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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.
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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)
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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
∫ 10
ρ0 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∫ 10
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 =
∫ 10
e−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,
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4 David Kinderlehrer, Micha�l Kowalczyk
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.
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Diffusion mediated transport and the flashing ratchet 5
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
∫ 1/20
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)
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6 David Kinderlehrer, Micha�l Kowalczyk
(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+1xi
ρ(x, T ) ≈ µi =k∑
j=1
µ∗j
∫ xi+1xi
g(x, σTdiff ; aj) dx
In matrix notation,
µ = µ∗P (τ)
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Diffusion mediated transport and the flashing ratchet 7
where for convenience we have set τ = σTdiff and matrix P (τ) is
defined by
P (τ) =
∫ x2x1
g(τ, a1) dx∫ x3
x2g(τ, a1) dx . . .
∫ xk+1xk
g(τ, a1) dx∫ x2x1
g(τ, a2) dx∫ x3
x2g(τ, a2) dx . . .
∫ xk+1xk
g(τ, a2) dx
. . .∫ x2x1
g(τ, ak) dx∫ x3
x2g(τ, ak) dx . . .
∫ xk+1xk
g(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 = (µs1, . . . , µ
sk) be the stationary vector for P (τ), i.e. µ
s = 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)
∫ 10
∫ 10
(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.
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8 David Kinderlehrer, Micha�l Kowalczyk
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 2π
2σTdiff −λTtr > log 2 then the following statements hold:
(1) Let ρ be a solution to (2) with ‖ρ(·, 0)‖H2 < 21/2+1
21/2−1 . 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 ρ̂(t) ∈ Rk is a vector defined by
ρ̂i(t) =∫ xi+1
xi
ρ(x, t) dx,
and P = P (τ) is the probability matrix defined in (8) then
|ρ̂(T ) − ρ̂(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 ρ̂s(t) be a
vector whosecomponents are defined by
ρ̂si(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 ρ̂siδai) ≤ K0ε,
‖ρs(·, T ) − ∑ki=1 ρ̂sig(·, ai, σTdiff)‖2L2 ≤ K0e−π2σTdiff ε.
(10)(2) There exists a constant κ = κ(σTdiff) > 0 such that we
have
|ρ̂s(T ) − µs(σTdiff)| ≤K0
κ(σTdiff)e−σπ
2Tdiff/2ε1/2.
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Diffusion mediated transport and the flashing ratchet 9
(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 µsi 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
1σ 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)
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10 David Kinderlehrer, Micha�l Kowalczyk
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 ρ̂i(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 {ρ̂(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∑ki=1
Vi=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.
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Diffusion mediated transport and the flashing ratchet 11
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 can
choose Tdiff such that if a < 1/4 then we have ρ̂s1 > 1/2
> ρ̂s2. Likewisefor a > 1/4 we have ρ̂s2 > 1/2 >
ρ̂s1.
Remark 3. Let a < 1/4. Observe that if σ > 0 and Ttr >
T ∗∗tr are fixed thenfor Tdiff = 0, or Tdiff = ∞ we get ρ̂s1 = ρ̂s2
= 1/2. Thus Theorem 4 impliesthat as we vary Tdiff from 0 to ∞ then
there exists time Tmaxdiff such that forTdiff = Tmaxdiff the
difference ρ̂s1 − ρ̂s2 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].
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12 David Kinderlehrer, Micha�l Kowalczyk
Proof. Multiplying (1) by ρ and integrating by parts we
obtain:
12
d
dt
∫ 10
ρ2 dx = −σ∫ 1
0
ρ2x dx −∫ 1
0
bρρx dx
≤ −12
∫ 10
b∂
∂xρ2 dx
=12
∫ 10
bxρ2 dx ≤ |bx|0
2
∫ 10
ρ2 dx.
Likewise, differentiating (1) with respect to x, multiplying by
ρx and thenintegrating over (0, 1) we obtain
12
d
dt
∫ 10
ρ2x dx ≤ [32|bx|0 + |bxxx|0]
∫ 10
[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
∫ 10
ρ2xx dx ≤ [52|bx|0 + 3|bxx|0 +
12|bxxx|0]
∫ 10
[ρ2 + ρ2x + ρ2xx] 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−π
2σTdiff (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−π
2σTdiff+λTtr/2 < 2−1/2, and
‖v(0)‖H2 ≤e−π
2σTdiff+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),
∫ 10
v = 0 we have
‖F(ρs + v) −F(ρs)‖H2 < ‖v‖H2 . (4)
-
Diffusion mediated transport and the flashing ratchet 13
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−π
2σTdiff+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∣∣∣∫ 10
ξ(x)ρ(x) dx − ∫ 10 ξ(y)ρ∗(y) dy∣∣∣ = ∣∣∣∫ 10 ∫ 10 [ξ(x) − ξ(y)]
dp(x, y)
∣∣∣≤ |ξ′|0
[∫ 10
∫ 10(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∗.
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14 David Kinderlehrer, Micha�l Kowalczyk
Choosing ξ = ρ and then ξ = ρ∗ we obtain
‖ρ − ρ∗‖2L2 ≤∣∣∣∫ 10 ρ2(x) dx − ∫ 10 ρ(y)ρ∗(y) dy
∣∣∣+
∣∣∣∫ 10 (ρ∗)2(x) dx − ∫ 10 ρ(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 < Cksuch 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+1xi
ρ∗ dx, i = 1, . . . , k.
Then
C−1k d(µ, µ∗)2 ≤ max |µj − µ∗j | ≤ ‖ρ − ρ∗‖L2 .
Proof. Obvious from Schwarz’s Inequality.
-
Diffusion mediated transport and the flashing ratchet 15
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),∫ 10
ζ0 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
∫ 10
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
∫ 10
ζ(·, t) = 1.We can solve (1) by the method of characteristics.
More precisely if we
define for x ∈ (0, 1), t > 0
s(x, t) ={
x, if x = xi or x = ai,s is determined from t = −
∫ xs
dyb(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)
-
16 David Kinderlehrer, Micha�l Kowalczyk
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+1xi
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+1xi
(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 ≤
∫ 10
∫ 10(x − y)2 dP (x, y)
=∫ 10
∫ 10(x − y)2 ∑ki=1 χ(xi,xi+1)(x)δ(y − ai) dµ(x)
=∫ 10
∑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+1xi
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+1i=1 [yi, zi] and ψxx > 0 otherwise.
We will prove the following lemma.
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Diffusion mediated transport and the flashing ratchet 17
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+1xi
(x − ai)2ζ(x) dx ≤ e−δt[
k∑i=1
∫ xi+1xi
(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+1xi
(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+1i=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+1xi
[(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+1xi
(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+1and bx(x) ≤ −c0/2 for x ∈ Eδ. From
(3) we get
∫ t0
∫ z′ixi
ζ(x, ξ) dxdξ =∫ t0
∫ z′ixi
sx(x, ξ)ζ0(s(x, ξ)) dxdξ
=∫ t0
∫ s(z′i,ξ)xi
ζ0(y) dydξ.
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18 David Kinderlehrer, Micha�l Kowalczyk
From the choice of δ it follows s(z′i, ξ) − xi ≤ (z′i −
xi)e−c0ξ/4 hence∫ t0
∫ s(z′i,ξ)xi
ζ0(y) dydξ ≤∫ ∞0
∫ xi+(z′i−xi)e −c0ξ/4xi
ζ0(y) dydξ
≤ − 14c0∫ z′i
xiζ0(y) ln( y−xiz′
i−xi ) dy
≤ − z′i−xi4c0
∫ 10
ζ0(xi + (z′i − xi)z) ln z dz
≤ |z′i−xi|2c0
‖ζ0‖C0(yi,zi)≤ δ
4c20‖ζ0‖C0(yi,zi).
(6)
Since an analogous estimate holds for the integral over (y′i+1,
xi+1), therefore∫ t0
∫Eδ
ζ dxdξ ≤ δc20
‖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
∫ 10
(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
∫ 10
(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∫ 10
(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:
-
Diffusion mediated transport and the flashing ratchet 19
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+1xi
(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 ≤ ∑ki=1 ∫ xi+1xi (x − ai)2ρ(x, t) dx≤
∑ki=1 {∫ xi+1xi (x − ai)2ζ(x, t) dx
+∫ xi+1
xi(x − ai)2[ρ(x, t) − ζ(x, t)] dx
}≤ ∑ki=1 ∫ xi+1xi (x − ai)2ζ(x, t) dx
+∫ 10[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 if
Ttr > T∗tr then
d(ρ(·, Ttr),k∑
i=1
µ∗i δai)2 ≤ R0(1 + c1)
log2 TtrT 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∗trT∗tr
< δ1 and set δ = 2 log TtrTtr . Estimate (1) follows
nowimmediately from Corollary 2. The proof is complete.
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20 David Kinderlehrer, Micha�l Kowalczyk
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),∫ 10
u0 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)‖2L2
= e 2σπ2(t0−Ttr)e−2σπ
2(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],
-
Diffusion mediated transport and the flashing ratchet 21
then by Lemma 2 and Proposition 1 it follows from (3) with t =
T
‖ρ(·, T ) − ρ∗(·, T )‖2L2 ≤ Ke 2σπ2(t0−Ttr)e−2σπ
2Tdiff 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,∣∣∣∣∣
∫ ba
[ρ(x, T ) − ρ∗(x, T )] dx∣∣∣∣∣2
≤ ‖ρ(·, T ) − ρ∗(·, T )‖2L2 ,
therefore by (2) we obtain
|ρ̂(T ) − ρ̂(0)P |2 ≤ kM(σ, Ttr, Tdiff). (5)
In particular if ρ ≡ ρs, the periodic orbit, then ρ̂s(T ) =
ρ̂s(0) = ρ̂s, hence
|ρ̂s − ρ̂sP |2 ≤ kM(σ, Ttr, Tdiff),
with ρ in the definition of M(σ, Ttr, Tdiff) replaced by ρs.
Consequently
|ρ̂s − 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
Ttr+ min{σ1/2eλTtr/4, 1}
], (7)
Combining (4) and (7) we obtain statement (1) of Theorem 2,
whereas from(5) it follows
|ρ̂(T ) − ρ̂(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∗ ≡ ρ̂s.Next we note that from estimates (7),
(8) and (6) we get
|ρ̂s − µs| ≤ K0κ(σTdiff)
e−σπ2Tdiff
[log Ttr
Ttr+ min{σ1/2eλTtr/4, 1}
]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.
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22 David Kinderlehrer, Micha�l Kowalczyk
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∫ 10
ρ dx = 1, for t > 0.
Then
d
dtd(ρ, 1)2 = −2σ
∫ 10
(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).
-
Diffusion mediated transport and the flashing ratchet 23
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 =∫ 10[−2(x − F )Ftρ + (x − F )2ρt] dx
=∫ 10[2σ(x − F )ρxρ + σ(x − F )2ρxx] dx
=∫ 10[−2σ(x − F )ρxρ − σ ∂∂x (x − F )2ρx] dx
=∫ 10{−2σ(x − F )ρxρ − 2σ[(x − F )(1 − ρ)ρx]} dx
= −2σ∫ 10(x − F )ρx dx
= 2σ∫ 10(1 − ρ)ρ dx
= −2σ∫ 10(ρ2 − 1) dx = −2σ
∫ 10(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∫ 10
(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
|ρ̂s − µs| ≤ C0κ(σTdiff)
e−σπ2Tdiff
Proof. To complete the proof of the Theorem, we need to estimate
the dif-ference |ρ̂s−µs|. This is accomplished in a manner similar
to the preceding,namely we will estimate |ρ̂s − ρ̂sP |. 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
|ρ̂(T ) − ρ̂(0)P |2 ≤ 2CKe−σπ2Tdiff
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24 David Kinderlehrer, Micha�l Kowalczyk
and in particular, for the periodic orbit ρs,
|ρ̂s − ρ̂sP |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 − P11 − P2 P2
),
hence the stationary vector µs of P is given by
µs1 =1 − P2
2 − (P1 + P2), µs2 =
1 − P12 − (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.
-
Diffusion mediated transport and the flashing ratchet 25
We will compute now ∆P = P1−P2 = κ(τ)(µs1−µs2). 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+a1−2n+a G(z, τ) dz =
∑∞−∞
∫ −1/2+2n+a−1+2n+a G(z, τ) dz
=∑∞
−∞∫ 1−2n−a1/2−2n−a G(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) + 12 ∂
2∆P∂a2
∣∣a=1/4 (a − 1/4)2
+ 16∂3∆P∂a3
∣∣a=1/4 (a − 1/4)3 + 124 ∂
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 ā ∈ [−1, 1] we
have∣∣∣∣∣∞∑−∞
G′′′(2n + ā)
∣∣∣∣∣ ≤ C∫ ∞−∞
|z|(τ + z2)e−z2/4ττ7/2
dz ≤ Cτ−3/2
and therefore ∣∣∣∣∂4∆P∂a4 |a=a∗∣∣∣∣ ≤ Cτ−3/2.
-
26 David Kinderlehrer, Micha�l Kowalczyk
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 −z2/4ττ7/2 dz≤
Cτ−3/2.
Furthermore∞∑−∞
[G(34− 2n) − G(1
4− 2n)] =
∞∑−∞
[12G′(−2n) + 1
4G′′(−2n)] + O(τ−3/2)
=14
∞∑−∞
G′′(−2n) + O(τ−3/2).
If we set zn = |2n|/τ1/2 then∞∑−∞
G′′(−2n) = 116π1/2τ
∞∑−∞
e−z2n/4(−2 + z2n)
τ1/2.
Given ε > 0 there exists Nε such that
∑|n|≥Nε
e−z2n/4(−2 + z2n)
τ1/2≤ ετ−1/2. (2)
On the other hand we observe that since zn → 0 as τ → 0
therefore∑
|n| 1/4.
-
Diffusion mediated transport and the flashing ratchet 27
In terms of the vector µs we then have
µs1 > µs2 +
cκ(τ) (1/4 − a)τ−1[1 + o(1)], for a < 1/4,
µs2 > µs1 +
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
ρ̂s1 = µs1 + (ρ̂s1 − µs1)> µs2 +
cκ(τ) (1/4 − a)τ−1[1 + o(1)] − �(Ttr, Tdiff , σ)
≥ ρ̂s2 + cκ(τ) (1/4 − a)τ−1[1 + o(1)] − �(Ttr, Tdiff ,
σ),(5)
where�(Ttr, Tdiff , σ) = K0κ(σTdiff)e
−σπ2Tdiff/2[
log TtrTtr
+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
Ttr+ min{σ1/2eλTtr/4, 1}
]1/2<
c
8(1 − 4a)eπ2τ/2τ−1, τ = σTdiff ,
(6)which implies that ρ̂s1 > 1/2 > ρ̂s2. Thus we have
proved the Theorem inthe 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 µ
s1 + µ
s2 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
-
28 David Kinderlehrer, Micha�l Kowalczyk
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 näıve sense of fairness
might be
E =14(2p − 1) + 3
4(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).
-
Diffusion mediated transport and the flashing ratchet 29
0.7
0.8
0.9
1
–4 –3 –2 –1 0 1 2Fig. 3. Ratchet efficiency. Comparison of
transport, ρ̂s1 + ρ̂
s2, 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.Then
ρi+1k = pk−1ρik−1 + (1 − pk+1)ρik+1, where
pk = probability of success from position k.
Since this may be rewritten as the difference equation
ρi+1k − ρik =12(ρik−1 − 2ρik + ρik+1) +
12(bk+1ρik+1 − bk−1ρik−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 − ρik =12(ρik−1 − 2ρik + ρik+1)
-
30 David Kinderlehrer, Micha�l Kowalczyk
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]