that approximately:

ρ(·, Ttr) ≈k∑

i=1

µ∗i δai

In fact, this approximation becomes more accurate when transport is thedominating effect in (3), or in other words as Ttr increases and σ decreases.Turning the potential off and diffusing for a period of time Tdiff leads to asolution to (2) ρ(·, T ) which satisfies

ρ(·, T ) ≈k∑

i=1

µ∗i g(·, σTdiff ; ai)

where g(x, t, a) is the Green’s function for the heat operator with Neumannboundary conditions on (0, 1) at time t with pole at a at t = 0. Thus at theend of one period cycle distribution of mass between the wells of ψ will beapproximately given by a vector µ = (µ1, . . . , µk), where

∫ xi+1

xi

ρ(x, T ) ≈ µi =k∑

j=1

µ∗j

∫ xi+1

xi

g(x, σTdiff ; aj) dx

In matrix notation,

µ = µ∗P (τ)


where for convenience we have set τ = σTdiff and matrix P (τ) is defined by

P (τ) =

∫ x2

x1g(τ, a1) dx

∫ x3

x2g(τ, a1) dx . . .

∫ xk+1

xkg(τ, a1) dx∫ x2

x1g(τ, a2) dx

∫ x3

x2g(τ, a2) dx . . .

∫ xk+1

xkg(τ, a2) dx

. . .∫ x2

x1g(τ, ak) dx

∫ x3

x2g(τ, ak) dx . . .

∫ xk+1

xkg(τ, ak) dx

. (8)

(For simplicity we will write g(x, τ, a) = g(τ, a) whenever it does not createconfusion).

In probabilistic terms P is a probability matrix and (1.2) is one step ina Markov chain whose transition probabilities are the entries of P . Givenτ let µs = (µs

1, . . . , µsk) be the stationary vector for P (τ), i.e. µs = µsP (τ).

We observe that since Pij(τ) > 0 for all i, j, P is ergodic and the stationaryvector µs(τ) is unique.

We would like to show that the periodic solution ρ = ρs is close to thestationary state µs considered as the measure

µs =k∑

i=1

µsi δai

For this we need a convenient metric defined on the space of probabilitydistributions. This naturally leads to the concept of Wasserstein distanceon the space of probability measures.

We recall that the Wasserstein metric is defined as follows

d(µ, ν)2 = infP(µ,ν)

∫ 1

0

∫ 1

0

(x − y)2 dp(x, y),

where P(µ, ν) is the set of probability measures p in (0, 1) × (0, 1) withmarginals µ and ν. It is well known that the weak* topology on the spaceof probability measures P on (0, 1) is metrized by d.

We will describe now a few important features of this metric. Given twopositive probability densities ρ, ρ∗ ∈ L1(0, 1), there is a monotone increasingφ, φ(0) = 0 and φ(1) = 1, with

d(ρ, ρ∗)2 =∫ 1

0

[x − φ(x)]2ρ(x)dx,

i.e., the minimizing joint distribution is dp(x, y) = δ{y−φ(x)}ρ(x)dx. Thisfunction, the solution of the Monge-Kantorovich mass transfer problem, isknown explicitly to be

φ(x) = (F ∗)−1(F (x)),

where F ∗ and F are the distribution functions of ρ∗, ρ, respectively [8,9,16].

We are now in the position to state our next theorem.


Theorem 2. There exist constants K0, T∗tr, T

∗diff > 0 depending on the po-

tential ψ only such that if σ > 0, Ttr > T ∗tr, Tdiff > T ∗

diff and 2π2σTdiff −λTtr > log 2 then the following statements hold:

(1) Let ρ be a solution to (2) with ‖ρ(·, 0)‖H2 < 21/2+121/2−1

. Set

µ∗i =

∫ xi+1

xi

ρ(x, 0) dx.

and

ε = ε(Ttr, Tdiff , σ) =[log Ttr

Ttr+ min{σ1/2eλTtr/4, 1}

].

We have

d(ρ(·, Ttr),∑k

i=1 µ∗i δai

) ≤ K0ε,

‖ρ(·, T ) − ∑ki=1 µ∗

i g(ai, σTdiff)‖2L2 ≤ K0e

−π2σTdiff ε.(9)

(2) If ρ(t) ∈ Rk is a vector defined by

ρi(t) =∫ xi+1

xi

ρ(x, t) dx,

and P = P (τ) is the probability matrix defined in (8) then

|ρ(T ) − ρ(0)P (σTdiff)| ≤ K0e−σπ2Tdiff/2ε1/2.

Now combining Theorem 1 and Theorem 2 it is possible to obtain a gooddescription of the qualitative behavior of the periodic orbit by comparingρs with its discrete counterpart.

Theorem 3. Let ρs be the periodic orbit of (2), let ρs(t) be a vector whosecomponents are defined by

ρsi(t) =

∫ xi+1

xi

ρs(x, t) dx,

and µs(τ) be the stationary vector of the matrix P (τ).Under the assumptions of Theorem 2 the following statements are true:

(1) We have estimates:

d(ρs(·, Ttr),∑k

i=1 ρsiδai) ≤ K0ε,

‖ρs(·, T ) − ∑ki=1 ρs

ig(·, ai, σTdiff)‖2L2 ≤ K0e

−π2σTdiff ε.(10)

(2) There exists a constant κ = κ(σTdiff) > 0 such that we have

|ρs(T ) − µs(σTdiff)| ≤ K0

κ(σTdiff)e−σπ2Tdiff/2ε1/2.


(3) There exist a positive constant C0 depending on ψ only such that

d(ρs(·, Ttr),∑k

i=1 µsi δai

) ≤(K0ε

1/2 + C0e −π2σTdiff/2

κ(σTdiff)

)ε1/2,

‖ρs(·, T ) − ∑ki=1 µs

i g(·, ai, σTdiff)‖2L2 ≤ K0εe

−π2σTdiff/2

+C0e −π2σTdiff/2ε1/2

(σTdiff)1/2κ(σTdiff).

(11)

Remark 2. (a) The first assertion of the Theorem 2 provides rigorous frame-work of our intuitive understanding of the mechanism of diffusion medi-ated transport. If the transport phase is sufficiently long and the diffusionconstant sufficiently small then at the end of the transport phase ρ(·, Ttr)is very close (in the weak* topology) to the sum of weighted Dirac massesconcentrated at the wells of the potential. Furthermore at the end of thediffusion phase ρ(·, T ) is close to the sum of weighted Gaussians withthe same weights. The coefficients µ∗ represent the initial distributionof mass between the maxima of ψ (teeth of the ratchet).More concisely, given a probability measure

µ∗ =k∑

i=1

µ∗i δai ,

permitting it to diffuse for a time Tdiff , and then accumulating the re-sulting masses at the ai gives rise to a measure

µ =k∑

i=1

µiδai, with µ = µ∗P (τ), τ = σTdiff .

One can check that if the potential ψ were convex then the solution tothe transport problem would tend to the Dirac masses concentrated atthe wells ai at an exponential rate in time. Of the two components of theerror term in (9) the first one, log Ttr/Ttr, is due to nonconvexity of thepotential ψ. The second, min{σ1/2eλTtr/4, 1}, accounts for the diffusionacross the maxima of ψ during the transport phase. This term is smallprovided that log 1

σ is much smaller than λTtr.(b) It is possible to prove other estimates, similar to the second estimate in

(9). First, we have

d(ρ(·, T ),k∑

i=1

µ∗i g(·, σTdiff , ai)) ≤ d(ρ(·, Ttr),

k∑i=1

µ∗i δai). (12)

This estimate follows from the fact that the Wasserstein distance of twosolutions to the heat equation is bounded by the distance of their initialdata (see [22]) and is discussed here in Section 4. Secondly one can showthat

d(ρ(·, T ),k∑

i=1

µ∗i g(·, σTdiff , ai)) ≤ K0e

−π2σTdiff . (13)


Indeed, it can be proven that if ρ is a solution to the heat equationon (0, 1) with Neumann boundary conditions and with initial data aprobability measure then

d(ρ(·, t), 1)2 ≤ Ce−π2σt.

Applying now the triangle inequality we infer the required estimate. Ifmin{σ1/2eλTtr/4, 1} = 1 then this last estimate is basically equivalent to(9). Observe that (12) does not take into account the dissipative charac-ter of the diffusion phase and (13), being independent of the initial data,does not take into account the accumulative character of the transportphase of the flashing ratchet.We refer the reader to section 5 where, using these ideas, we give analternative proof of Theorem 3 (2).

(c) The remaining assertions of the theorem describe the transfer of massbetween the wells of the potential during one full cycle of the flashingratchet.In order to explain statement (2) in Theorem 2 we consider an assem-bly of particles undergoing Brownian motion on (0, 1) with reflectingboundary conditions and the drift potential given by h(t)ψ(x). The num-bers ρi(t) are the probabilities of finding a particle between the maximaxi, xi+1 of the ratchet ψ at time t. Informally we can say that the stochas-tic process {ρ(nT )}n=0,1,... is approximated by a Markov process withthe transition matrix P (σTdiff).It is important to notice that the entries Pij of P (σTdiff) depend only onthe distribution of the wells of ψ and the time of relaxation by diffusionrepresented by τ = σTdiff . Intuitively Pij(τ) is the probability that aparticle undergoing Brownian motion will diffuse in time τ from thej-th well of ψ, aj , to the interval (xi, xi+1).Assertion (3) of Theorem 3 simply states that the limiting, as number ofcycles of the ratchet goes to ∞, distribution of mass between the teethof the ratchet is approximated by the stationary state of the Markovprocess given by P (τ). From this point of view Theorem 3 is a statementabout the stability of this Markov process.

(d) The constant κ appearing in Theorem 3 is defined by

κ = inf∑k

i=1Vi=0

|V |=1

|V P (τ) − V |.

Observe that κ is simply equal to the spectral gap for the operatorP (τ)−id, i.e. κ(τ) is the biggest eigenvalue of P (τ)−id no bigger than 1.We further notice that κ(0) = 0, κ(τ) > 0 for τ > 0 and limτ→∞ κ(τ) =1. Since in the range of parameters described by Theorem 2 we haveσTdiff > log 2/(2π2) therefore we can replace κ above by some positive,uniform constant.


Theorem 2 can be used to prove unidirectional transport of mass if onecan determine the vector µs. In the next theorem we consider a 2-toothed,periodic and asymmetric potential ψ.

Theorem 4. Let ψ be a potential described above and satisfying addition-ally:

0 = x1 < a = a1 < x2 = 1/2 < 1/2 + a = a2 < x3 = 1.

For any fixed σ > 0 there exists T ∗∗tr such that for each Ttr > T ∗∗

tr we canchoose Tdiff such that if a < 1/4 then we have ρs

1 > 1/2 > ρs2. Likewise

for a > 1/4 we have ρs2 > 1/2 > ρs

1.

Remark 3. Let a < 1/4. Observe that if σ > 0 and Ttr > T ∗∗tr are fixed then

for Tdiff = 0, or Tdiff = ∞ we get ρs1 = ρs

2 = 1/2. Thus Theorem 4 impliesthat as we vary Tdiff from 0 to ∞ then there exists time Tmax

diff such that forTdiff = Tmax

diff the difference ρs1 − ρs

2 is maximized. We do not prove it herehowever in section 7 we present some numerical simulations which seem tojustify this claim.

This paper is organized as follows: in section 2 we prove Theorem 1 andestablish equivalence of L2 and Wasserstein metric on the set of trajectoriesof (2). Sections 3 and 4 are devoted to the proof of Theorem 2 and Theorem3; in particular in section 3 we study the deterministic ratchet. In section 5we give a different proof of Theorem 3 (2). Finally section 6 is devoted to theproof of Theorem 4. The results of some numerical simulations for 2-teethratchet are described in section 7. The Parrondo game and its relationshipwith the flashing ratchet is discussed in section 8.

We are indebted to Felix Otto, Peter Palffy-Muhoray, Bard Ermentrout,Shlomo Ta’asan, Noel Walkington, Chun Liu and Yasushi Okada for stim-ulating conversations. An earlier version of the results described here wasdiscussed by the authors in [15].

2. Periodic orbit for the flashing ratchet

2.1. Proof of Theorem 1

We begin with a simple lemma.

Lemma 1. Consider

ρt = σρxx + (bρ)x, x ∈ (0, 1), t > 0,ρx = 0, x = 0, 1, t > 0,ρ(x, 0) = ρ0(x), x ∈ (0, 1).

(1)

We have‖ρ(·, t)‖H2 ≤ eλt/2‖ρ(·, 0)‖H2 , (2)

where λ = [9|bx|0 + 6|bxx|0 + 3|bxxx|0].


Proof. Multiplying (1) by ρ and integrating by parts we obtain:

12

d

dt

∫ 1

0

ρ2 dx = −σ

∫ 1

0

ρ2x dx −

∫ 1

0

bρρx dx

≤ −12

∫ 1

0

b∂

∂xρ2 dx

=12

∫ 1

0

bxρ2 dx ≤ |bx|02

∫ 1

0

ρ2 dx.

Likewise, differentiating (1) with respect to x, multiplying by ρx and thenintegrating over (0, 1) we obtain

12

d

dt

∫ 1

0

ρ2x dx ≤ [

32|bx|0 + |bxxx|0]

∫ 1

0

[ρ2 + ρ2x] dx.

Observe that by a simple reflection argument we have ρxxx = 0 at x = 0, 1,hence differentiating the equation twice with respect to x, multiplying byρxx and integrating we get

12

d

dt

∫ 1

0

ρ2xx dx ≤ [

52|bx|0 + 3|bxx|0 +

12|bxxx|0]

∫ 1

0

[ρ2 + ρ2x + ρ2

xx] dx.

Estimate (2) follows now from Gronwall inequality. We are in position now to finish the proof of the theorem.Consider problem (2). Let F be a map ρ(·, 0) �→ ρ(·, T ). We will look

for a fixed point of F which is equivalent to finding the periodic orbit. Letρ(x, t) = 1 + v(x, t), in the sequel we will simply write v(x, t) = v(t).

Using Lemma 1 we have

‖v(T )‖H2 ≤ e−π2σTdiff‖v(Ttr)‖H2 ≤ e−π2σTdiff (1 + ‖ρ(Ttr)‖H2)

≤ e−π2σTdiff [1 + eλTtr/2(1 + ‖v(0)‖H2)]< R,

where the last inequality holds provided that 2π2σTdiff − λTtr > log 2, sothat e−π2σTdiff+λTtr/2 < 2−1/2, and

‖v(0)‖H2 ≤ e−π2σTdiff+λTtr/2 + e−π2σTdiff

1 − e−π2σTdiff+λTtr/2<

21/2

1 − 2−1/2≡ R. (3)

It follows that F(B(1, R)) ⊆ F(B(1, R)) where B(1, R) denote a ball withcenter at 1 and radius R in H2. Moreover, by parabolic regularity, F isa compact mapping from B(1, R) into itself. From Schauder fixed pointtheorem we infer now that F has a fixed point ρs ∈ B(1, R).

Observe that for any r > R we have F(B(1, r)) ⊂ B(1, r) and thus Fdoes not have a fixed point in H2 \ B(1, R).

We will show now that ρs is stable as a fixed point of F , that is for anyv ∈ H2(0, 1),

∫ 1

0v = 0 we have

‖F(ρs + v) −F(ρs)‖H2 < ‖v‖H2 . (4)


If by v(T ) we denote the solution to (2) with the initial data v(0) ≡ v then(4) is equivalent to ‖v(T )‖H2 < ‖v(0)‖H2 . By Lemma 1 we have

‖v(T )‖H2 ≤ e−π2σTdiff‖v(Ttr)‖H2 ≤ e−π2σTdiff+λTtr/2‖v(0)‖H2 < ‖v(0)‖H2 ,

hence (4) follows.Clearly from (4) it follows that ρs is the unique periodic orbit.By a similar argument we show that if ρ(·, t) is a solution to (2) then

there exists n0 such that ρn0(·, T ) ∈ B(1, R), hence by the above ‖ρs(·, T )−ρn(·, T )‖H2 → 0 as n → ∞. Parabolic regularity yields now (7).

Finally we observe that from (3) it follows that

‖ρs(0)‖H2 ≤ 1 + R =21/2 + 121/2 − 1

. (5)

This ends the proof of Theorem 1. Arguing as in the proof above we get:

Corollary 1. Let ρ(·, t) be a solution to (2) with ρ(·, 0) ∈ B(1, R). We thenhave

‖ρ(·, t)‖H2 ≤ e−π2σ(t−Ttr)+ [eλTtr/2(2 + R)], t ≥ 0.

2.2. Weak* and L2 topology on probability densities

We denote

R0 =21/2 + 121/2 − 1

.

In the sequel we restrict ourselves to those trajectories of (2) which initiallybelong to B(0, R0). As we have shown above (Corollary 1) those trajectoriesbelong to a certain bounded, weakly compact subset A of H2. This suggeststhat different norms on A are equivalent. In particular we are interested inthe relationship between weak* and L2 topology induced on A.

The following lemma shows that on a bounded subset of H2, weak* andL2 distance are in some sense equivalent.

Lemma 2. Let ρ, ρ∗ ≥ 0 be two probability distribution functions on (0, 1),‖ρ‖H2 , ‖ρ∗‖H2 ≤ M/2 for some M > 0. We have the following estimate

‖ρ − ρ∗‖2L2 ≤ Md(ρ, ρ∗).

Proof. For any function ξ ∈ C1(0, 1) we can write∣∣∣∫ 1

0ξ(x)ρ(x) dx −

∫ 1

0ξ(y)ρ∗(y) dy

∣∣∣ =∣∣∣∫ 1

0

∫ 1

0[ξ(x) − ξ(y)] dp(x, y)

∣∣∣≤ |ξ′|0

[∫ 1

0

∫ 1

0(x − y)2 dp(x, y)

]1/2

≤ |ξ′|0d(ρ, ρ∗),

where p is a probability measure on (0, 1)2 whose marginals are ρ and ρ∗.


Choosing ξ = ρ and then ξ = ρ∗ we obtain

‖ρ − ρ∗‖2L2 ≤

∣∣∣∫ 1

0ρ2(x) dx −

∫ 1

0ρ(y)ρ∗(y) dy

∣∣∣+

∣∣∣∫ 1

0(ρ∗)2(x) dx −

∫ 1

0ρ(y)ρ∗(y) dy

∣∣∣≤ (|ρx|0 + |ρ∗x|0)d(ρ, ρ∗) ≤ Md(ρ, ρ∗),

where the last inequality follows from Sobolev embedding. The proof iscomplete.

Note that the Wasserstein metric and ordinary Euclidean metrics can beused to control each other in finite dimensions. For two probability measures

µ =k∑

i=1

µiδai , and µ∗ =k∑

i=1

µ∗i δai ,

any joint distribution has the form

p =k∑

i,j=1

νijδai⊗ δaj

, νij ≥ 0,

k∑i,j=1

νij = 1.

From this it is not difficult to show that there are constants 0 < ck < Ck

such that

ck max |µj − µ∗j | ≤ d(µ, µ∗)2 ≤ Ck max |µj − µ∗

j |.

Above we saw how the L2 norm may be controlled by the Wassersteinmetric. In the next lemma, we want to say that understanding the L2 normleads to control of the convex combinations of Dirac measures that resultfrom transport.

Lemma 3. Let ρ and ρ∗ be probability distribution functions on (0, 1) anddefine

µ =∑k

i=1 µiδai, with µi =

∫ xi+1

xiρ dx, i = 1, . . . , k,

µ∗ =∑k

i=1 µ∗i δai , with µ∗

i =∫ xi+1

xiρ∗ dx, i = 1, . . . , k.

Then

C−1k d(µ, µ∗)2 ≤ max |µj − µ∗

j | ≤ ‖ρ − ρ∗‖L2 .

Proof. Obvious from Schwarz’s Inequality.


3. Deterministic ratchet

3.1. Limiting transport problem

In this section we will begin proving Theorem 2. We will first considera deterministic ratchet which can be thought of as a limit of the flashingratchet. Suppose that Ttr is given and let ρ(·, t), with ρ(·, 0) ∈ B(0, R0), bea solution to (2), restricted to the interval 0 < t < Ttr. We will derive alimiting problem for ρ as σ → 0.

Lemma 4. Let ζ be the solution to the following transport problem

ζt = (ζb)x, (x, t) ∈ (0, 1) × (0, Ttr),ζx(0) = 0 = ζx(1), t ∈ (0, Ttr),ζ(x, 0) = ζ0(x), x ∈ (0, 1),∫ 1

0ζ0 dx = 1, ζ0 ≥ 0.

(1)

If ζ0(x) = ρ(x, 0) then

ρ(·, t) ∗⇀ ζ(·, t),

for t ∈ (0, Ttr).

Proof. Multiplying (1) by a test function φ ∈ C∞((0, 1)× (0, Ttr)), φx(0) =0 = φx(1) and integrating over (0, 1) × (0, t), 0 < t < Ttr, we obtain

∫ t

0

∫ 1

0

ρ(φt + σφxx − bφx) dxdt = 0.

Since ρ is bounded in L∞(0, Ttr;L2) therefore we can pass to the limit asσ → 0 above.

In terms of Wasserstein distance the conclusion of the above lemma canbe stated

limσ→0

d(ρ dx, ζ dy) = 0. (2)

We shall now describe some properties of the solutions to (1). Observefirst that we have

∫ 1

0ζ(·, t) = 1.

We can solve (1) by the method of characteristics. More precisely if wedefine for x ∈ (0, 1), t > 0

s(x, t) ={

x, if x = xi or x = ai,

s is determined from t = −∫ x

sdy

b(y) , otherwise,

thenζ(x, t) = sx(x, t)ζ0(s(x, t)). (3)

Observe that st = ψx(s) and therefore ddtψ(s) = [ψx(s)]2 ≥ 0. In fact outside

of a small neighborhood of the set of critical points of ψ we have that s(·, t)


converges at an exponential rate to the set of maxima of ψ, {x1, . . . , xk+1}.Heuristically

ζ∗⇀

k∑i=1

ζiδai , ζi =∫ xi+1

xi

ζ0 dx.

On the other hand, as σ → 0 any solution to (2) restricted to (0, Ttr)converges weak* to the corresponding solution of the transport problem(1). This suggests that at the end of the transport phase of the flashingratchet we have

ρ(Ttr) ≈k∑

i=1

µ∗i δai , µ∗

i =∫ xi+1

xi

ρ(x, 0) dx.

In order to make this statement precise we first need a technical lemma.

Lemma 5. Let µ be a probability measure on (0, 1) and µi = µ((xi, xi+1)),i = 1, . . . , k. We have

d(µ,

k∑i=1

µiδai)2 ≤

k∑i=1

∫ xi+1

xi

(x − ai)2 dµ.

Proof. We define a probability measure p on (0, 1) × (0, 1) by

p(x, y) =k∑

i=1

µ(x)χ(xi,xi+1)(x)δ(y − ai),

where χ(a,b)(x) is the characteristic function of the interval (a, b). It is easyto check that the marginals of p are µ and

∑ki=1 µiδai . Therefore we have

d(µ,∑k

i=1 µiδai)2 ≤

∫ 1

0

∫ 1

0(x − y)2 dP (x, y)

=∫ 1

0

∫ 1

0(x − y)2

∑ki=1 χ(xi,xi+1)(x)δ(y − ai) dµ(x)

=∫ 1

0

∑ki=1(x − ai)2χ(xi,xi+1)(x) dµ(x)

as claimed. We want to estimate the Wasserstein distance between a solution to (2)

ρ(t), with ρ(·, 0) ∈ B(0, R0) at time t = Ttr and the measure µ∗ defined by

µ∗ =k∑

i=1

µ∗i δai , µ∗

i =∫ xi+1

xi

ρ(x, 0) dx.

We will first estimate d(ζ,∑k

i=1 ζiδai), where ζ is the solution of the trans-port problem (1) corresponding to ρ.

Let [yi, zi], i = 1, . . . , k+1 be the disjoint intervals such that ψxx(x) ≤ 0,if x ∈ ∪k+1

i=1 [yi, zi] and ψxx > 0 otherwise. We will prove the following lemma.


Lemma 6. There exists a constant δ1 > 0 depending on ψ only such thatfor any solution ζ to (1) with initial condition ζ(0) = ζ0, and any δ ∈ (0, δ1]we have

k∑i=1

∫ xi+1

xi

(x − ai)2ζ(x) dx ≤ e−δt

[k∑

i=1

∫ xi+1

xi

(x − ai)2ζ0(x) dx

]

+δ2(λ + c0

c30

)k∑

i=1

‖ζ0‖C0(yi,zi), (4)

where constant c0 satisfies c0 = inf∪k+1i=1 [yi,zi]

|bx|.

Proof. We fix i, 1 ≤ i ≤ k + 1. Multiplying (1) by (x− ai)2 and integratingover (xi, xi+1) we get

∫ xi+1

xi

(x − ai)2(ζψx)x = −2∫ xi+1

xi

(x − ai)ψxζ =d

dt

∫ xi+1

xi

(x − ai)2ζ.

For each δ > 0 we set Eδ = {x ∈ (xi, xi+1) | ψx(x)(x− ai) < δ(x− ai)2/2}.Observe that if δ is chosen sufficiently small then Eδ ⊂ ∪k+1

i=1 [yi, zi]. It isalso easy to see that for any x ∈ Eδ we have

max{|x − xi|, |x − xi+1|} <δ

2c0.

From∫ xi+1

xi

[(x − ai)ψx − δ(x − ai)2/2]ζ dx ≥∫

Eδ

[(x − ai)ψx − δ(x − ai)2/2]ζ dx

we get∫

Eδ

[(x − ai)ψx − δ(x − ai)2/2]ζ dx ≥ −δ(λ

c0+ 1)

∫Eδ

ζ dx,

and thus

d

dt

∫ xi+1

xi

(x − ai)2ζ dx ≤ −2δ

∫ xi+1

xi

(x − ai)2ζ dx + δ(λ

c0+ 1)

∫Eδ

ζ dx (5)

We will estimate the last integral in (5). Taking δ smaller if necessary we canalways achieve Eδ = (xi, z

′i) ∪ (y′

i+1, xi+1) with some z′i < zi, y′i+1 > yi+1

and bx(x) ≤ −c0/2 for x ∈ Eδ. From (3) we get

∫ t

0

∫ z′i

xiζ(x, ξ) dxdξ =

∫ t

0

∫ z′i

xisx(x, ξ)ζ0(s(x, ξ)) dxdξ

=∫ t

0

∫ s(z′i,ξ)

xiζ0(y) dydξ.


From the choice of δ it follows s(z′i, ξ) − xi ≤ (z′i − xi)e−c0ξ/4 hence

∫ t

0

∫ s(z′i,ξ)

xiζ0(y) dydξ ≤

∫ ∞0

∫ xi+(z′i−xi)e

−c0ξ/4

xiζ0(y) dydξ

≤ − 14c0

∫ z′i

xiζ0(y) ln( y−xi

z′i−xi

) dy

≤ − z′i−xi

4c0

∫ 1

0ζ0(xi + (z′i − xi)z) ln z dz

≤ |z′i−xi|2c0

‖ζ0‖C0(yi,zi)

≤ δ4c2

0‖ζ0‖C0(yi,zi).

(6)

Since an analogous estimate holds for the integral over (y′i+1, xi+1), therefore

∫ t

0

∫Eδ

ζ dxdξ ≤ δ

c20

‖ζ0‖C0(ai,bi)

and thus we finish the proof of the lemma by applying Gronwall’s inequalityin (5).

Lemma 7. Let ρ be a solution to (2) with ‖ρ(·, 0)‖H2 ≤ R0 and let ζ be thesolution to the transport equation (1) with ζ(·, 0) ≡ ρ(·, 0). Then

∫ 1

0

(ρ − ζ)+ dx ≤ min{Cλσeλt/2, 1}, t ∈ [0, Ttr], (7)

where Cλ is a constant depending on λ only.

Proof. Let E+ = {ρ−ζ ≥ 0}. Subtracting (2), and (1) and then integratingthe resulting expression over E+ we obtain

d

dt

∫ 1

0

(ρ − ζ)+ dx ≤ σ‖ρ‖H2 .

By Corollary 1 ‖ρ(t)‖H2 < (1+R0)eλt/2 hence applying Gronwall’s inequal-ity we get ∫ 1

0

(ρ − ζ)+ dx ≤ σ2(1 + R0)

λeλTtr/2.

Since in addition we have trivially∫ 1

0

(ρ − ζ)+ dx ≤∫ 1

0

ρ dx = 1

therefore the proof of the lemma follows. We will denote

c1 = (λ + c0

c30

).

We have the following:


Corollary 2. There exists δ1 > 0 depending on ψ only such that for anyδ ∈ (0, δ1] and any solution ρ to (2) with ‖ρ(·, 0)‖H2 ≤ R0 the followingestimate holds

d(ρ(·, t),k∑

i=1

µ∗i δai

)2 ≤ e−δt

[k∑

i=1

∫ xi+1

xi

(x − ai)2ρ(x, 0) dx

]

+c1δ2‖ρ(·, 0)‖H2 + min{Cλσeλt/2, 1}, (8)

for each t ∈ [0, Ttr], where we have set

µ∗i =

∫ xi+1

xi

ρ(x, 0) dx.

Proof. From Lemma 5 we get

d(ρ(·, t),∑ki=1 µ∗

i δai)2 ≤ ∑k

i=1

∫ xi+1

xi(x − ai)2ρ(x, t) dx

≤ ∑ki=1

{∫ xi+1

xi(x − ai)2ζ(x, t) dx

+∫ xi+1

xi(x − ai)2[ρ(x, t) − ζ(x, t)] dx

}≤ ∑k

i=1

∫ xi+1

xi(x − ai)2ζ(x, t) dx

+∫ 1

0[ρ(x, t) − ζ(x, t)]+ dx.

Estimate (8) follows now from Lemma 6 and Lemma 7.

4. Proof of Theorem 2 and Theorem 3

4.1. Following a solution in the transport phase

Combining Corollary 2 and Proposition 1 we get:

Corollary 3. Let ρ be a solution to (2) with ‖ρ(·, 0)‖H2 < R0.There exists a positive constant T ∗

tr depending only on ψ such that ifTtr > T ∗

tr then

d(ρ(·, Ttr),k∑

i=1

µ∗i δai)

2 ≤ R0(1 + c1)log2 Ttr

T 2tr

+ min{CλσeλTtr/2, 1}. (1)

Proof. Let δ1 be the constant given in the statement of Lemma 6. ChooseT ∗

tr such that 2 log T∗tr

T∗tr

< δ1 and set δ = 2 log TtrTtr

. Estimate (1) follows nowimmediately from Corollary 2. The proof is complete.


4.2. Following a solution in the diffusion phase

From Otto’s result ([22] Proposition 1, estimate (133) and the argumenttherein we have):

Proposition 1. Let ρ, ρ∗ be two solutions of

ut = σuxx, in (0, 1) × (0,∞),ux = 0, at x = 0, 1, t > 0,u(x, 0) = u0(x) ≥ 0, in (0, 1),∫ 1

0u0 dx = 1.

Then for any t0, t1, 0 ≤ t0 ≤ t1 we have

d(ρ(·, t1), ρ∗(·, t1)) ≤ d(ρ(·, t0), ρ∗(·, t0)).

4.3. Conclusion of the proof

Let ρ∗(·, t) be a solution of

ρ∗t = σρ∗xx, in (0, 1) × (Ttr, T ),

ρ∗x = 0, at x = 0, 1, t > Ttr,

ρ∗(x, Ttr) =∑k

i=1 µ∗i δai

, in (0, 1), t = Ttr.

We then have an explicit formula for ρ∗

ρ∗(x, t) =k∑

i=1

µ∗i g(x, σ(t − Ttr), ai). (2)

Let ρ be a solution to (2) described in the statement of Theorem 2. By thestandard parabolic estimate we get for any t > t0 > Ttr

‖ρ(·, t) − ρ∗(·, t)‖2L2 ≤ e−2σπ2(t−t0)‖ρ(·, t0) − ρ∗(·, t0)‖2

L2

= e 2σπ2(t0−Ttr)e−2σπ2(t−Ttr)‖ρ(·, t0) − ρ∗(·, t0)‖2L2 .

(3)From Corollary 1 we obtain

‖ρ(·, t0)‖H2 ≤ e−π2σ(t0−Ttr)+λTtr/2(1 + R0).

By explicit calculation we also get

‖ρ∗(·, t0)‖H2 ≤ C[σ(t0 − Ttr)]−5/4.

If we set

K = K(σ, Ttr, Tdiff) = 2[e−π2σt0+λTtr/2(1 + R0) + C(σt0)−5/4],


then by Lemma 2 and Proposition 1 it follows from (3) with t = T

‖ρ(·, T ) − ρ∗(·, T )‖2L2 ≤ Ke 2σπ2(t0−Ttr)e−2σπ2Tdiff d(ρ(·, t0), ρ∗(·, t0))

≤ Ke 2σπ2(t0−Ttr)e−2σπ2Tdiff d(ρ(·, Ttr), ρ∗(·, Ttr))

≡ M(σ, Ttr, Tdiff).(4)

Since for any 0 ≤ a < b ≤ 1 we have, as we already noted in Lemma 3,∣∣∣∣∣∫ b

a

[ρ(x, T ) − ρ∗(x, T )] dx

∣∣∣∣∣2

≤ ‖ρ(·, T ) − ρ∗(·, T )‖2L2 ,

therefore by (2) we obtain

|ρ(T ) − ρ(0)P |2 ≤ kM(σ, Ttr, Tdiff). (5)

In particular if ρ ≡ ρs, the periodic orbit, then ρs(T ) = ρs(0) = ρs, hence

|ρs − ρsP |2 ≤ kM(σ, Ttr, Tdiff),

with ρ in the definition of M(σ, Ttr, Tdiff) replaced by ρs. Consequently

|ρs − µs|2 ≤ kM(σ, Ttr, Tdiff)κ(σTdiff)

. (6)

Taking now T ∗diff such that 2π2σT ∗

diff − λT ∗tr > log 2, setting t0 = Ttr +

log 2/(2π2σ) and choosing a positive constant K0 appropriately we obtainfrom Corollary 3

d(ρ(·, Ttr),k∑

i=1

µ∗i δai) ≤ K0

[log Ttr


], (7)

Combining (4) and (7) we obtain statement (1) of Theorem 2, whereas from(5) it follows

|ρ(T ) − ρ(0)P | ≤ K0e−σπ2Tdiff/2d(ρ(·, Ttr),

k∑i=1

µ∗i δai

). (8)

This completes the proof of Theorem 2.To finish the proof of Theorem 3 we first observe that (10) is simply (9)

with ρ ≡ ρs and µ∗ ≡ ρs.Next we note that from estimates (7), (8) and (6) we get

|ρs − µs| ≤ K0

κ(σTdiff)e−σπ2Tdiff

[log Ttr


]1/2

.

Finally from Lemma 1, Statements (1) and (2) of Theorem 3 and triangleinequality applied to Wasserstein distance and L2 distance, respectively weobtain (3).

The proof of Theorem 3 is complete.


5. An alternative proof of the main theorem

Our verification of transport in the previous section (Theorem 3 (2))consisted in following the periodic solution and the discrete ratchet througha period. It relied on Otto’s estimate in Proposition 1. In this section, wegive a different proof based on estimating the decay to equilibrium of anysolution of the diffusion equation with probability measure initial conditions.

Note that for µ ∈ P, we may associate ρµ, the solution of (4) with initialvalue µ. We may express this solution as

ρµ(x, t) = 1 +∑

ckφk(x)e−σπ2k2t, x ∈ (0, 1), t > 0,

ck = 〈µ, φk〉, where |ck| ≤√

2,

since φk(x) =√

2 cos πkx are uniformly bounded. Thus in any domain(0, 1) × (t,∞), t > 0 the family of {ρµ} is a normal family.

First we calculate the variation of the Wasserstein metric, a special caseof [16].

Lemma 8. Let ρ be a solution of

ρt = σρxx, in (0, 1), t > 0,

ρx = 0, for x = 0, 1, t > 0,

ρ ≥ 0, and∫ 1

0ρ dx = 1, for t > 0.

Then

d

dtd(ρ, 1)2 = −2σ

∫ 1

0

(ρ − 1)2 dx.

Proof. Let F (x, t) denote the distribution function of ρ(x, t). The dis-tribution function of the limit state 1 is just F ∗(x) = x, hence

d(ρ, 1)2 =∫ 1

0

[x − F (x, t)]2ρ(x, t) dx.

To differentiate this note that

Ft(x, t) =∫ x

0

ρt(x′, t) dx′ = σ

∫ x

0

ρxx(x′, t) dx′ = σρx(x, t).


Using this, integrating by parts, and collecting terms, here recalling thatF (0, t) = 0 and F (1, t) = 1, we find that

ddtd(ρ, 1)2 =

∫ 1

0[−2(x − F )Ftρ + (x − F )2ρt] dx

=∫ 1

0[2σ(x − F )ρxρ + σ(x − F )2ρxx] dx

=∫ 1

0[−2σ(x − F )ρxρ − σ ∂

∂x (x − F )2ρx] dx

=∫ 1

0{−2σ(x − F )ρxρ − 2σ[(x − F )(1 − ρ)ρx]} dx

= −2σ∫ 1

0(x − F )ρx dx

= 2σ∫ 1

0(1 − ρ)ρ dx

= −2σ∫ 1

0(ρ2 − 1) dx = −2σ

∫ 1

0(ρ − 1)2 dx.

Theorem 5. Given µ ∈ P, let ρ denote the solution of

ρt = σρxx, in (0, 1), t > 0,

ρx = 0, for x = 0, 1, t > 0,

with initial condition µ. Then there is a constant K, independent of µ, suchthat

d(ρ, 1) ≤ Ke−σπ2t.

Proof. To use the lemma, simply observe that∫ 1

0

(ρ2 − 1) dx =∑

c2ke−2σπ2k2t

and integrate from t to ∞.

Theorem 6. Under the assumptions of Theorem 2 there exist a positiveconstant C0 such that

|ρs − µs| ≤ C0

κ(σTdiff)e−σπ2Tdiff

Proof. To complete the proof of the Theorem, we need to estimate the dif-ference |ρs−µs|. This is accomplished in a manner similar to the preceding,namely we will estimate |ρs − ρsP |. Given ρ, as before, let ρ∗ be given by(2). Then

‖ρ(·, T ) − ρ∗(·, T )‖2L2 ≤ Cd(ρ(·, T ), ρ∗(·, T ))

≤ C[d(ρ(·, T ), 1) + d(1, ρ∗(·, T ))]

≤ 2CKe−σπ2Tdiff ,

or|ρ(T ) − ρ(0)P |2 ≤ 2CKe−σπ2Tdiff


and in particular, for the periodic orbit ρs,

|ρs − ρsP |2 ≤ 2CKe−σπ2Tdiff .

Setting C0 = 2CK ends the proof. Note that in this form of the analysis, the diffusion time may be rather

longer than in the preceding proof, however we may replace ρ∗ above by themeasure determined by the periodic solution to show that all distributionsapproach the periodic state in the weak* topology.

6. 2-Toothed ratchet

We will begin with an explicit formula for g, the Green’s function forthe heat operator with Neumann boundary conditions on (0, 1).

Let

Γ (x, τ, a) =e−(x−a)2/4τ

2(πτ)1/2.

For brevity we will usually suppress x dependence and simply write Γ (τ, a).It is a matter of straightforward calculation to check that

g(τ, a) =12

∞∑−∞

[Γ (τ, 2n−a)+Γ (τ,−2n−a)+Γ (τ, 2n+a)+Γ (τ,−2n+a)].

(1)We assume that Ttr > T ∗

tr, Tdiff > T ∗diff so that the assertions of Theorem

2 are satisfied.We set

P1(τ) =∫ 1/2

0

g(x, τ, a) dx, P2(τ) =∫ 1

1/2

g(x, τ, 1/2 + a) dx,

and

P (τ) =(

P1 1 − P1

1 − P2 P2

),

hence the stationary vector µs of P is given by

µs1 =

1 − P2

2 − (P1 + P2), µs

2 =1 − P1

2 − (P1 + P2).

Intuitively P1 represents the probability of a particle undergoing Brownianmotion which at time t = 0 occupies the well a to remain in (0, 1/2) in timeτ ; a similar interpretation can be given to P2.

By a simple calculation we get that the spectral gap of P (τ)− id is givenby

κ(τ) = 2 − (P1 + P2).

Notice that κ is a monotonically increasing function of τ , κ(τ) > 0 for τ > 0,and limτ→∞ κ(τ) = 1.


We will compute now ∆P = P1−P2 = κ(τ)(µs1−µs

2). Using the formulafor g(τ, a) we obtain

∆P =∞∑−∞

{∫ 1/2−2n+a

−2n+a

−∫ 3/2−2n+a

1−2n+a

}G(z, τ) dz,

where G(z, τ) = Γ (z, τ, 0). Since G(z, τ) = G(−z, τ) therefore∑∞−∞

∫ 3/2−2n+a

1−2n+aG(z, τ) dz =

∑∞−∞

∫ −1/2+2n+a

−1+2n+aG(z, τ) dz

=∑∞

−∞∫ 1−2n−a

1/2−2n−aG(z, τ) dz.

We then have

∆P =∞∑−∞

{∫ 1/2−2n+a

−2n+a

−∫ 1−2n−a

1/2−2n−a

}G(z, τ) dz.

Clearly ∆P (1/4, τ) = 0; also ∆P (τ, a) is for each fixed τ > 0 a smoothfunction of a. Expanding ∆P (τ, a) in power series in terms of a we get

∆P (τ, a) = ∂∆P∂a

∣∣a=1/4 (a − 1/4) + 1

2∂2∆P∂a2

∣∣a=1/4 (a − 1/4)2

+ 16

∂3∆P∂a3

∣∣a=1/4 (a − 1/4)3 + 1

24∂2∆P∂a4 (a − 1/4)4 |a=a∗ ,

with some a∗ ∈ (0, 1/2).Coefficients of the consecutive terms in the above polynomial satisfy (for

simplification we write G(·, τ) = G(·)):∂∆P∂a

∣∣a=1/4 =

∑∞−∞[G(1/2 − 2n + a) − G(−2n + a)+G(1 − 2n − a) − G(1/2 − 2n − a)]

∣∣a=1/4

= 2∑∞

−∞[G(3/4 − 2n) − G(1/4 − 2n)],

∂2∆P∂a2

∣∣|a=1/4 =∑∞

−∞[G′(1/2 − 2n + a) − G′(−2n + a)

−G′(1 − 2n − a) + G′(1/2 − 2n − a)]∣∣a=1/4

= 0,

∂3∆P∂a3

∣∣a=1/4 = 2

∑∞−∞[G′′(3/4 − 2n) − G′′(1/4 − 2n)],

∂4∆P∂a4 |a=a∗ =

∑∞−∞[G′′′(1/2 − 2n + a∗) − G′′′(−2n + a∗)

−G′′′(1 − 2n − a∗) + G′′′(1/2 − 2n − a∗)].

One can show that for any constant a ∈ [−1, 1] we have∣∣∣∣∣∞∑−∞

G′′′(2n + a)

∣∣∣∣∣ ≤ C

∫ ∞

−∞

|z|(τ + z2)e−z2/4τ

τ7/2dz ≤ Cτ−3/2

and therefore ∣∣∣∣∂4∆P

∂a4|a=a∗

∣∣∣∣ ≤ Cτ−3/2.


We also have for some a∗ ∈ [1/4, 3/4]

G′′(3/4 − 2n) − G′′(1/4 − 2n) =12G′′′(a∗ − 2n),

hence ∣∣∣∂3∆P∂a3

∣∣a=1/4

∣∣∣ ≤ C∫ ∞−∞

|z|(τ+z2)e −z2/4τ

τ7/2 dz

≤ Cτ−3/2.

Furthermore∞∑−∞

[G(34− 2n) − G(

14− 2n)] =

∞∑−∞

[12G′(−2n) +

14G′′(−2n)] + O(τ−3/2)

=14

∞∑−∞

G′′(−2n) + O(τ−3/2).

If we set zn = |2n|/τ1/2 then

∞∑−∞

G′′(−2n) =1

16π1/2τ

∞∑−∞

e−z2n/4(−2 + z2

n)τ1/2

.

Given ε > 0 there exists Nε such that

∑|n|≥Nε

e−z2n/4(−2 + z2

n)τ1/2

≤ ετ−1/2. (2)

On the other hand we observe that since zn → 0 as τ → 0 therefore

∑|n|<Nε

e−z2n/4(−2 + z2

n)τ1/2

=12

∑|n|<Nε

|zn+1 − zn|e−z2n/4(−2 + z2

n)

= 2∑

|n|<Nε

|zn+1 − zn|d2

d2ze−z2/4|z=zn

= 2∫ Nε

−Nε

d2

dz2e−z2/4 dz + o(1)

= −2Nεe−N2

ε /4 + o(1), as τ → 0. (3)

From (2), (3) it follows that there exists a constant c > 0 such that

1/4∞∑−∞

G′′(−2n) ≤ −cτ−1

for all sufficiently large τ .It follows that as τ → ∞ we have

∆P (τ, a)

{> c(1/4 − a)τ−1[1 + o(1)], for a < 1/4,

< −c(a − 1/4)τ−1[1 + o(1)], for a > 1/4.


In terms of the vector µs we then have

µs1 > µs

2 + cκ(τ) (1/4 − a)τ−1[1 + o(1)], for a < 1/4,

µs2 > µs

1 + cκ(τ) (a − 1/4)τ−1[1 + o(1)], for a > 1/4.

(4)

Assuming that a < 1/4 and applying Theorem 3 we conclude that

ρs1 = µs

1 + (ρs1 − µs

1)> µs

2 + cκ(τ) (1/4 − a)τ−1[1 + o(1)] − ε(Ttr, Tdiff , σ)

≥ ρs2 + c

κ(τ) (1/4 − a)τ−1[1 + o(1)] − ε(Ttr, Tdiff , σ),(5)

whereε(Ttr, Tdiff , σ) = K0

κ(σTdiff)e−σπ2Tdiff/2

[log Ttr

Ttr

+min{σ1/2eλTtr/4, 1}]1/2

.

Clearly for each σ > 0 and Ttr we can chose Tdiff sufficiently large such that

K0

[log Ttr


]1/2

<c

8(1 − 4a)eπ2τ/2τ−1, τ = σTdiff ,

(6)which implies that ρs

1 > 1/2 > ρs2. Thus we have proved the Theorem in

the case a < 1/4.The case a > 1/4 is similar, we omit the details.

7. Some results of numerical simulations

This section contains results of simulations. Figure 2 shows a periodicsolution ρ(x, t), x ∈ (0, 1), at t = Ttr and t = T where the potential ψ ispiecewise linear and asymmetric with 4 wells located at ai = (i−1)/4+1/16,i = 1, . . . , 4.

To construct Figure 3, we ran simulations for a sequence of Tdiff andplotted transport µ∗

1 + µ∗2 vs. log2 Tdiff . We then computed µs

1 + µs2 of the

discrete ratchet for essentially the same σ. Our estimate (4) only gives thelong time behavior of µ1 as a function of τ = σTdiff .

8. Parrondo’s game: losing strategies can win

The Parrondo Paradox is a pair of coin toss games, each of which is fair,or even losing, but a strategy of playing them in alternation can becomewinning. The result is directed motion of capital, although this outcomeis favored by neither constituent game. It has been proposed as a discreteanalog to the flashing ratchet example of a Brownian motor. As a specificexample, consider a pair of games, an a game and a b game [11]. The a gameis single toss of a fair coin, to win or to lose a dollar. The b game has two


0

2

4

6

8

10

12

14

16

0.2 0.4 0.6 0.8 1x

Fig. 2. Snapshots of a periodic solution for a potential with four teeth at t = Ttr,the end of a transport phase (’spiked’ curve), and t = T , the end of a diffusionphase

parts. If the present capital is divisible by 4, then a coin with very unfairprobability p is played. Otherwise a coin with quite favorable probability p′

is played. The numbers p and p′ are chosen so that the expectation of the bgame is zero, or perhaps even less than zero. What does ’fair’ mean in thissituation? A naıve sense of fairness might be

E =14(2p − 1) +

34(2p′ − 1) = 0.

However when playing the b game, not all integers mod4 occur with thesame frequency, so ’fair’ could mean fair with respect to its equilibriumdistribution, or

E = ρ1(2p − 1) + (ρ2 + ρ3 + ρ4)(2p′ − 1) = 0,ρk = Prob(capital mod 4 = k − 1 in equilibrium).


0.7

0.8

0.9

1

–4 –3 –2 –1 0 1 2Fig. 3. Ratchet efficiency. Comparison of transport, ρs

1 + ρs2, predicted by the

Markov chain (upper curve) and by solving the PDE (lower curve) vs. log2 Tdiff

in any event, let ρik denote the probability density of capital k at play i.

Thenρi+1

k = pk−1ρik−1 + (1 − pk+1)ρi

k+1, wherepk = probability of success from position k.

Since this may be rewritten as the difference equation

ρi+1k − ρi

k =12(ρi

k−1 − 2ρik + ρi

k+1) +12(bk+1ρ

ik+1 − bk−1ρ

ik−1),

with

bk = −(2pk − 1) ={−(2p − 1), k = 0 mod 4,−(2p′ − 1), k = 1, 2, 3 mod 4,

and the fair coin a game may be written as the diffusion difference equation

ρi+1k − ρi

k =12(ρi

k−1 − 2ρik + ρi

k+1)


playing the two games in alternation has the appearance of the flashingratchet.

Further analysis reveals that this analogy is incorrect, as we point outin [12]. Although this Parrondo game has many interesting features, in-cluding a novel ratchet-like mechanism and an interesting dependence onthe detailed balance properties of game b, the essential feature we wish tobring out here is that winning or losing depends essentially on the poten-tial difference of the potential generated by the bk. This potential is justthe piecewise linear function whose slopes are the bk and is not generallyperiodic. The direction of the flashing ratchet, on the other hand, dependson the geometry of the potential landscape, in particular, its asymmetry.These need not coincide and it is possible to give a winning Parrondo gamewhose flashing ratchet analog moves distribution to the left.

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Department of Mathematical SciencesCarnegie Mellon University

Pittsburgh, PA 15213e-mail: [email protected]

and

Department of MathematicalSciences

Kent State UniversityKent, OH 44242

e-mail: [email protected]

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