Localization, Escape Rate and Delocalization in Kinked Ratchet Potentials
by
Sang Hyun Choi
Submitted to the Department of Physics in partial fulfillment of the Requirements for the Degree
of
BACHELOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
ARCHI\ES
June, 2015
MASSEME T
@2015 SANG HYUN CHOI
All Rights Reserved
INSTITUTE
LG Y
LO-HO
S,
AUG 10 2015
LIBRARIES
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part
Author
Signature redacted
Certified by
Signature redacted
-
Department of Physics
,*ay 8, 20,15
Alfredo Alexander-Katz
Thesis Supervisor, Department of Materials Science and Engineering
Signature redacted
/11
Jeremy England
Thesis Co-Supervisor, Department of Physics
Accepted by
Signature redacted
___
'Professor Nergis Mavalvala
Senior Thesis Coordinator, Department of Physics
I
Localization, Escape Rate and Delocalization in Kinked Ratchet Potentials
by
Sang Hyun Choi
Submitted to the Department of Physics
on May 8, 2015, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
The particle localization in ratchet potentials with segments of reverse directions, or kinked
ratchets, is computationally studied. Kinks localize a particle on on-off pulsating ratchet potentials, forming stable points in the effective potential. Analogous Kramers rate for transition
betwen kinks is derived through simulations under different values of parameters defining the
system. Adding tilting to the system with a proper choice of tilting frequency induces stochastic
resonance. Delocalization of a particle is observed in the resonant activation regime.
Thesis Supervisor: Alfredo Alexander-Katz
Title: Associate Professor of Materials Science and Engineering
Thesis Co-Supervisor: Jeremy England
Title: Assistant Professor of Physics
3
4
Acknowledgments
I would like to thank Professor Alfredo Alexander-Katz for giving me the opportunity to work
in his group, and for his guidance and support. I appreciate all the insightful discussions we had
and his consistent encouragement. I would like to thank all the members of the Alexander-Katz
group for being wonderful colleagues and making my research experience fruitful. I would also like
to thank Professor Jeremy England for co-supervising my thesis and Bobby Marsland from the
England group for answering my questions on non-equilibrium statistical physics. Finally, I would
like to thank Professor Udo Seifert and David Hartich from the Institute for Theoretical Physics
at the University of Stuttgart for introducing me to the research on non-equilibrium dynamics.
5
6
Contents
1
Introduction
11
11
Study of Transport by Thermal Noise
Basics of Stochastic Dynamics .....
1.2.1 Langevin Equation.......
1.2.2 Fokker-Planck Equation . . .
Overview of the Ratchet System . .
1.3.1 Pulsating Ratchet . . . . . .
1.3.2 Tilting Ratchet . . . . . . . .
1.3.3 Application of Ratchets . . .
Outline of the Paper . . . . . . . . .
12
12
13
14
14
15
17
18
Method and System Description
19
3
Localization in Kinked Ratchets
3.1 Single Kink . . . . . . . . . . . . . .
3.1.1 Effective Potential . . . . . .
21
21
21
. . . . . . . . . . . . .
.
22
23
24
.
.
2
.
1.4
.
.
.
.
1.3
.
.
1.1
1.2
3.3
3.2.1 Effective Potential . . . . . .
Periodic Kink Lattice . . . . . . . .
.
Double Kink
Effective Rate Theory
4.1 Kramers Problem . . . . . . . . . . . . . . . . . .
4.2 Parameter Dependence of Effective Energy Barrier AG..
4.2.1 Dependence on the Fraction of Time during which the Ratchet is Turned Off
4.2.2 Dependence on the Ratchet Height . . . .
4.2.3 Dependence on the Ratchet Asymmetry .
4.2.4 Dependence on the Distance between Kinks
4.3 Effective Diffusion Coefficient . . . . . . . . . . .
4.4 Mean First Passage Time . . . . . . . . . . . . .
25
Stochastic Resonance and Delocalization by Resonant Activation
5.1 Stochastic Resonance . . . . . . . . . . . . . . . . . . .
5.2 Stochastic Resonance in Kinked Ratchets . . . . . . .
5.3 Resonant Activation . . . . . . . . . . . . . . . . . . .
5.4 Double Kink . . . . . . . . . . . . . . . . . . . . . . .
5.5 Periodic Kink Lattice . . . . . . . . . . . . . . . . . .
35
Conclusion and Future work
43
.
29
.
.
.
.
31
32
38
39
.
7
30
35
37
38
.
6
27
27
28
.
5
26
.
.
4
3.2
Appendix A MATLAB code
47
A.1 M ain Simulation Code ...................................
A.2 Ratchet Potentials ........
.....................................
47
49
A.2.1
A.2.2
Single Kink .............................................
Double Kink .............................................
49
50
A.2.3
Periodic Kink Lattice ......................................
50
8
List of Figures
1.1
1.2
The potential of a pulsating on-off ratchet system. When a potential is turned off,
there is no underlying potential in the system like the dot-dashed horizontal line at
zero potential. In this case, a particle, drawn as a circle in the figure, undergoes
free diffusion. When a potential is turned on, the particle is driven by the ratchet
potential drawn in a solid line. Net drift to the right direction is resulted in this case
as shown by the blue arrow.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The tilting ratchet potential. (a) Before tilting or at the half period, the ratchet potential is the same as the one in Figure 1.1, resembling a sawtooth. (b) The ratchet
potential with the maximum tilting in the positive direction. (c) The ratchet potential with the maximum tilting in the negative direction. Adapted from Astumian
1997 [2]
3.1
3.2
3.3
3.4
3.5
4.1
15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
The ratchet potential with single kink. Dotted line at x = 0 denotes the location of
the kink. As indicated in the arrow, a particle under this potential is driven toward
the kink, being trapped there. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) The particle distribution under the single kink ratchet potential in linear scale.
Dashed line at x = 0 indicates the location of the kink. (b) The distribution in
logarithmic scale. Dashed line shows the envelope of the overall distribution. The
distribution was generated with Ag = 0.2, L = 0.7, and f = 0.95. . . . . . . . . .
The ratchet potential with two kinks. Dashed vertical lines indicate the location of
kinks. The direction of the sawtooth changes at the kinks and the halfway between
the kinks. Because of the direction of the segments of the ratchet, particles get
trapped at the kinks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) The particle distribution in linear scale for Ag = 0.2, L = 0.7, f = 0.95 and Ax
= 3.7. Dashed vertical lines at x = 3.7 denote the location of kinks, and it is where
the particle has most frequently existed. (b) The distribution in logarithmic scale.
Dashed lines are an envelope of the distribution. . . . . . . . . . . . . . . . . . . .
(a) The particle distribution in a periodic kink lattice in linear scale for Ag = 0.2,
L = 0.7, f = 0.95 and Ax = 3.7. Dashed line indicates the location of a kink.
The system is in a periodic boundary condition, and one period is shown. (b) The
logarithmic scale of (a). Dashed lines are the linear envelope of the distribution. (c)
The distribution showing five periods. Dashed lines indicate kink locations. . . . .
. 21
. 22
. 23
. 23
. 24
The effective potential of a double kink ratchet system. Due to thermal fluctuation,
a particle is able to transfer from one stable point to another. Since the location of a
minimum point and that of a kink coincide, the half-distance Ax between the kinks
is the half-distance between the minima. AG is the effective energy barrier.
. . . . 25
9
4.2
4.3
4.4
4.5
4.6
4.7
4.8
5.1
5.2
5.3
5.4
(a) The particle distribution for various fractions of time of turning off the ratchet
f. Dashed lines are the logarithmic scale of the particle distribution and the solid
lines are linear envelopes of the distributions. (b) The relation between AG/Ax
and 1 - f. Dashed line is the linear regression of the slopes of the envelopes in (a).
Error bars indicate 95% confidence bound. . . . . . . . . . . . . . . . . . . . . . .
(a) The particle distribution under various ratchet heights Ag. Dashed lines show the
logarithmic scale of the distribution and the solid lines show their linear envelopes.
(b) The relation between 3AG/Ax and Ag. Dashed line is the linear regression of
the slopes of the envelopes in (a). Error bars indicate 95% confidence bound. . . .
(a) The particle distribution under various ratchet asymmetry parameters L. Dashed
lines show the logarithmic scale of the distribution and the solid lines show their
linear envelopes. (b) The relation between slope of the envelopes and 2L - 1. The
dot-dashed line is a quadratic fitting and the dashed line is a linear fitting. Error
bars indicate 95% confidence bound .
. . . . . . . . . . . . . . . . . . . . . . . .
(a) The particle distribution under various values of Ax. Dashed lines are the logarithmic scale of the distribution while solid lines are their linear envelopes. (b) The
relation between 3AG/Ax and Ax. Error bars indicate 95% confidence bound. For
comparison, the data from a single kink ratchet is also plotted as Ax = 0 case. . .
Effective diffusion coefficients relative to the free diffusion coefficient for under different values of (a) f and (b) Ag. Error bars indicate one standard deviation obtained
by error propagation [6].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The time series of the location of a particle for Ag = 0.2, Ax = 3.7, f = 0.95, and
L = 0.7. The horizontal solid line marks the kink locations or the minimum points
of Veff while the horizontal dashed lines mark the boundary of the kink region. . .
(a) The distribution of time intervals for Ag = 0.2, L = 0.7, f = 0.95 and Ax =
3.7. (b) Linearly fitting MFPT on the collective variable (1 - f)Ag(2L - 1)Ax/Teff.
Error bars indicate one standard deviation. . . . . . . . . . . . . . . . . . . . . . .
27
.
28
.
29
.
31
.
32
.
33
.
33
The schematic of the stochastic resonance. A particle is in a bistable potential
connected to a heat bath. A suitable amount of noise in the system causes the
particle to hop to a global minimum. Adapted from Gammaitoni 1998 [15] . . . . .
The trajectory of a particle in a double kink ratchet with tilting with frequency
W = rr/T. The parameters are Ag = 0.2, L = 0.7, f = 0.95, Ax = 3.7 and A 0 =
0.4Ag/L. The horizontal lines near x = 0 indicate the location of kinks. . . . . . . .
Finding the tilting frequency that induces rapid transition. Simulations were performed for Ag = 0.2, L = 0.7, Ax = 3.7, f = 0.95 and Ao = 0.4 Ag/L. Error bars
indicate one standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The particle trajectory in the presence of tilting. Tilting frequency W = 0.01 and
tilting amplitude A 0 = 0.4 Ag/L. Other parameters are set as Ag = 0.2, L = 0.7,
f
5.5
.
= 0.95, and A x = 3.7.
36
37
38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
(a)The particle distribution in a periodic kink lattice with resonant activation for
Ag = 0.2, L = 0.7, f = 0.95, Ax = 3.7, w = 0.01 and A 0 = 0.4 Ag/L. It is showing
one period. (b) The particle distribution in a periodic kink lattice with the same
condition for five periods. (c) The particle distribution in a double kink ratchet for
the same condition. Dashed lines in each figure indicate the location of kinks. Solid
lines in (a) and (b) are the distributions for the same condition without tilting.
. .
10
41
Chapter 1
Introduction
1.1
Study of Transport by Thermal Noise
Imagine a pollen fallen on a surface of water. Even though you keep everything intact, the pollen
would drift around with a random trajectory. The movement is called Brownian motion after a
botanist Robert Brown who observed the phenomenon.
Einstein and Langevin independently
worked on the theory of Brownian motion [12] [21]. Langevin hypothesized that a particle, or the
pollen in the example, is subject to two forces, namely a fluctuating force with frequent change in
direction and magnitude and a viscous drag force. While the fluctuating force drives the random
motion of the pollen, the drag force slows it down. The fluctuating force is often called thermal
noise. As its name suggests, thermal noise originates from a heat bath in contact with the system
of interest. Although the amplitude of thermal noise is in the order of kBT, which is very small, it
has a non-negligible effect on a small system like a pollen suspended on water. At equilibrium, it is
obvious that useful work cannot be extracted from thermal noise since the fluctuation is symmetric
and thus the effect averages out to zero. However, when the system is driven far from equilibrium,
one may suspect the possibility of doing useful work out of the thermal fluctuation. Indeed, this
motivation led many physicists to working on the problem of biased Brownian motion, especially
driven by unbiased perturbation. Such directional transport by Brownian motion turns out to be
realized by breaking the spatial inversion symmetry of the system [27]. The system with broken
symmetry is called a ratchet system, and is our central interest in this paper.
11
1.2
Basics of Stochastic Dynamics
As mentioned in the previous section, small systems like a pollen on water are subject to
thermal fluctations from a heat bath. In this regime, usual Newtonian mechanics and equilibrium
statistical mechanics are not sufficient to describe the system.
Therefore, stochastic dynamics
involving random fluctuations is introduced. The dynamics of individual systems like a particle
in Brownian motion follows a Langevin equation while the ensemble probability follows a FokkerPlanck equation [30].
Since the system studied in this paper is one-dimensional, the equations
introduced are one-dimensional as well.
1.2.1
Langevin Equation
If a small particle of mass m is in a fluid, a frictional force or a damping force due to the medium
will act on the particle. Friction is written as
f
= -myv where -y is the damping constant and v is
the velocity of the particle. However this expression is valid only when the velocity due to thermal
fluctuation is negligible. Since the particle is assumed to be very small, the velocity due to thermal
energy Vthermal = VkBT/m is no longer negligible. In other words, when the mass of the particle
m is not much bigger than that of molecules of the medium, the effect of each collision between
the particle and the fluid molecule becomes important. Because there are too many molecules in
the fluid, it is impossible to solve all the coupled equations exactly. So the force by the collisions
is treated as a stochastic term. As a result, the equation of motion for the particle contains the
random force term as well as the friction term.
v = -yv - V'(x)/m +
Here, V(x) is the potential of the system and
(t)
(1.1)
(t) is the random force, or the Langevin force,
which is the fluctuating force divided by the mass m. Eq 1.1 is called a stochastic differential
equation or the Langevin equation. It is identical to Newtonian force except that it contains a
stochastic term. The Langevin force is assumed to be white noise, which has zero average and the
6 correlation [29] as
<
(t) >= 0, <
(t)(t') >= (2-ykBT/m)6(t - t')
12
(1.2)
where kB is the Boltzmann constant and T is the temperature.
When the particle is very small
and submerged in a liquid medium, the inertial force on the particle becomes negligible [26]. So
the system becomes overdamped. For this overdamped limit, the inertial term 'b is neglected, and
Eq 1.1 is reduced to
-yv
-V'(x) +
=
(1.3)
(t)
The Langevin equation is used to simulate the movement of the particle under the ratchet
potential in this paper.
1.2.2
Fokker-Planck Equation
As mentioned in the previous section, the microscopic interaction in the system is treated as a
fluctuating stochastic term. The Fokker-Planck equation describes how the distribution function
of fluctuating macroscopic variables evolves. One-dimensional Brownian motion is expressed as
aw
at
=[
aD(1)(x)
ax
+
D(2)(X) W
ax2
(1.4)
X
where W(x, t) is the distribution function for the Brownian motion, D(2) (x) > 0 is the diffusion
coefficient, and DC')(x) is the drift coefficient [29].
When Brownian particles are in an external field, the time evolution of the distribution function
in position and velocity space W(x, v, t) can be expressed as
aw=
at
--
09x
V+ a
9V
v
M
+ yW
M
av2I
(1.5)
where F(x) is the force from the potential of the system. This special form of the Fokker-Planck
equation is called the Klein-Kramers or Kramers equation [29]. In the large friction limit where
the Langevin equation reduces to Eq 1.3, the Fokker-Planck equation for the distribution function
in position W(x, t) becomes
t=
at
m-Y _9
- -F(x) + kBT a2
axx2
W
(1.6)
Eq 1.6 is called the Smoluchowski equation [29]. Since the system we are working on is assumed
to be in the large friction limit, Eq 1.6 is used to the analytical expression for W(x, t) in this paper.
13
By solving for W(x, t) from the Fokker-Planck equation, one can get the mean value of macroscopic variables. The Fokker-Planck equation is applicable to not only systems near the equilibrium
but also systems far from the equilibrium. Since the equation deals with fluctuations, it is mostly
used for small systems. When the system becomes large enough to be able to neglect small fluctuations, the stochastic part, which is the diffusion term, becomes negligible, and the equation, as
well as the system, becomes deterministic.
1.3
Overview of the Ratchet System
At equilibrium, no work can be done from a noise. Feynman designed a microscopic "ratchet
and pawl" device to discuss the possibility of utilizing thermal noise along with the anisotropy to
drive a motor [14] and showed that the anisotropy of the ratchet's teeth alone can not generate net
motion. Although a thermal gradient can make the device do net work, it is unrealistic to implement
the thermal gradient enough to do a non-negligible work in the real system [2]. Theoretical work
by Magnasco showed that if the particle is subject to an external force with time correlations,
detailed balance is broken, which enables the particle to have a nonzero net drift [23]. This led to
many theoretical works on the "ratchet potential", which is the asymmetric periodic potential that
resembles saw teeth as shown in Figure 1.1. The ratchet system, or the Brownian motor, requires
satisfaction of all three conditions to generate the net drift of a particle, which are presence of
thermal noise, anisotropy of the medium, and time-dependent energy supplied either by external
variations of the constraints on the system or by chemical reactions far from equilibrium [2]. The
diffusive transport by thermal noise is called the ratchet effect [27]. The third condition for the
ratchet effect, the external perturbations of the system, is achieved in various ways, but the most
common way is to either pulsate or tilt the ratchet.
1.3.1
Pulsating Ratchet
In a pulsating ratchet scheme, a set of potentials, usually two, appears alternately with certain
transition rates. The alternating potentials can be oscillating between V(x) - AV and V(x) + AV
or more drastically between 0 and V(x). The latter is called the on-off ratchet because the potential
is "turned on" and "turned off'. The on-off ratchet is how the biased Brownian motion is induced
14
in our study.
y(t)
,-( (0))[ 1+f (t) I +((t)
=
(1.7)
Eq 1.7 shows the overdamped system with an on-off ratchet [27]. Here, V(x) is the ratchet
potential and f(t) is a function of time which takes the value of either 1 or -1. f(t) = 1 describes
the case when the potential is "turned on" while f(t) = -1
is the case when the potential is
"turned off". In a more general case, f(t) is assumed to be an unbiased time periodic function or
a stochastic stationary process [27].
0.5an
-Turned
----- Turned off
0.40.3
L
0.2
'*
0
0.1g
-0 .1 -3
-3
-2
1
0
1
-1
1
1
2
3
Position
Figure 1.1: The potential of a pulsating on-off ratchet system. When a potential is turned off,
there is no underlying potential in the system like the dot-dashed horizontal line at zero potential.
In this case, a particle, drawn as a circle in the figure, undergoes free diffusion. When a potential
is turned on, the particle is driven by the ratchet potential drawn in a solid line. Net drift to the
right direction is resulted in this case as shown by the blue arrow.
Figure 1.1 shows the ratchet effect in the on-off ratchet. When the ratchet is turned on, a
particle feels force due to the potential and moves towards a location of lower energy, tending to
stay at the local minimum of the ratchet. When the potential is turned off, the particle starts
to diffuse freely. As shown in Figure 1.1, the asymmetry of the sawtooth (L > 1 - L) makes the
particle more likely to end up moving to the minimum on its right side than to the one on its left
side when the potential is turned on again. Therefore, after a long run, the net movement of the
particle is in the right direction, which is precisely the ratchet effect.
1.3.2
Tilting Ratchet
In a tilting ratchet scheme, the overall slope of the ratchet is changing as if somebody is tilting
the x-axis of the ratchet shown in Figure 1.2 (a).
15
(a)
U = Usa
saw
r~p~r\
(b)
(C)
U= Usaw + XlFmaX I
U sawXImaxI
t
U = Usa - x 1Fmax
(t
Figure 1.2: The tilting ratchet potential. (a) Before tilting or at the half period, the ratchet
potential is the same as the one in Figure 1.1, resembling a sawtooth. (b) The ratchet potential
with the maximum tilting in the positive direction. (c) The ratchet potential with the maximum
tilting in the negative direction. Adapted from Astumian 1997 [2]
In Figure 1.2, Fmax is a tilting force. When Eo/L < IFmax| < Eo/(1 - L), the potential energy
V(x), or U in the figure, decreases monotonically in the case depicted in Figure 1.2 (c) because
the tilting aids the drift due to the anisotropy. When the ratchet enters the regime of Figure 1.2
(b), a particle is driven to flow in the "reverse" direction of the ratchet. However, because of the
constraint in the magnitude of the force, there still remain minima to trap the particle. Therefore,
tilting induces net flow of particles even without thermal noise [2]. In the presence of noise, net
drift of a particle occurs even for IFmaxl < Eo/L [23]. In fact, increase in thermal noise increases
the flow [2].
ys(t)
-V(x(t)) + y(t) + (t)
(1.8)
Eq 1.8 shows the overdamped system with a tilting ratchet [27]. The function y(t) is the tilting
force. In general, the equation for the potential is V(x, t) = Vo(x) +xy(t). y(t) is either an unbiased
periodic function, in which case it is called a "rocking ratchet", or an unbiased stationary random
process, in which case it is called a "fluctuating force ratchet" [27]. In this paper, we focus on the
rocking ratchet.
16
1.3.3
Application of Ratchets
A ratchet potential has applications in various areas of science. Here its two most common
applications are introduced. First one is a molecular motor in biology. Eukaryotic cells, which are
cells with nucleus, have cytoskeleton which is a filamental structure organizing and interconnecting
other organelles in a cell [27]. It is a path for intracellular transport of nutrients, wastes, proteins
and others. One major type of this filament is made of proteins called microtubuli consisting of
tubulin. Motor proteins such as kinesin move along this path and do the transporting job. Kinesin
is processive, which means that it remains bound to the substrate for multiple rounds of activity [8],
and travels only in one direction due to the asymmetric protein structure of the microtubule [27]
and kinesin motor domain [18]. Surrounding environment serves as a heat bath, causing energy
dissipation as well as providing reactants and taking products from chemical reactions. Kinesin
experiences friction not only with the solvent but also itself and microtubule, and keeps undergoing
chemical reactions with ATP to generate energy. Therefore, damping and fluctuation depend on
its position, making the resulting effective potential "rough" as well as the system is coupled to
non-equilibrium enzymatic chemical reactions. Working out the model with more biochemical and
mathematical rigor shows that the molecular motor system of kinesin is modeled with a pulsating
ratchet scheme [27] [3].
Second application is its use in modeling quantum devices. An one-dimensional rocking ratchet
is realized in an asymmetric SQUID (Superconducting Quantum Interference Device) threaded by
a magnetic flux [32] and an one-dimensional array of Josephson-junctions with alternating critical
currents [13]. The asymmetric SQUID consists of a ring with two Josephson junctions in series
in one arm and one junction in the other arm. When the ring is threaded by a magnetic flux
that is not integer multiple of half the flux quanta, the total phase across the ring experiences an
effective potential of a ratchet. Then when an external AC current is given to the device, only one
sign of the phase velocity is favored [32], which shows the characteristics of a rocking ratchet. In
a parallel array of Josephson-junctions with alternating critical currents, a quasi-particle called a
fluxon moves along the pinning potential. Appropriately tuning the asymmetry of the array gives
a sawtooth-like potential, and indeed fluxons show directional movement [13].
Ratchet potentials are used in numerous other systems including transport of DNA [4], transport
17
of bacteria [19] and quantum ratchets [28], not discussed in above examples. Such versatility makes
the study on ratchet potentials both interesting and important.
1.4
Outline of the Paper
Previous works on biased Brownian motion have been focused on the directional transport of
particles [23] [25] [11] [9].
In this paper, localization of particles by biased Brownian motion is
studied, which is obtained by inserting "kinks" in the ratchet. However, a particle is still able to
overcome the localization and escapes the trap. Delocalization is obtained by tilting the system.
I will ultimately induce stochastic resonance and resonant activation, showing the tunability of
localization in a biased Brownian motion. In Chapter 2, the simulation method used in the study
is reviewed. Chapter 3 discusses the localization of a particle in various kinked ratchets. Chapter
4 focuses on the escape of the particle from kinks by studying the transition rate. In Chapter 5,
stochastic resonance is realized by using the escape rate obtained in Chapter 4, and delocalization
by resonant activation is discussed. Finally, Chapter 6 concludes our discussion and suggests future
works.
18
Chapter 2
Method and System Description
The system of interest in this paper is composed of a Brownian particle under a given potential
V(x, t). For simplicity, the system is studied only for one-dimensional case. The stochastic process
of a Brownian particle is denoted as its spatial location, or its trajectory, x(t). Assuming the friction
is large, as in a liquid medium, we use the overdamped Langevin equation shown in Eq 1.3 as the
equation of motion for the particle. The study was done computationally through simulations in
MATLAB. The random force
(t) is implemented with MATLAB function randn, which generates
a random number from the normal distribution.
The spatial potential of the system V(x) is defined to be a ratchet with various conditions
specified by parameters.
The parameters used in the simulation are the height of the ratchet
(Ag), the half-distance between kinks (Ax), the fraction of time the ratchet is turned off (f) and
the asymmetry parameter of the ratchet (L).
pictorially.
Figure 1.1 and Figure 3.3 shows each parameter
Here, Ax and L are dimensionless parameters since they actually mean the ratio of
half-distance between kinks to the ratchet tooth length and the ratio of longer segment of the
ratchet tooth to the whole ratchet tooth length respectively. However, the ratchet tooth length is
set to be 1 as shown in Figure 1.1, so AZx and L are equivalently used as the half-distance and the
length of the longer segment as in the figure.
In the first part of this study, we are interested in a pulsating on-off ratchet. During simulations, a particle moves according to Eq 1.3 while the ratchet potential V(x, t) is turned on and off
periodically. The particle undergoes free diffusion while V(x, t) is turned off and is drived while
V(x, t) is turned on.
19
In the second part of the study, in addition to turning on and off the ratchet, the same system
is being tilted periodically. The tilting of the ratchet is implemented by addition of a sine function
in the force of the system. The frequency of tilting that induces stochastic resonance and resonant
activation is explored.
During both parts, the location of the particle x(t) is recorded after each time step.
Each
simulation is performed with 3 x 108 time steps. Constants such as -y, kB and m are set as 1. The
MATLAB code for the simulations is attached in the Appendix.
20
Chapter 3
Localization in Kinked Ratchets
Single Kink
3.1
As discussed in the Introduction, the ratchet potential induces an unidirectional motion of a
particle. If the "grain" of the ratchet is reversed in the middle of the track, the directional flows
converge at the point of reversal as shown in Figure 3.1. This point is called a "kink" throughout
the paper. The presence of a kink significantly perturbs the directional motion of a particle [17].
Due to the collision of flow, the probability of a particle to exist at the kink becomes much higher
than other locations on the track. Thus a particle gets trapped or localized at the kink.
0.4
-
0.3
10.2~.0.1
0L
-4
-3
-2
-1
0
Position
12
3
4
Figure 3.1: The ratchet potential with single kink. Dotted line at x = 0 denotes the location of the
kink. As indicated in the arrow, a particle under this potential is driven toward the kink, being
trapped there.
3.1.1
Effective Potential
Figure 3.2 (a) shows the histogram of the trajectory of a particle under the single kink ratchet
potential. The higher the bin is, the more frequently the particle has existed at that location.
21
Therefore, this histogram is equivalent to the probability distribution of the particle. Each peak in
the distribution indicates a local minimum in the ratchet track. As expected, the highest peak is
at the kink, which confirms our assumption that the particle would be localized.
(a)
(b)
7 x106
16
14
6
61-
212
-14
305-
10-
E
22
6-
1
0
-20
-A41
-10
0
Partide Position
10
20
-20
-10
0
Partide Position
10
20
Figure 3.2: (a) The particle distribution under the single kink ratchet potential in linear scale.
Dashed line at x = 0 indicates the location of the kink. (b) The distribution in logarithmic scale.
Dashed line shows the envelope of the overall distribution. The distribution was generated with
Ag = 0.2, L = 0.7, and f = 0.95.
The effective potential of the system affects the overall shape, or the envelope, of the distribution.
Taking the logarithmic scale of the distribution shows linear envelopes as in Figure 3.2 (b). Solving
the Smoluchowski equation (Eq 1.6) for stationary state [29] gives the expression of probability
distribution
W t (x) oc exp[-V(x)/kBT]
(3.1)
where V(x) is the underlying potential of the system. The stationary probability distribution shown
in Figure 3.2 (b) is Nexp[-alxI, for some positive constant a and N. From Eq 3.1, the effective
potential of a single kink ratchet system is thus deduced as Veff(x) = ajxI, having a V-shape.
3.2
Double Kink
Now, as shown in Figure 3.3, the ratchet potential has two kinks. Since there are two kinks,
the distance between them becomes an additional parameter of the system. In this paper, it is
parametrized by the half-distance between the kinks, which is denoted as Ax. As in the single kink
case, the reversal of the ratchet direction causes collision of particle flow and thus localization of
the particle at the kinks.
22
0.4W
0.3-
0
L
0-
-0.1 -6
4
6
4
2
0
Position
-2
Figure 3.3: The ratchet potential with two kinks. Dashed vertical lines indicate the location of
kinks. The direction of the sawtooth changes at the kinks and the halfway between the kinks.
Because of the direction of the segments of the ratchet, particles get trapped at the kinks.
3.2.1
Effective Potential
Due to the presence of two localizing points, the overall particle distribution has two peaks, each
of which is located at the kinks as shown in the histogram of the particle trajectory in Figure 3.4
(a). As in the case of single kink ratchets, the effective potential has its minima at kinks. It is
analogous to a two-well or bistable potential.
(b)
(a) 5x 106
_14-
4-
12
c3
C
E
E
1
0
-30
---
16
86-
-20
10
0
-10
Particle Position
20
4
-30
30
-20
10
0
-10
Particle Position
20
30
Figure 3.4: (a) The particle distribution in linear scale for Ag = 0.2, L = 0.7, f = 0.95 and Ax =
3.7. Dashed vertical lines at x = t3.7 denote the location of kinks, and it is where the particle has
most frequently existed. (b) The distribution in logarithmic scale. Dashed lines are an envelope of
the distribution.
Figure 3.4 (b) shows that the envelope of the distribution is exponential. It makes sense because
a double kink ratchet is superposition of two single kink ratchets. By Eq 3.1, each of the peaks is
described by Wt oc exp[-adx/kBT]. Placing the resulting V-shaped potentials next to each other,
we get a W-shaped effective potential of a double kink ratchet system.
23
3.3
Periodic Kink Lattice
As a double kink ratchet system has two localizing points, a N-kink ratchet has N localizing
points. It means that we can construct a lattice made of kinks in a ratchet.
(a)
(b)
2 "106
15
1.5
14-
(
a-
0 13
E 12
z
E0.5z
00
2
0
11
4
6
10
8
Partice Position
2
4
6
8
Particle Position
3 -106
(C)
0
E
0
10
20
Particle Position
30
40
Figure 3.5: (a) The particle distribution in a periodic kink lattice in linear scale for Ag = 0.2, L
= 0.7, f = 0.95 and Ax = 3.7. Dashed line indicates the location of a kink. The system is in a
periodic boundary condition, and one period is shown. (b) The logarithmic scale of (a). Dashed
lines are the linear envelope of the distribution. (c) The distribution showing five periods. Dashed
lines indicate kink locations.
Figure 3.5 shows the particle distribution in a periodic kink lattice, in which kinks are placed
periodically. The period of kink placement is 2Ax, and the system is constructed with a periodic
boundary condition. Figure 3.5 (a) shows the distribution for one period. As in the case of a single
kink ratchet and a double kink ratchet, a particle is localized at kinks, forming a highest peak in
the distribution. Figure 3.5 (b) shows that the envelope of the distribution is exponential as in
the previous cases. So the effective potential of the system becomes a symmetric sawtooth-shaped
potential with its minima located at kinks. Figure 3.5 (c) shows five periods, which is essentially
alignment of 5 copies of (a). A particle tends to be localized at kinks, forming high peaks of the
distribution there. The difference among peak heights will disappear with more statistics.
24
Chapter 4
Effective Rate Theory
Our discussion on the double kink ratchet concludes that the effective potential is a W-shaped
bistable potential shown in Figure 4.1. As time evolves, a particle is not only confined to one
minimum point of Veff
(x) but also transfers to another minimum point. Indeed, such transition is
obvious in Figure 3.4 where the probability of a particle to exist in either kink is almost the same.
AX
00
AG
Position
Figure 4.1: The effective potential of a double kink ratchet system. Due to thermal fluctuation, a
particle is able to transfer from one stable point to another. Since the location of a minimum point
and that of a kink coincide, the half-distance Ax between the kinks is the half-distance between
the minima. AG is the effective energy barrier.
Tuning the parameters of a ratchet changes the effective energy barrier AG. An energy barrier
is related to the transition rate of a particle to move from one potential well to another by Kramers
relation discussed soon. Therefore, probing AG gives us information about the transition rate in
the kinked ratchet system.
25
4.1
Kramers Problem
When a particle is on a potential that has maxima and minima, it tends to stay in one of the
minima because they are energetically more favorable. However, random fluctuation in the energy
can induce the particle to jump over the energy barrier and move to another minimum. Transition
rate is most commonly studied in chemical reactions, for example, from molecule A to molecule
B, including biochemical reactions. Non-equilibrium dynamics is concerned about the transition
rate of a Brownian particle to overcome a potential barrier. This problem is called the Kramers
problem [33]. One way to approach this problem is to use the mean first-passage time (MFPT),
which is defined as the average time elapsed until a stochastic process starting at some point xo
leaves a prescribed domain of the state space for the first time [16].
Solving the Fokker-Planck equation gives the MFPT r as
T(X) = k+]
b
eu(IkBT
dz e-Utz)IkBT
(4.1)
where a is a reflecting point and b is an absorbing point in the potential U(x) [16]. Kramers problem
is solved using Eq 4.1.
The absorbing barrier b is Xmax where U(xmax) is the maximum of the potential. The reflecting
barrier a is set to be -oo.
kBT is assumed to be small, which implies a high barrier limit. In this
condition, the integral over z is dominated by the potential near the minimum, whose expression
is obtained by quadratic expansion as U(z)
=
Umin + 1w 2 .(z
-
xmin) 2 . Likewise, the integral
over y is dominated by the potential near the maximum with quadratic expansion as U(y)
Umax - 2
2
mnax(Y - Xmax)
[33]. Calculating Eq 4.1 with these approximations gives the expression
for T. Then the transition rate or the Kramers escape rate k is just a reciprocal of
k = r-
=
WminWmax
7rm-Y
I
T,
which is [20]
Umax - Umin oc exp[-AU/kBT]
kBT
I
(4.2)
It is notable that k from Eq 4.2 reproduces the Arrhenius equation, k oc exp[-AE/kBT] [1]
where AE is the activation energy, or the energy barrier, which is Umax - Umin in our case.
26
Parameter Dependence of Effective Energy Barrier AG
4.2
By Eq 3.1, the probability distribution is expected to be exp[-Veff(x)/kBT. From Figure 3.4,
it is also exp[-3ajxj] where
3
is 1/kBT.
a is the slope of the linear segment of Veff(x), so
a = AG/Ax. In other words, Veff(x) = (AG/Ax)x. Using this relation, the dependence of AG
on various parameters of the system, which are
f,
Ag, L, and Ax, is investigated by measuring the
slope of Veff(x).
4.2.1
Dependence on the Fraction of Time during which the Ratchet is Turned
Off
When the ratchet is turned off, a particle undergoes free diffusion, which is much faster than
the movement of the particle interfered by a potential. Therefore, the longer the potential is turned
off (bigger
the farther the particle reaches. A broader distribution of a particle is interpreted
f),
f,
as lower AG. To verify the relation between AG and
simulations are performed for
0.900, 0.925, 0.950 and 0.975. Other parameters are set as Ag
(a)
18
(b)
f=0.875
X f =0.90
f = 0.925
* f = 0.95
0 f = 0.975
16-
0.2, L
f
0.7 and Ax
= 0.875,
3.7.
--
2.5
-+
2-
01.5
0
62
12-
_ _ _ _ _ _ _ _ _ --0.5-
___fi
-20
-0.5-
100-
-15
-10
0.02
-5
0.04
0.06
0.08
0.1
0.12
0.14
1-f
Particle Position
Figure 4.2: (a) The particle distribution for various fractions of time of turning off the ratchet
f. Dashed lines are the logarithmic scale of the particle distribution and the solid lines are linear
the
envelopes of the distributions. (b) The relation between #AG/Ax and 1 - f. Dashed line is
bound.
confidence
95%
linear regression of the slopes of the envelopes in (a). Error bars indicate
Figure 4.2 (a) is superposition of the particle distributions shown in Figure 3.4 (b) under various
values of
f.
As expected, a higher value of
f
yields a broader distribution or weaker localization,
27
implying lowering of the effective potential barrier. Figure 4.2 (b) shows the linear relation between
AG and
of
f
(1 -
f.
In the plot, the variable is (1 -
f)
instead of
f.
Reminding ourselves that the function
we are looking for is going to be an exponent of the probability distribution convinces us that
f)
is a better variable since f = 1, which means the potential is turned off all the time, gives
us uniform distribution without biasedness. In other words,
conclude that AG oc (1 -
f
= 1 should mean AG = 0. So we
f).
However, extrapolating the linear fitting in Figure 4.2 (b) gives AG = 0 before f =1.
implies different physics at
f
~ 1, which is a regime close to free diffusion. For now, we take the
asymptotic linear relation between AG and (1 -
4.2.2
This
f).
Dependence on the Ratchet Height
It is uncertain whether AG would depend on Ag because a particle travels mostly during the
time the potential is turned off. To see the possible relation, simulations are performed for Ag
0.15, 0.2, 0.25, 0.3, 0.35, and 0.4. Other parameters are set as f = 0.95, L = 0.7 and Ax = 3.7.
18
16
14
0
X
EO =
E =
EO =
-+
*EO =
0 EO =
EO =
(b) 2
0.15
02
025
0.3
0.35
0,4
1.8 --
1.6
-
(a)
1.4S12
1.2
CL
10
E
o. 0.8
-ii
0.6
6
0.4
4
2
0.2
-14
-12
-10
-8
-6
Particle Position
-4
0
0.1
-2
0.2
0.3
0.4
0.5
Ag
Figure 4.3: (a) The particle distribution under various ratchet heights Ag. Dashed lines show the
logarithmic scale of the distribution and the solid lines show their linear envelopes. (b) The relation
between 3AG/Ax and Ag. Dashed line is the linear regression of the slopes of the envelopes in
(a). Error bars indicate 95% confidence bound.
Figure 4.3 shows that a bigger value of Ag gives a steeper slope of the envelope. This implies a
bigger value of AG and thus stronger localization. As shown in Figure 4.3 (b), AG depends linearly
28
on Ag. However, as in the case for
f dependence,
extrapolation of the linear fitting gives AG = 0
before Ag falls to zero, suggesting the possible entrance to the new regime. For a regime where a
ratchet effect is still robust, it is valid to say AG oc Ag.
4.2.3
Dependence on the Ratchet Asymmetry
One of the conditions necessary for the ratchet effect is asymmetry in a ratchet. As shown in
Figure 1.1, if L = 1 - L, the probability for a particle to land on a trap on either side becomes
identical, and the biasedness in its motion disappears. One can speculate that the more asymmetric
the ratchet is, the more robust the directional motion is. More robust ratchet effect results in
stronger localization. So this implies the possible correlation between AG and L. The simulations
are performed for L
=
(a) 18
0.6, 0.65, 0.7, 0.75, and 0.8. Other parameters are set as f
0.95, Ag
=
3.7.
0
(b)
L =0.60
1-X L = 0.65
+ L0.9
* L = 0.75
14-
=
1
09
-
0.2, and Ax
=
0.8-
0 L = 0.80
0.7
12
CL
10
V/
%/
0.6
e
6
- 0.5
n
;0.3
S0.4
4
-20
0.2
21
-15
-10
Partide Position
0.11
0.1
-5
0.2
0.3
0.4
0.6
0.5
0.7
2L-1
Figure 4.4: (a) The particle distribution under various ratchet asymmetry parameters L. Dashed
lines show the logarithmic scale of the distribution and the solid lines show their linear envelopes.
(b) The relation between slope of the envelopes and 2L - 1. The dot-dashed line is a quadratic
fitting and the dashed line is a linear fitting. Error bars indicate 95% confidence bound.
Figure 4.4 shows that bigger values of L gives a steeper slope of the envelopes. This implies
that the more asymmetric the ratchet is, the stronger the localization is. The variable on the x-axis
is set to be 2L - 1 rather than L so that the biasedness in the motion vanishes at L = 0.5, which is
the symmetric case, by the same argument why the proper variable was 1 case for the time fraction dependence of AG.
29
f
instead of
f
in the
Figure 4.4 (b) suggests either fAG oc (2L - 1)2 or /AG oc (2L - 1). The data points themselves
seem that quadratic relation is better. However, collecting all the parameters together in the later
section suggests the other option. Therefore, the data points for L close to 0.5, which gives free
diffusion, are thought to be in the regime between free diffusion and a ratchet effect suggested for
other parameters. The entrance to the regime happens earlier than other parameters, given that
asymmetry is a necessary condition for a ratchet effect.
There are two earlier studies supporting the linear relation in our regime of study.
First,
for a steep ratchet, the force to stop the movement of a particle in the reverse direction to the
ratchet effect is F8 tp = (Qy/ft)/(L - 1/2) [2] [22] where t is the duration time. Hence the net
force causing the directional motion is F8 tp oc (2L - 1). F8 tp comes from the effective potential
of the system by Ftop = -xVeff(x).
So Veff(x) oc (2L - 1) but then Veff(x) oc AG and thus
AG oc (2L - 1). Second, for a two-state pulsating ratchet with low switching rate, velocity of a
particle v = (L - (1 - L))/2Tir 2 [9] where -r and
T2
are duration times for each potential of the two
states. In an overdamped condition like our system, force F oc v, which means that F 0c (2L - 1).
The same argument as before concludes that AG cc (2L - 1).
4.2.4
Dependence on the Distance between Kinks
When the distance between the two kinks is long, the transition between them is expected to be
more difficult simply because a particle has to travel farther. Then, the effective potential barrier
is larger. In order to verify the hypothesis, simulations are performed under various half-distances
between kinks Ax = 1.7, 2.7, 3.7, 4.7, and 5.7. Other parameters are set as
f
= 0.95, Ag = 0.2
and L = 0.7.
Figure 4.5 (a) shows that the envelopes of particle distributions are almost parallel to each
other. The data from a single kink ratchet is included in the plots as Ax = 0 case for comparison,
and its slope is parallel, too. Figure 4.5 (b) suggests independence of the slopes of the envelopes
from Ax.
Here again, the data points of double kink ratchets have similar values as a single
kink ratchet, which confirms that the slopes are indeed independent of Ax. The slopes indicate
OAG/Ax. Therefore, we conclude that AG oc Ax.
30
(a) 16
(b) 0.44
V dx = 0.0
0 dx =1.7
X
15 1
14
dx = 2.7
0.43
I
+| dx = 3.7
dx = 4.7t0.2
0 dx = 5-7
04
X
-
13-
0
012-
1 1
0.41
0.4-
10.39
10
0.38
9
0.37
0o.36~
8
-15
-10
0
0
-5
Figure 4.5: (a) The particle distribution
mic scale of the distribution while solid
3AG/Ax and Ax. Error bars indicate
single kink ratchet is also plotted as Ax
4.3
4
2
6
AX
Particle Position
under various values of Ax. Dashed lines are the logarithlines are their linear envelopes. (b) The relation between
95% confidence bound. For comparison, the data from a
= 0 case.
Effective Diffusion Coefficient
Before collecting all the parameters for AG, we should consider the effective temperature of the
system because, as Eq 4.2 and our discussion on the logarithmic slope of particle distributions show,
AG accompanies kBT. Diffusion coefficient D is proportional to temperature T by the Einstein
relation D = kBT-y [12]. Likewise, Teff oc Deff where the subscript "eff" means "effective". In
the presence of a ratchet, Deff is expected to be different from D, the free diffusion coefficient, in
a sense that the potential landscape is "rough" due to the sawtooth of the ratchet compared to the
complete flat one for free diffusion. In order to verify the effect of a ratchet on diffusion, Deff is
measured through simulations according to the following equation [29].
D=
1
1
lim - < [x(t)
2 t-+oo t
-
-
x(O)] 2 >
Figure 4.6 shows the effective diffusion coefficient Deff under different values of Ag and
(4.3)
f.
Def f
is measured in a symmetric ratchet (L = 0.5) not to consider the drift due to the ratchet effect.
Since the roughness of the potenital landscape is what affects diffusion, Deff does not depend on
the asymmetry parameter L.
31
(a)
(b)
14
I
0.9
-
12
0.80
0.7-
0
)
0.8
0.6-
0
0.6
0.5-
0.40.2
0.86
0.40.88
0.9
0.92
0.94
0.96
0.1
0.98
0.0033.4
0.2
0.3
0.4
0.
0.5
fAg
Figure 4.6: Effective diffusion coefficients relative to the free diffusion coefficient for under different
values of (a) f and (b) Ag. Error bars indicate one standard deviation obtained by error propagation
[6].
Figure 4.6 shows that Def f depends linearly on
f
and Ag. The dependence makes sense because
the movement of a particle would be disturbed more as the potential persists longer (f decreases)
and as the ratchet becomes higher (Ag increases). So the presence of a ratchet indeed decreases
Deff. The value of Deff/D for f = 0.975 in Figure 4.6 (a) is larger than 1, but this data point
has a large error bar, which is presumably caused by that the behavior at this point is effectly free
diffusion.
So Deff ID ~1 at this point. However, Deff
4 D for most of the conditions in the
presence of a ratchet.
In the later section, we collect all the parameters and fit the data into the Kramers reaction
theory model. While the Kramers theory deals with quasi-equilibrium dynamics, our system is in
non-equilibrium.
Therefore, temperature is modified from the "true temperature" T in
effective temperature Teff = TDeff /D.
#
to the
It turns out to help fitting our data into the equilibrium
picture of Kramers.
4.4
Mean First Passage Time
Collecting all the parameter dependences gives the relation AG oc (1 - f)Ag(2L - 1)Ax. By
Eq 4.2, mean first passage time (MFPT)
T
oc exp[AG/kBTeff]. Therefore, fitting the model for
AG in directly measured MFPT is expected to confirm our model.
Figure 4.7 is a trajectory of a particle under a double kink system. Distributions like Figure 3.4
(a) are histograms of trajectories like this. The trajectory shows that a particle stays at one kink-
32
I
IIIII
I
I
15
10
0
Lit
5
LLKLi~L
7
77
0
a-
-5
L~6 hkiI~LI
7~7~-
V WL
f-,
I
-10
-15
-20
0
1
2
6
5
Time
4
3
7
10
9
8
5
x10
Figure 4.7: The time series of the location of a particle for Ag = 0.2, Ax = 3.7, f = 0.95, and
L = 0.7. The horizontal solid line marks the kink locations or the minimum points of Veff while
the horizontal dashed lines mark the boundary of the kink region.
trap for certain amount time and then transfers to the other trap. Therefore, taking the average
of the duration time the particle stays at one trap gives MFPT.
(a)
(b) 16
70
o
Parameter: Ag
* Parameter: 1-f
g Parameter: 2L-1
X Parameter: Ax
15
60
14
--4
50
13
'40
12
a,
-
30
20
1
9C
X
10
U
--
10
5
Time Interval
M
I
-
M
0
0
15
0.02
0.04
0.06
0.08
0.1
0.12
(1-f)(Ag)(2L-1)(Ax)/Teff
,104
Figure 4.8: (a) The distribution of time intervals for Ag = 0.2, L = 0.7, f = 0.95 and Ax = 3.7.
(b) Linearly fitting MFPT on the collective variable (1- f)Ag(2L - 1)Ax/Teff. Error bars indicate
one standard deviation.
Figure 4.8 (a) is the distribution of time intervals during which a particle stays in one kink
before moving to another kink. As shown with the dashed line, the histogram of the intervals is
well fitted in an exponential distribution. MFPT is obtained by taking the mean of the distribution.
Figure 4.8 (b) shows the relation between MFPT and all the parameters discussed before. Here
the collective variable (1 - f)Ag(2L - 1)Ax is divided by the relative effective diffusion coefficients
33
Deff /D because by Eq 4.2, the exponent of r is not just AG but AG/kBTeff as discussed in the
previous section. By Einstein's relation [12], Teff O( Deff.
The linear fit in Figure 4.8 (b) is reasonably well fitted in the "middle" and "upper" region.
The data is not as well fitted in the small (1 - f)Ag(2L - 1)Ax region, where the constraint due to
ratchets is weak enough that the particle movement is close to free diffusion. From the divergence
of the data points, it is suspected that this is a separate regime that requires another further study.
Especially, Ax dependence in this regime needs reexamination.
The error bars in Figure 4.8 (b) are obtained by the standard deviation of mean and the error
propagation.
The standard deviation of a mean is gmean = o/V(N) [6] where o is standard
deviations of the interval distribution from Figure 4.8 (a) and N is the number of samples. Since
the y-axis is the logarithm of r , the uncertainty in ln(T) is gin(,) =
01,/
[6].
Overall, logarithmic scale of MFPT has a linear relation with the collective variables. So this
establishes an analogous expression for the Kramers escape rate k by taking reciprocal of MFPT
T
k
=T
k cx exp[-AG/kBTeff] = exp[-Go(1 - f)(2L - 1)AxAg/kBTeff]
where Go is a constant factor for AG.
34
(4.4)
Chapter 5
Stochastic Resonance and
Delocalization by Resonant Activation
5.1
Stochastic Resonance
So far the system is subject to periodic turning on and off of the ratchet potential. It becomes
curious what happens if another periodic external force, which is tilting in this case, is applied to
the system. A careful choice of the tilting frequency is expected to realize stochastic resonance.
Stochastic resonance is a phenomenon that a signal is amplified by noise in the system. It was
originally introduced by Benzi to study the climage change of periodic coming of ice age [5]. The
climate was modeled as a two-well potential and one of the wells represented ice age. Short-term
climate fluctuations are Gaussian white noise and if they are synchronized with weak purturbations
the Earth is subject to, stochastic resonance occurs and a giant change in the climate comes. This
study showed that the coupling between an internal stochastic mechanism and an external periodic
forcing causes the large amplitude, long-term alternations in temperature and the absense of either
fails to reproduce the phenomenon.
In a more general setting, if a particle is under a double-well potential that is connected to a
heat bath, it is constantly subject to random fluctuations. The fluctuations cause a particle in one
well to jump over an energy barrier and move to the other well, and its transition rate follows the
Kramers rate from Eq 4.2. Until now, the system is the same as our previous systems.
35
Then a
[N
ago
#10
*go
Figure 5.1: The schematic of the stochastic resonance. A particle is in a bistable potential connected
to a heat bath. A suitable amount of noise in the system causes the particle to hop to a global
minimum. Adapted from Gammaitoni 1998 [15]
weak periodic force is applied.
yv = -V'(x) + Aosin(wt) + (t)
(5.1)
Eq 5.1 describes the overdamped system with periodic tilting where W is a tilting frequency and
AO is a tilting amplitude [27] [15]. With this periodic force, the once symmetric bistable potential
becomes asymmetric.
As shown in Figure 5.1, the energy barrier between two wells raises and
lowers periodically in an antisymmetric manner
[15].
In fact, the lowering of the energy barrier due to tilting itself is too small for a particle to
overcome the barrier because we are assuming the force in Eq 5.1 to be weak. However, noiseinduced transtion described by Eq 4.2 can be synchronized with this force. The synchronization
happens when the time-scale matching condition for stochastic resonance [15] [24] is satisfied as
T, = 2T
36
(5.2)
where T is MFPT from Eq 4.2 for noise-induced transtion and T, is the period of forcing, which is
obtained by T, = 2ir/w.
5.2
Stochastic Resonance in Kinked Ratchets
1400
V
1200
1000-
800
-
0
0
600-
400
-
200
0
2
4
6
Time
8
12
10
x104
Figure 5.2: The trajectory of a particle in a double kink ratchet with tilting with frequency w =r/T.
The parameters are Ag = 0.2, L = 0.7, f = 0.95, Ax = 3.7 and AO = O.4Ag/L. The horizontal
lines near x = 0 indicate the location of kinks.
and
Adding tilting to the system according to Eq 5.1 gives a combination of a pulsating ratchet
observe
a rocking ratchet. Especially, tilting with frequency w according to Eq 5.2 enables us to
on a tilted
stochastic resonance in kinked ratchet potentials. Figure 5.2 is a trajectory of a particle
gigantic
double kink ratchet potential with w = 7r/T. The trajectory looks like a sine function with
at
amplitude. The amplitude is so big that the location of kinks seems not to affect the trajectory
all. In fact, the particle behaves as if a ratchet does not exist. The ignorance of the ratchet suggests
that the system is undergoing stochastic resonance. The input tilting amplitude was 0.4 Ag/L =
without
0.11, which is small enough that a particle would be still trapped in a ratchet minimum
noise. However, coupling the tilting with thermal noise results in amplification of the output signal,
which means that the system behaves as if huge tilting is applied. The effective tilting is so large
force
that the underlying ratchet potential becomes relatively flat and the particle does not feel the
is indeed
by the ratchet. Inspecting the period of the sinusoidal signal in Figure 5.2 confirms that it
the external tilting that we applied. Therefore, we conclude that the coupling between thermal
noise in the system and the signal, or tilting we applied, has given rise to stochastic resonance.
37
Resonant Activation
5.3
Having a periodic perturbation in the system can not only amplify the input perturbation
through stochastic resonance but also increase the Kramers transition rate. Doering and Gadoua
named the latter as resonant activation [10]. It is not to be confused with stochastic resonance.
While stochastic resonance is concerned with the amplification of input signal by interference with
the noise in the system, resonant activation is about the increase in the Kramers escape rate.
In this scheme, the height of energy barrier fluctuates between E
+ AE and E - AE. A careful
choice of the fluctuation frequency brings about "resonance" between the fluctuation and the noiseinduced transtion, leading to decrease in MFPT. The exact analytical expression for MFPT is very
demanding to get, so it has been studied with approximations [10] [7] [31].The exact expression for
MFPT will enable us to get the "resonance frequency" wr which induces the resonant activation
with minimum MFPT.
Double Kink
5.4
The effective potential for a double kink ratchet is W-shaped or bistable. Tilting the ratchet
raises and lowers the effective energy barrier AG in an antisymmetric manner as in Figure 5.1,
and a particle is in one of the potential wells. In the perspective of the particle, therefore, the
barrier AG is fluctuating, which is the setting for resonant activation. For various values of tilting
frequency w, MFPT is measured to see how the transition rate of a particle depends on it.
12000
--
10000-
-
8000
E-
uL 600040002000
____
104
_
103
__
__10-2
101'
100
Tilting Frequency
Figure 5.3: Finding the tilting frequency that induces rapid transition. Simulations were performed
for Ag = 0.2, L = 0.7, Ax = 3.7, f = 0.95 and AO = 0.4 Ag/L. Error bars indicate one standard
deviation.
38
Figure 5.3 shows MFPT for various tilting frequencies w. The value of MFPT decreases and
increases again as it passes a certain value of w of order 10-2. MFPT at w = 0.01 is 399.57 which is
significantly smaller than MFPT for the same condition without tilting, which is r = 1.9418 x 104.
Such huge decrease in MFPT signifies that the resonance frequency w, is of order 10-2.
30
-20
10
-30
0
2
4
6
lime
8
10
12
x104
Figure 5.4: The particle trajectory in the presence of tilting. Tilting frequency w = 0.01 and tilting
amplitude Ao = 0.4 Ag/L. Other parameters are set as Ag = 0.2, L = 0.7, f = 0.95, and Ax =
3.7.
Rapid transition between the kinks implies that delocalization of a particle in a kinked ratchet
has been realized. As shown in Figure 5.4, a particle rapidly moves between the kinks. It is not
confined to one trap but rather delocalized between them. Tilting the system with different w far
from the resonance frequency does not delocalize the particle. Therefore, it shows the tunability of
localization of a particle in kinked ratchet systems.
5.5
Periodic Kink Lattice
In a periodic kink lattice, resonant activation induces delocalization of a particle not only around
the region between two kinks but at everywhere. Figure 5.5 shows the comparison between the
particle distribution with resonant activation for a double kink ratchet and a periodic kink lattice.
Figure 5.5 (a) is the distribution for a periodic kink lattice for one period. Since there are
multiple kinks in the system, the particle is completely delocalized. The probability for a particle
to exist at the local minima of a ratchet is still higher than other places, which is demonstrated
by the peaks at the local ratchet minima.
However, the particle is delocalized in a sense that
kinks are no longer the location for the highest peak in the distribution. All the minima in the
39
ratchet are equally stable points. Delocalization is more obvious when compared to the distribution
without tilting which is drawn with a solid line on the same plot. It is more apparent in Figure 5.5
(b) which shows five periods of the kink lattice.
Indeed, due to delocalization, the particle is
effectively uniformly distributed throughout the five periods. Note that it is almost impossible to
distinguish kinks from other minima on the ratchet without the dashed lines indicating them. It is
contrasting with the distribution without resonant activation drawn with solid lines, which shows
localization of the particle at kinks. On the other hand, Figure 5.5 (c) shows that a particle in a
double kink ratchet is mostly around the kinks and the region between the kinks. Although the
particle occasionally goes farther than the kink region, possibly due to its ballistic-like motion, it
moves between kinks for the most of the time. So the overall shape of the distribution looks like a
Gaussian distribution. It is contrasting to the distribution in the absence of tilting which is highly
peaked at kinks. Comparison between Figure 5.5 (a), (b) and Figure 5.5 (c) shows the difference in
the extent of delocalization. Overall, Figure 5.5 suggests the tunability of delocalization in kinked
ratchet systems.
40
(a) 2
6
x 10
1
0
0
UJ
Q)
1
2
4
5
6
7
8
20
25
30
35
40
10
0
10
Particle Position
20
30
40
3
C3
~ (b) 3 x 10
I
a..
t+--
0
2
E
1
.....
Q)
..c
::I
z
0
0
5
10
15
(c) 6 x 10
4
2
oL__
_.___.-.aiitm•
-40
30
20
Figure 5.5: (a)The particle distribution in a periodic kink lattice with resonant activation for ~g
= 0.2, L = 0.7, f = 0.95, ~x = 3.7, w = 0.01 and Ao = 0.4 ~g/ L. It is showing one period.
(b) The particle distribution in a periodic kink lattice with the same condition for five periods.
(c) The particle distribution in a double kink ratchet for the same condition. Dashed lines in each
figure indicate the location of kinks. Solid lines are the distributions for the same condition without
tilting.
41
42
Chapter 6
Conclusion and Future work
In conclusion, localization, escape rate and delocalization in kinked ratchets have been discussed.
When the direction of a ratchet potential reverses, the point of reversal, which is called a kink,
becomes the most probable region for a particle to exist because of the ratchet effect. Therefore,
localization of a partcle is achieved. All the three kinds of kinked ratchets, a single kink ratchet,
a double kink ratchet and a periodic kink lattice, exhibit localization. The particle distribution
shows that the effective potential at kinks is V-shaped. In the case of a double kink ratchet, the
dependence of escape rate k from one kink to the other on various parameters, Ag, L,
studied and the analogous Kramers rate turns out to be k oc exp[-Go(1 -
f)(2L -
f
and Ax, is
1)AxAg/kBTeff].
Then, tilting the kinked ratchet with frequency w = 7rk gives rise to stochastic resonance. In
this regime, the originally very weak tilting amplitude is magnified by coupling with the thermal
noise in the system. The output "signal" is so large that the underlying ratchet is ignored.
In both a double kink ratchet and a periodic kink lattice, delocalization of a particle around
kinks is observed for the resonant activation regime. Because kinks are placed periodically in a
kink lattice, it is more apparent that the preference on kinks disappears and a particle becomes
equally likely to exist at all minima in the ratchet.
There still remain unsolved questions. First of all, our model of the analogous Kramers rate
does not work well for small (1 -
f)(2L
- 1)AxAg, which is a regime close to free diffusion. It
seems that the transition between free diffusion and a robust ratchet effect does not occur smoothly
with our model. Therefore, further study on this regime is needed to understand the whole system.
Second, we do not have an exact analytical expression for MFPT when tilting is added. The order
43
of the resonance frequency
wr
was obtained by simulations, but derivation of the expression will give
us the exact value of wr for the minimum MFPT. Finally, it will be worth exploring the ignorance
of ratchets in the stochastic resonance regime in other conditions.
Our study has demonstrated that placing kinks on a ratchet potential localizes particles in the
system. However, additionally tilting the potential causes stochastic resonance, resonant activation
and consequential delocalization. It suggests tunability of localization and thus more control over
the system. Experimental demonstration of this work is needed but is expected to be feasible.
Kinked ratchet potentials will open more possibilities on applications and studies on the systems
with ratchet potentials.
44
Bibliography
[1] S. Arrhenius.
Uber die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch
Sduren. Z. Phys. Chem., 4:226-248, 1889.
[2] R. Astumian. Thermodynamics and kinetics of a brownian motor. Science, 276:917-922, 1997.
[3] R. Astumian and I. Der6nyi. A chemically reversible brownian motor: Application to kinesin
and ncd. Biophys. J., 77:993-1002, 1999.
[4] J. Bader et al. DNA transport by a micromachined brownian ratchet device. Proc. Natl. Acad.
Sci. U.S.A., 96:13165-13169, 1999.
[5] R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani. Stochastic resonance in climatic changes.
Tellus, 34:10-16, 1982.
[6] P. Bevington and D. Robinson. Data Reduction and Error Analysis for the Physical Sciences.
McGraw-Hill, 2003.
[7] M. Bier and R. Astumian. Matching a diffusive and a kinetic approach for escape over a
fluctuating barrier. Phys. Rev. Lett., 71:1649-1652, 1993.
[8] S. Block. Leading the procession: New insights into kinesin motors. J. Cell. Biol., 140:1281-
1284, 1998.
[9] J. Chauwin, A. Ajdari, and J. Prost. Current reversal in asymmetric pumping.
Europhys.
Lett., 32:373-378, 1995.
[10] C. Doering and J. Gadoua. Resonant activation over a fluctuating barrier. Phys. Rev. Lett.,
69:2318-2321, 1992.
[11] C. Doering, W. Horsthemke, and J. Riordan. Nonequilibrium fluctuation-induced transport.
Phys. Rev. Lett., 72:2984-2987, 1994.
[12] A. Einstein. Uber die von der molekularkinetischen Theorie der Wdrme geforderte Bewegung
von in ruhenden Fliissigkeiten suspendierten Teilchen. Ann. Phys., 17:549-560, 1905.
[13] F. Falo, P. Martinez, J. Mazo, and S. Cilla. Ratchet potential for fluxons in Josephson-junction
arrays. Europhys. Lett., 45:700-706, 1999.
[14] R. Feynman, R. Leighton, and M. Sands. The Feynman Lectures on Physics. Addison-Wesley,
1966.
[15] L. Gammaitoni, P. Hdnggi, P. Jung, and F. Marchesoni.
Phys., 70:223-287, 1998.
45
Stochastic resonance.
Rev. Mod.
[16] P. Hinggi and P. Talkner. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys.,
62:251-341, 1990.
[17] T. Harms and R. Lipowsky. Driven ratchets with disordered tracks. Phys. Rev. Lett., 79:2895-
2898, 1997.
[18] U. Henningsen and M. Schliwa. Reversal in the direction of movement of a molecular motor.
Nature, 389:93-96, 1997.
[19] S. Hulme, W. DiLuzio, S. Shevkoplyas, L. Turner, M. Mayer, H. Berg, and G. Whitesides.
Using ratchets and sorters to fractionate motile cells of Escherichia coli by length. Lab Chip,
8:1888-1895, 2008.
[20] H. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions.
Physica, 7:284-304, 1940.
[21] P. Langevin. Sur la thdorie du mouvement brownien. Comptes Rendues, 146:530-533, 1908.
[22] D. Liu, D. Astumian, and T. Tsong. Activation of Na+ and K+ pumping modes of (Na,K)ATPase by an oscillating electric field. J. Biol. Chem, 265:7260-7267, 1990.
[23] M. Magnasco. Forced thermal ratchets. Phys. Rev. Lett., 71:1477-1481, 1993.
[24] B. McNamara and K. Wiesenfeld. Theory of stochastic resonance. Phys. Rev. A, 39:4854-4869,
1989.
[25] J. Prost, J. Chauwin, L. Peliti, and A. Ajdari. Asymmetric pumping of particles. Phys. Rev.
Lett., 72:2652-2655, 1994.
[26] E. Purcell. Life at low reynolds number, 1976.
[27] P. Reimann. Brownian motors: noisy transport far from equilibrium. Phys. Rep., 361:57-265,
2002.
[28] P. Reimann, M. Grifoni, and P. Hsinggi. Quantum ratchets. Phys. Rev. Lett., 79:10-13, 1997.
[29] H. Risken. The Fokker-Planck Equation. Springer, 1996.
[30] U. Seifert. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep.
Prog. Phys., 75, 2012.
[31] C. Van den Broeck. Simple stochastic model for resonant activation. Phys. Rev. E, 47:4579-
4580, 1993.
[32] I. Zapata, R. Bartussek, F. Sols, and P. Hdnggi. Voltage rectification by a SQUID ratchet.
Phys. Rev. Lett., 77:2292-2295, 1996.
[33] R. Zwanzig. Nonequilibrium Statistical Mechanics. Oxford Universiy Press, 2001.
46
Appendix A
MATLAB code
A.1
Main Simulation Code
% Ratchet potential random walk
%% Parameters and Variables
t = 0.0;
dt = 0.01;
tmax = 1000000;
E0 = 0.2; % barrier height of the ratchet
LO = 0.7; % adjust slope
deltax = 3 + LO; % distance from the origin to the "minimum"
x = 0.0; % position of the particle
v = 0.0; % velocity of the particle
i = 0;
timeseries = zeros (1,tmax/dt);
timestep = zeros (1,tmax/dt);
passage = [J;
turnOff = 0.95; % ratio of the time the potential is turned off
turningPeriod = 500;
where = 0;
AO = 0.4*EO/LO;
w = 0.01;
%w = pi/(1.9285*10^4); % tilting freq.
%tau = 1.9285*10^4 for EO = 0.2, LO = 0.7, turnOff = 0.95, deltax
=
3.7;
%% Simulation
while t < tmax
i = i + 1;
if mod((100*t),turningPeriod) < turnOff*turningPeriod
% potential turn off. Free diffusion. Flip according to the flip
% general case
+ randn;
%
dv = -v
%
%
assume gamma = 1, 2*ganuma*kB*T = 1 for
v =v + dv*dt;
47
simplicity
rate
% overdamped case without
tilting
v = randn;
% overdamped case with tilting
v = randn + AO*sin(w*t);
x = x
+ v*dt;
timeseries(i) = x;
.
else % potential turn on. Force present
% general case
dv = -v + ratchet-force(x,EO,LO, deltax) + randn;
%
v =v + dv*dt;
%
% overdamped case without tilting
%
v = ratchet-force (x,EO,LO, deltax)
randn
+
% overdamped case with tilting
v = ratchet-force (x,EO,LO,deltax)
x = x + v*dt;
timeseries(i) = x;
+ randn;
+ A*sin(w*t );
end
t = t + dt;
timestep(i)
=t
if x <= (deltax+LO)
&& x > (deltax-LO)
&& where
-=
1
where = 1;
passage = cat (2, passage , timestep ( i ));
elseif x <= (-deltax+LO) && x > (-deltax-LO) && where ~= -1
where = -1;
passage = cat (2, passage , timestep ( i
));
end
end
%% Time Series Plot
timeseries-runltrunc = timeseries (1:5:end);
timestep-runl-trunc = timestep (1:5:end);
figure ()
hold on
plot (timestep-runltrunc , timeseries-runl-trunc , 'k')
hlinel = refline (0, deltax);
set(hlinel, 'Color', 'r','LineWidth',2)
hLinell = refline(0, deltax-LO);
set (hlinell , 'Color', 'b' , 'LineWidth' ,2)
hline12 = refline(0, deltax+LO);
set (hline12 , 'Color', 'b' , 'LineWidth' ,2)
hline2 = refline (0, -deltax);
48
set (hline2 , 'Color ' , 'r ' , 'LineWidth' ,2)
hline2l = refline (0, -deltax+LO);
set (hline2l , 'Color ' , 'b' , 'LineWidth' ,2)
hline22 = refline(0, -deltax-LO);
set (hline22 , 'Color' , 'b' , 'LineWidth' ,2)
xlabel ('Time')
ylabel ('Particle Position ')
hold off
%% Particle
[ counts ,
Distribution
centers]
,
figure ()
hold on
bar(centers , counts, 'w')
'r--',
plot ([ deltax , deltax ] , ylim
plot ([-deltax,-deltax] , ylim , 'r--'
xlabel ('Particle Position ')
ylabel('Number of Particles
hold off
%
%
%
%
%
%
200);
= hist (timeseries
'LineWidth', 2)
, 'LineWidth',2)
figure ()
semilogy (centers ,counts , 'LineWidth' ,2)
hold on
xlabel ('Position of Particles ')
ylabel ('Number of Particles ')
hold off
%% Mean First Passage Time (Tau)
t-interval = zeros(1,length(passage)-1);
for j = 1:length(passage)-1
t-interval(j) = passage(j+1)-passage(j);
end
[tcounts,
tcenters]
,30);
= hist(t-interval
% histogram of time intervals
%MFPT is mean( t-interval).
% uncertainty obtained by std (tinterval)/
figure ()
hold on
bar(tcenters , tcounts , 'b')
xlabel ('Time Interval ')
ylabel ('Frequency')
hold off
49
sqrt (length ( t-interval))
A.2
Ratchet Potentials
A.2.1
Single Kink
%% Force from one-kink ratchet
function f = one-kink-ratchet-force(x,EO,LO)
if abs(x)-floor(abs(x)) < LO
if x > 0.0
f = -EO/LO;
else
f = EO/LO;
end
else
if x > 0.0
f = EO/(1.0-LO);
else
f = -EO/(1.0-LO);
end
end
end
A.2.2
Double Kink
%% Force from the Ratchet Potential
function f = ratchet -force (x,EO, LO, deltax)
if x>0.0 && x <= deltax
if x-floor(x) < LO
f = EO/LO;
else
f = -EO/(1.0-LO);
end
elseif
x <= -deltax
if (abs(x)-deltax)-floor (abs(x)-deltax) < LO
f = EO/LO;
else
f = -EO/(1.0-LO);
end
elseif x<=0.O && x>-deltax
if abs(x) - floor(abs(x)) < LO;
f = -EO/LO;
else
f = EO/(1.0-LO);
end
else
if (x-deltax)-floor (x-deltax) < LO
f = -EO/LO;
else
f = EO/(1.0-LO);
end
50
end
end
A.2.3
Periodic Kink Lattice
%% Force of periodic kink lattice
function f = kink-latticeforce(x,EO,LO,deltax)
x = mod(x,2*deltax);
if x < deltax
if x-floor(x) < LO
f = EO/LO;
else
f = -EO/(1-LO);
end
else
if (x-deltax)-floor (x-deltax) < LO
f = -EO/LO;
else
f = EO/(1-LO);
end
end
51