Computer simulation of Feynman’s ratchet and pawl system
Jianzhou Zheng, Xiao Zheng, GuanHua Chen
Department of Chemistry, University of Hong Kong, Hong Kong, China
(Dated: September 22, 2009)
In this work, we introduce two different designs of ratchet and pawl systems. Molecular dynamics
simulations for both particle driven and Langevin force driven systems have been carry out to
examine the detailed behavior of these systems. We find out that the ratchet will rotate as designed
when the temperature of the pawl chamber is lower than the ratchet side. Several parameters and
configurations are tested. The results show that there is a maximum efficiency when optimum load
is applied. The correlation between ratchet and environment of particle driven system is much
lower than that of Langevin force driven system. After comparing results between different designs
of ratchet, we find out that the efficiency of the ratchet is greatly depended on the design.
I.
INTRODUCTION
Maxwell demon was introduced by James Clerk
Maxwell in this book, Theory of Heat, 1871 [1, 2] The basic setting is that, there are two chambers which divided
by a board with a hole. Maxwell demon is an intelligent
living that guards at the hole, which only let hot particles go to left and let cool particles go to right, otherwise
the demon will shut the door.
In this design, after a long run, hot particles will be in
left chamber as well as cool particles in right, which results in a temperature difference between these two chambers.
According to the second law, there should not be such a
demon which sit around the hole in equilibrating system
and work without any energy consumption. However,
when the system is out of equilibrium, the demon of such
kind should work in some special conditions.
There have been many debates on whether such a demon exists and whether it would function as designed.
Most recently, with the advent of nanomachinery devices, the prospect of testing fundamental hypotheses of
thermodynamics and realizing Maxwell’s demons at the
nanoscale has been contemplated [3–5].
In 1912 Smoluchowski pointed out that thermal fluctuations would prevent any automatic device from operating successfully as a Maxwell demon in a lecture entitled “Experimentally Verifiable Molecular Phenomena
that Contradict Ordinary Thermodynamics.” [9] Most of
investigations on Maxwell’s demon have been thought experiments. Despite intense, lasting interest on Maxwell’s
demons and incessant quest for them as documented
in the literature, only a few microscopic simulations
have been carried out to examine the Maxwell’s demons.
Zhang and Zhang [6] formulated a set of sufficient conditions for the survival of a Maxwell’s demon: (1) a device
can be used to generate and sustain a robust momentum
flow inside an isolated system; (2) a system with an invariant phase volume is capable to support such a flow.
[∗] Email: [email protected]
However, they provided no realistic models and simulations. In 1992 a trap-door device was studied for the first
time via numerical simulation by Skordos and Zurek [7]
showing that the trap door, acting only as a pump, cannot extract useful work from the thermal motion of the
molecules. Skordos later [8] analyzed a membrane system and stated that the second law of thermodynamics
requires the incompressibility of microscopic dynamics or
an appropriate energy cost for compressible microscopic
dynamics.
In 1962, Richard Feynman introduced a ratchet and
pawl system about an apparent perpetual motion machine of second kind [10]. The system consists of two
chambers. In one chamber, an asymmetric ratchet rotates freely in one direction but is prevented by a pawl
from rotating in the other direction, and it is immersed
in a bath of molecules at temperature T1 . In the other
chamber, there is a paddle wheel which also immersed in
a bath of molecules with temperature T2 . The ratchet
and the paddle wheel is connected by a massless and
frictionless rod. In this condition both the ratchet and
the pad undergo random Brownian motions with a mean
kinetic energy that are determined by the temperature.
Since the pawl only allows motion in one direction,
when a molecule collides with a paddle, it only impulses
a torque to the ratchet in the expected direction. After many random collisions, the net effect should be the
ratchet rotating continuously in the expected direction.
In this case, the motion of a ratchet can be used to
do work on other systems (The expected work extracted
from this device should be very small, but there are still
a small amount of net work). The energy to do this work
is extracted from the heat bath.
This is contradicted from the second law of thermodynamics, which states that:
It is impossible for any device that operates on a cycle
to receive heat from a single reservoir and produce a net
amount of work.
In the lecture, Feynman demonstrated that it should
not work. Because the weight of the ratchet and pawl
should be small enough to response by an individual collision from a single molecule. In that scale, the pawl
should also in a thermal equilibrium. In another word,
the pawl will have the same temperature as the environ-
2
ment. As a result, the pawl itself will vibrate randomly
and fail to operate as designed.
After that, many researchers have derived some simplified model to test this system. Jarzynski and Mazonka [11] introduced a simple realization of Faynman’s
ratchet-and-pawl system, and indicated that their model
can act both as a heat engine and as a refrigerator. However, there is no molecular dynamics result to support
their conclusion. Meurs and his coworkers [12] have mentioned several models to show rectifications of thermal
fluctuations, but only analytical results were presented.
Other previous attempts include employing the overdamped Fokker-Planck equations with stochastic variables to simulate Feynman’s ratchet and pawl [13] and
modeling asymmetric Brownian particles immersed in
thermal baths made of gaseous molecules that are treated
explicitly [14, 15]. The former one overly simplifies the
gas-particle dynamics, and the latter one has no welldefined Maxwell’s demon. Velasco and his coworkers
[16] carefully considered the efficiency of the Feynman’s
ratchet, but the model is too simple to represent Feynman’s system and there is no simulation result to support
them. Schneider and his coworkers [17] have simulated
Brownian motion of interacting particles in a periodic
potential and driven by an external force. However, in
their simulation, the Brownian particles are driven by
Langevin force and their system is quite different from
Feynman’s settings.
In our previous work [18], we have introduced a simplified trap-door based Maxwell demon, and molecular
dynamics calculations have been carried out to evaluate
detailed behavior of such system. Results show that, no
temperature differentiation can happen when the trap
door and the chamber particles are in thermal equilibrium. As well as when the trap door is cooled down by
placing it into a chamber with low temperature, as expected, the trap-door creates a temperature and density
gradient between the two chambers of the box it divides,
and the cooled trap door creates more readily a density
gradient than that of temperature.
After dramatic improvement of nano technology ,
bio-fabrication techniques make the construction of
structures with features smaller than 10 nm become
possible[19–22]. However, the construction of functional
nanoelectromechanical systems (NEMS) is hindered by
the inability to provide motive forces to power NEMS devices. Researchers are using biomolecular motors to do
their construction. For example, kinesin [25, 26], myosin
[28], RNA polymerase [27], and adenosine triphosphate
(ATP) synthase [23, 24, 29, 30] function as nanoscale
biological motors. As we know, in nano scale region,
thermal fluctuation plays a very important role. However, in most of these cases, thermal fluctuation is often
neglected. That will greatly impede the improvement
of nano technology. The molecular dynamics simulation
of devices in nano scale will give us an intuitive picture
of the thermal fluctuation. Since Feynman’s ratchet and
pawl system is well studied and there is no realistic molec-
FIG. 1: An illustration of the ratchet and pawl system. Left:
Top view. Right: Side view. Both chambers are filled with
gas particles, a spring attached inside one end of each pawl,
the ratchet in lower chamber can rotate freely
ular dynamics simulation in literature, We set up a simplified Feynman’s system to do molecular dynamics simulation. Benefited from the improvement of the computer
technology, the realistic molecular dynamics simulations
of such system can be carry out to evaluate their statistical behavior.
There are some research results in literature. Marcelo
[35] have considered a Brownian particle in a periodic potential under heavy damping, and get analytical results
by using Fokker Planck Equation. They found out that
the Brownian particle can not get any net drift speed,
even if the symmetry of the potential is broken; And if
applied an external force with time correlations, the particle can exhibit a nonzero net drift speed. Magnascol
and Stolovitzky [36] later presented an analytical result
to test Feynman’s ratchet and pawl system. However, in
order to get analytical solution of Fokker Planck equation, they worked in overdampped region. Astumian
[37] and Bier also did similar analyze on heavily damped
Brownian particle in a periodic potential and found out
that even the net force is zero, there is a net flow. Risken
and Voigtlaender have solved the Fokker-Planck equation for the extremely underdamped Brownian motion in
a double-well potential[31]. Schneider and his coworkers
[17] have simulated Brownian motion of non-interacting
particles in a periodic potential and driven by and external force. The Fokker-Planck equation is solved, but the
potential applied to the Brownian particle is too simple
and it is quite different from Feynman’s system.
Up to now, many researchers did similar simulations
in over damped region. However, there is no work which
considers underdamped region and has both ratchet and
pawl setup. In our work, two models which are very
close to Feynman’s ratchet and pawl are built for the
simulation.
3
II.
FIRST MODEL
The system we have simulated contains two identical
chambers which filled with random distributed gas particles. Several pawls are placed in upper chamber and a
ratchet consist of a simple stick and another perpendicular joint stick. The angle of this two sticks are fixed. As
showed in Figure 1. The gas molecules are hard dishes
moving inside a two-dimensional space and colliding with
each other. All collisions of gas molecules are energy- and
momentum-conserving. After particle-wall collisions, the
particle momentum reverses its component perpendicular to the wall. Geometrically, the pawl, the ratchet and
the perpendicular stick are line segments of zero width
impenetrable by the molecules (the mass of the perpendicular stick is zero). A spring with a force constant Kφ
is placed in one end of pawl. In the equilibrium position,
the pawl angle is φ0 .
The ratchet is a free rotor. It will only change its angular velocity during collision. The change of angular
velocity of ratchet is calculated in Appendix A. The Lagrangian of the pawl can be written as
N
L=
1X
1
1
mgas vi2 + Ipawl φ̇2 − Kφ (φ − φ0 )2 − Vint (1)
2 i=1
2
2
where vi is the velocity magnitude of the ith gas particle,
φ (φ0 ) is the pawl angle (the equilibrium pawl angle),
φ̇ is angular velocity, Kφ is the elastic constant of the
pawl spring, mgas and Ipawl are the mass of the gaseous
particles and the momentum of inertia of the pawl, respectively, and Vint is the interaction potential energy
among all rigid bodies in the system, which is constant
except when particle-pawl and ratchet-pawl collisions occur. The third term on the right hand side of equation describes the interaction between the spring and the pawl.
Equations of motion for the particle velocities and the
pawl angle φ are then derived from the Lagrangian. For
instance, in the absence of collisions, the pawl angle φ(t)
follows
"s
#
Kφ
φ(t) = φ0 + [φ(ti ) − φ0 ] cos
(t − ti )
Ipawl
s
"s
#
Ipawl
Kφ
+ φ̇(ti )
sin
(t − ti )
(2)
Kφ
Ipawl
for t > ti , where ti denotes the starting time.
As showed in Figure 24, where θ is the rotation of
the ratchet, φ (φ0 ) is the angle of pawl (the equilibrium
angle). Kφ is the force constant. Details are explained
in Appendix A.
A Fortran code is written for this simulation. Figure 2 is the flowchart of the program. The whole program consists of four major subroutines (read parameters, initialize, main loop and finalize). ”read parameter”
is for reading the simulation conditions and initial parameters. ”initialize” is for initializing the gas particles with
FIG. 2: Flowchart of the program. Left: Structure of main
program. Right: Detailed structure of main loop procedure
Maxwell distributed velocities. ”finalize” is used for saving simulation parameters in order to resume interrupted
trajectories. The detail of main loop is show in the right
part of Figure 2. After entering main loop, the forces, acceleration, velocities and coordinates are updated with a
given time step. Then, by comparing with the last step, a
subroutine called ”find collision” is called to search colliding events occurred in current step. If there are collisions,
all updated information are restored to previous step and
the program enters the ”small time step updating loop”.
In the ”small time step updating loop” (in the right middle part of Figure 2), firstly, the program updating all
valuables with small time step, the ”find collision” subroutine is called and it records all collisions happened in
this step. finally, the ”treat collision” subroutine is called
subsequently to treat the recorded colliding events. And
after the ”small time step updating loop”, the program
goes back to main loop and updates with original time
steps.
In the simulation, a set of parameters is chosen as follows: the number of particles in each chamber N = 8,
the number of pawls NP = 6, the radius of each chamber R = 3.5, the equilibrium ratchet angle φ0 = 0.8,
the mass of the gas molecules in lower chamber mgas1 =
20 , the mass ofpthe gas molecules in upper chamber
mgas2 = mgas1 × T2 /T1 (mass of particle in pawl chamber should be scaled down to have similar velocities as
particles in ratchet chamber), the length of the ratchet
RB = 2.0, the length of the pawl RL = 0.8, the mass of
the ratchet mB = 20, the mass of the pawl mL = 10, the
distance between the fixed point of pawl and the center
of chamber RP = 2.4, the temperature of ratchet chamber T1 = 1000, and the force constant Kφ = 3500. Our
simulation takes typically several hours using a standard
Pentium 4 processor, and the fluctuation in the total energy of the system is kept below 10−10 .
The gas molecules are given random initial positions,
and their velocities are initialized according to the Boltzmann distribution as discussed in our previous work. The
velocity Verlet algorithm is use in these simulations. The
4
time evolution of the system of the molecules and the
ratchet is simulated by adopting the following algorithm:
to minimize numerical errors, adaptable time steps are
used. There are two kinds of time steps in this simulation.
Our program iterates a main cycle which starts from t0
to the end of simulation with a time step ∆t = 0.0001.
Within each cycle, the velocities and positions of all particles and the trap door are updated assuming no collision
takes place, a subroutine is then called to examine collision events that may have been overlooked. Four types
of collision events are involved: particle-particle collisions, particle-wall collisions, particle-ratchet collisions
and particle-pawl collisions. Information of the previous step is recorded so that we can restore a current
state to the one that precedes it. The collision criteria
are: 1) particle-particle collisions: the distance between
two particles is less than the diameter of a particle; 2)
particle-ratchet collisions(particle-pawl collisions is the
same): the distance between particle and ratchet is less
than the radius of a particle; 3) ratchet-pawl collisions:
the rotating end of ratchet cross the pawl;
We take a particle-particle collision as an example. At
the end of each time step, the center-of-mass distance
between two particles is measured to determine whether
a collision has taken place. If there is no collision, simulation carries on to the next time step. Otherwise, collisions need to be accounted for. In this case, the simulation is restored to the recorded previous state, and
the time step is reduced to ∆t0 = 10−3 dt. After the
restoration, the program performs 103 steps using ∆t0 .
It means one regular step is divided into 103 steps in
a collision event. After the collision, the simulation returns to the main loop with the original time step ∆t.
If a three-body collision is to occur, our program records
all the relevant information and pauses. However, such
a three-body collision event has not been encountered in
our system which is composed of 16 particles with a time
step ∆t0 as small as 10−3 dt.
We have carefully considered various collision events
including particle-particle collisions, particle-wall collisions, particle-ratchet collisions, particle-pawl collisions
and ratchet-pawl collisions. Details on how these collision events are handled are explained in Appendix A.
For a particle-wall collision, the normal component of the
particle’s velocity reverses its direction while the tangential counterpart remains unchanged.
Firstly, we set no temperature difference between the
ratchet and pawl chamber. Figure 3 shows 1000 individual runs of this model. In these trajectories. Some
of them rotate clockwise as well as some others rotate
counter clockwise.
In a individual trajectory, there are two visual processes. First, the rotor get enough energy from hot particles, and cross the barrier of bending potential of the
angle φ. Second, the vibrating pawl hit the rotor and
the rotor bounce back. These two processes are all dominated in our simulation.
Figure 4 shows the averaged results. The solid line is
FIG. 3: 1000 Individual runs of the model.
FIG. 4: Averaged rotation of the ratchet in one chamber system.
the net rotation of the rotor, and the dashed line represents the standard deviation of these trajectories. After
averaged over 1000 individual trajectories, the net rotation of the ratchet is zero.
When applying lower temperature to pawl’s chamber
(T2 ). As we expected, the averaged result of the rotor
rotates clockwise as showed in Figure 5.
In individual trajectory of this case, there are also two
visual processes. First, the rotor get enough energy from
hot particles, and cross the barrier of bending potential
of the angle φ. Second, the leg of the rotor hit the pawl
and bounce back. After applying lower temperature to
the pawl’s chamber, the vibrations of pawl are greatly
depressed by colliding with cool particles. Therefore the
first process becomes dominant, and the ratchet goes to
designed direction.
After fitting with a line, we can calculate the angular
velocity of the ratchet. Figure 5 shows three different
5
FIG. 5: Rotation of ratchet with different temperature configurations.
FIG. 7: Efficiency of the ratchet and pawl system when applying different load
FIG. 6: Angular velocity of ratchet with different temperature
configurations.
FIG. 8: Effect of different number of gas particle
results of the ratchet with different T1 /T2 . The angular
velocity become larger when higher temperature difference applied.
After testing several combinations of T1 /T2 , as showed
in Figure 6, when T1 /T2 goes to 1, the ratchet has no
net rotation. Angular velocities of the ratchet increased
when T1 /T2 become larger. When T1 /T2 reaches 103 ,
the effect of the temperature differences become smaller,
hence the curve goes flat in higher difference region.
To test the efficiency of the ratchet, temperature difference T1 /T2 = 103 is used in later simulations. We apply
a constant torque against the desired rotating direction
as a load applied to the ratchet. The heat flow from the
ratchet chamber to pawl chamber is collected by summarising the energy change of ratchet (or pawl) on each
ratchet-pawl collision.
Figure 7 shows the efficiencies of the system when
different loads are applied. The result shows that the
maximum efficiency is 0.262% when the load is equal to
1.5 × 10−4 . When the load exceeds 3.1 × 10−4 , the rotor
rotates to opposite direction, hence, the efficiency goes
negative.
The number of particles included in each chamber is
considered. Figure 8 shows that when large number of
gas particles is used, the angular velocity of the ratchet
will decrease. It is because that the rotation will be heavily blocked by gas particles when high density of particles
is applied.
After introducing and simulating this simplified
ratchet and pawl system, a clear picture of such system is showed. When there is no temperature difference between ratchet and pawl chamber, no net rotation
6
rotate around the fixed point with a spring attached. The
equilibrium position of the pawl is φ0 . There are uncorrelated Gaussian type random force on both the ratchet
and the pawl.
The related Langevin equation can be written as,
ω̇θ = −γωθ − fθ (θ, φ, θ̇, φ̇) + Γθ (t)
θ̇ = ωθ
ω̇φ = −γωφ − fφ (θ, φ, θ̇, φ̇) + Γφ (t)
φ̇ = ωφ
FIG. 9: An illustration of the simplified Feynman’s ratchet
and pawl system for Fokker Planck equation
(4)
Here, the random forces acted on θ and φ are uncorrelated and both are distributed in Gaussian type, we
have
2γθ kTθ
δ(t − t0 )
Iθ
2γφ kTφ
δ(t − t0 )
< Γφ (t)Γφ (t0 ) > =
Iφ
< Γθ (t)Γφ (t0 ) > = 0
< Γθ (t)Γθ (t0 ) > =
will be observed. After apply lower temperature to pawl
chamber, the vibrations of pawls are greatly depressed,
hence the ratchet rotates as expected. Different temperature configurations are tested, we found out that ,
in lower temperature difference region (T1 /T2 < 103 ),
higher temperature difference will result in higher rotating speed, and when temperature difference which is
larger than 103 , higher temperature difference will not
affect the angular velocity. The efficiency of the system
has been tested. The maximum efficiency is 0.262% when
the external load is equals to 1.5 × 10−4 . External load
which exceeds 3.1 × 10−4 will result in opposite rotation
direction of ratchet.
III.
LANGEVIN SOLUTION
Langevin equation is commonly used to solve the motion of Brownian particles in a potential. It is a stochastic
differential equation. The general form of the Langevin
equation for Brownian motion is as follows,
dv
= −γv + F (x) + Γ(t)
dt
(3)
Here, v is velocity, x is position, F (x) is the force
acted on the particle, and Γ is a Langevin force which
is a stochastic term and satisfies < Γ(t) >= 0 and
< Γ(t)|Γ(t0 ) >= qδ(t − t0 ) . Here, q is correlation constant.
To solve this equation, one may count many individual
propagation via time t, or introduce a distribution density function and solve the related Fokker Planck Equation.
Figure 9 is an illustration of the simplified model of
Feynman’s ratchet and pawl system. The ratchet and
pawl are both non-penetrable straight sticks with zero
width and certain mass, mθ and mφ . The ratchet can
rotate freely around the center as well as the pawl can
(5)
k is Boltzmann constant, Tθ and Tφ are the temperature of the ratchet and pawl chamber, γθ and γφ are
friction coefficients, and Iθ and Iφ are the momentum
inertia of the ratchet and pawl respectively.
The related Fokker-Planck Equation can be written as
follows[38],
∂
W (θ, φ, θ̇, φ̇, t) =
∂t
(
∂
∂
kT ∂ 2
∂
−
θ̇ +
(fθ + γθ θ̇) + γθ
−
φ̇
∂θ
Iθ ∂ θ̇2
∂φ
∂ θ̇
)
∂
kT ∂ 2
+ (fφ + γφ φ̇) + γφ
W (θ, φ, θ̇, φ̇, t) (6)
Iθ ∂ θ̇2
∂ φ̇
Here, fθ = fθ (θ, φ, θ̇, φ̇) and fφ = fφ (θ, φ, θ̇, φ̇) are
forces acted on ratchet and pawl. There are sharp values
in these functions when ratchet collides to pawl. ( Here,
fθ = δ θ̇Iθ and fφ = δ φ̇Iφ − kφ (φ − φ0 ) .) Detailed definition of δ θ̇ and δ φ̇ are in Appendix A. Equation 4.4 is
a four variable Fokker time dependent Planck Equation.
There is no analytical solution [38]. To numerically solve
the equation and determine the distribution function, one
way is to use finite grid space, guess an initial distribution density function and solve it by iteration. But for a
four variable Fokker Planck equation, This is very time
consuming. A easier way is to statistically solve the related Langevin Equation [38] (calculate a large number
of individual trajectories), and calculate the distribution
function of the system.
The setting in this simulation are similar to the particle
driven model. The major different between them is that
the particles in chambers are replaced by Langevin force.
7
FIG. 10: Net rotation in different T1 and T2 .
Compare to the particle driven model, the correlation between environment and ratchet (or pawl) is much
higher. The Langevin force is continuously acted on the
ratchet (or pawl), while in previous case, the gas particle
only collide on ratchet (or pawl) occasionally. Therefore,
the mass of the ratchet and pawl is tuned to much smaller
in order to have similar angular velocities. In our case,
the mass of the ratchet and pawl are 100 times smaller.
In this simulation, to achieve similar results, a set of
parameters is chosen as follows: the number of pawls
NP = 6, the radius of each chamber R = 3.5, the equilibrium ratchet angle φ0 = 0.8, RB = 2.0, the length of
the pawl RL = 0.8, the mass of the ratchet mB = 0.2, the
mass of the pawl mL = 0.1, the temperature of ratchet
chamber T1 = 109 , the distance between the fixed point
of pawl and the center of chamber RP = 2.4, the force
constant Kφ = 3500, and the friction constant of the
ratchet and pawl are γθ = 0.5 , γφ = 5.
In order to have comparable results, the integrating
time step and total simulation time is set to ( dt = 0.0001
and 500 ). Another Fortran code is written to perform
this simulation. There are only ratchet-pawl collision.
The structure of the program is similar to the previous
one. The Langevin force is applied to the system on all
major time steps. In later results, if not pointed out, 1000
individual trajectories are counted to determine averaged
properties.
Firstly, we test the condition with no temperature difference T1 = T2 = 1000. The result shows no different as
in particle driven system, As the dashed line showed in
Figure 10. No net rotation is observed.
When the temperature of the two chamber is set to
T1 = 1000, T2 = 1. The ratcheting effect occurs, the
solid line in Figure 10 shows that there is net rotation
when lower pawl temperature is applied.
To observe the distribution density function of the system, we have counted the distribution of angular velocity of the ratchet (θ̇). Figure 11 is the distribution of
θ̇. As it shows, the central point of the distribution is
FIG. 11: Distribution of θ̇
FIG. 12: Angular velocity with different length of ratchet
in −0.02, the average angular velocity of ratchet is negative. Therefore, the ratchet rotates to one direction as
expected.
When tuning the parameter, different length of ratchet
( other parameters are fixed ) is tested. Intuitively thinking, there should be no ratcheting effect when the ratchet
cannot reach the pawl which is in equilibrium position.
That means, in our configuration of the device, the length
of ratchet should larger than 1.843 in order to collide with
the pawl. However, we found out that even the ratchet
is shorter than 1.843, net rotation occurs. Figure 12
shows the angular velocity calculated in different length
of ratchet. It is clear that, the ratchet rotates to desired
direction when the length is longer than 1.75. The reason is that the pawls are vibrating during the simulation,
and there is a distribution of angle of the pawl. Figure
13 is the angle distribution of the pawl in our simulation. It shows that the pawl have visible distribution in
0.9 > φ > 0.7. This result shows that there may still
be ratcheting effect even when the length of ratchet is
shorter than expected.
8
FIG. 13: Angle distribution of the pawl
FIG. 15: Angular velocity and different friction coefficient
FIG. 14: Angular velocity in different temperature difference
FIG. 16: Efficiency when applying external load
The relation between angular velocity and different
T1 /T2 is also tested. Results is showed in Figure 14.
It is similar to the one in particle driven model. When
the temperature difference larger than T1 /T2 = 103 , The
curve goes flat.
Different friction coefficients of ratchet (γθ ) have been
tested. Result is showed in Figure 15. The angular
velocity increase when friction coefficient become larger.
But when the friction coefficient is larger than 2.0, the
curve become flat.
Friction coefficient (γ) is a property which describe
the correlation between the object and the environment.
Larger γ means larger Langevin force but also larger friction. As showed in Langevin equation, the Langevin force
is in the order of square root of γ as well as the friction force is in the order of γ. Therefore, in low damping region (lower γ region), Langevin force is dominant.
The angular velocity of ratchet increases as γ increase.
However, in large γ region (over damping region), both
Langevin force and friction are dominant. The driving
force (Langevin force) was partially cancelled by friction.
Hence, the curve became flat.
The efficiency of this system is also tested for comparison. Result is showed in Figure 16. There is a peak
efficiency ( 0.152%) when load is equals to 5.5 × 10−4 .
In this part, we derived the Fokker Planck equation of
the ratchet and pawl system. Because there is no analytical solution, we solve the problem by statistically counting a large number of individual trajectories of Langevin
equation. We get similar results between this Langevin
force driven system and previous particle driven system, by using higher temperature and smaller mass of
the ratchet and pawl. The ratchet rotates only when
lower pawl temperature is applied. Distribution of angular velocity of ratchet is observed to understand how the
ratchet rotates. After testing different configurations of
parameters, we found that the ratchet can rotates to desired direction even when the length of ratchet is shorter
than expected, and it is because the angle of pawl is distributed in certain range. Different friction coefficients
are also tested. In low damping region, larger friction coefficient can results in larger angular velocity, and in large
9
EOMs for θ and φ are
"
mL RL RB
sin φ 2
θ̈ =
sin φ φ̇ θ̇ +
φ̇
Iθ
2
(8)
#
I2 sin φ 2 cos φ Kφ
Kφ
+
θ̇ +
(φ − φ0 ) +
(φ − φ0 )
2IL
2IL
Iθ
Ã
!
mL RL RB sin φ
2
2
φ̈ = −
I1 θ̇ + I2 φ̇ + 2I2 φ̇ θ̇
(9)
2IL Iθ
−
I1
Kφ (φ − φ0 ).
IL Iθ
Here, the moments of inertia are
FIG. 17: An illustration of the ratchet and pawl system. Left:
Top view. Right: Side view.
IB =
IL =
damping region, friction coefficient can only slightly affect the angular velocity of the ratchet. Efficiency of this
Langevin force driven system has been tested. The result
is similar to the particle driven system. There is peak efficiency ( 0.152% ) when load is equals to 5.5 × 10−4 .
IV.
SECOND MODEL
There are many different designs of ratchet and pawl
systems. In order to compare the properties of those
different designs. In this part, we introduce a different
design of ratcheting system. In this design, we built the
asymmetry in the rotating part. There are two components in the rotor (branch and leg). The pawls are only
in-penetrable points.
The system contains two identical circular chambers,
placed with several pawls and a ratchet showed in Figure
17. The gas molecules are hard dishes moving inside a
two-dimensional space and colliding with each other. All
collisions of gas molecules are energy- and momentumconserving. After particle-wall collisions, the particle
momentum reverses its component perpendicular to the
wall. The rotor consisted of a branch, a leg and a rod.
The branch and the leg are connected by the perpendicular rod which constrains the two joint points of branch
and leg to be in same positions in their chamber. Geometrically, the branch, the leg and the rod are line segments of zero width impenetrable by the molecules, and
the rod between these two chamber is of zero mass. A
spring with a force constant Kφ is placed in the rod between the branch and the leg. In equilibrium state, the
branch-leg angle is φ0 .
The potential energy of the ratchet is
P E(ratchet) =
1
Kφ (φ − φe )2 .
2
(7)
I1 =
I2 =
Iθ =
1
2
mB RB
3
1
2
mL RL
3
2
+ mL RL RB cosφ
IB + IL + mL RB
1
IL + mL RL RB cosφ
2
I1 IL − I22
.
IL
(10)
The detailed derivation is showed in Appendix B.
In the simulation, a set of parameters is chosen as follows: the number of particles in each chamber N = 8, the
radius of each chamber r = 3.5, the equilibrium branchleg angle φ0 = 0.8, the length of the branch RB = 2.0,
the length of the leg RL = 0.8 the length between the
pawl and centre point RP = 2.6, the mass of the branch
mB = 20, the mass of the leg mL = 30, the mass of
gas particle in branch chamber mgas1 = 50, the
p mass
of gas particle in leg chamber mgas2 = mgas1 × T2 /T1
(mass of particle in leg chamber should be scaled down
to have similar velocities as particles in branch chamber),
the size of gas particle rgas = 0.1, time step dt = 0.0001,
temperature of branch chamber T1 = 1000, and the force
constant Kφ = 3500.
The simulation is performed by another program of
similar structure. In this program, collision events are include particle-particle, particle-ratchet, particle-leg, and
leg-pawl collisions. All special cases of collisions are considered. (Special cases: The gas particle is colliding with
the joint point of the branch and leg, or colliding with the
tail of the leg) The gas molecules are given random initial
positions, and their velocities are initialized according to
the Boltzmann distribution. The time evolution algorithm of the system is the same as in previous part.
As our previous results, the ratchet has no net rotation
when applying same temperature to both branch and leg
chambers, as well as it rotates when lower temperature in
leg chamber is applied. As showed in Figure 18. When
T1 /T2 = 1000, the ratchet rotates clockwise as expected.
Different temperature configurations are tested. As
showed in Figure 19, It is similar as the one in Feynman’s
10
FIG. 18: Rotation of ratchet on different temperature applied
FIG. 20: Efficiency of ratchet when applying different load
FIG. 19: Angular velocities on different temperature configurations
FIG. 21: Rotation of ratchet on different temperature applied
setting. When T1 /T2 > 1000, the ratchet rotates at similar speed even when higher temperature differences are
applied.
After applying different load to the ratchet, as showed
in Figure 20, there is a maximum efficiency (1.12×10−5 )
when the load is equals to 0.0008. This maximum efficiency is much lower than the one in Feynman’s setting.
The related Langevin equation can be written as,
ω̇θ = −γωθ − fθ (θ, φ, θ̇, φ̇) + Γθ (t)
θ̇ = ωθ
ω̇φ = −γωφ − fφ (θ, φ, θ̇, φ̇) + Γφ (t)
φ̇ = ωφ
(11)
Where the random forces acted on θ and φ. Here,
fθ = fθ (θ, φ, θ̇, φ̇) and fφ = fφ (θ, φ, θ̇, φ̇) are interacting
force between branch and leg. There are sharp values in
these functions when ratchet collide to pawl. Detailed
equations of these forces ( with collision and without collision ) are in Appendix B.
The parameters used to do this simulation are similar
as particle driven system except that the mass of the
branch and leg are tuned to have comparable angular
velocities ( mB = 0.2, mL = 0.3 ) temperature of branch
chamber is set to T1 = 107 , and the friction coefficients
of branch and leg are set to 0.5 and 5.
The ratchet has no net rotation when applying same
temperature to both branch and leg chambers, as well as
it rotates clockwise when lower temperature is applied to
leg chamber. As showed in Figure 21. When T1 /T2 =
1000, the ratchet rotates to desired direction.
Different temperature configurations are tested. As
showed in Figure 22, It has similar behaviour as previous setting. When T1 /T2 > 1000, the ratchet rotates
11
FIG. 24: For both θ and φ, the most convenient choice for
range would be (−π, π).
FIG. 22: Angular velocities on different temperature configurations
(or friction) in leg chamber hinder the rotation of the
branch, and hence heat transfered directly from branch
to gas particles in leg’s chamber with no contribution to
asymmetric rotation of the ratchet.
V.
FIG. 23: Efficiency of ratchet when applying different load
clockwise at similar speed.
After applying different load to the ratchet, as showed
in Figure 23, when the load is equals to 4 × 10−4 , there
is a maximum efficiency (1.79 × 10−5 ).
In this part, we have simulated another design of
ratchet and pawl system, We have tested the rotating
behavior by introducing both particle driven system and
Langevin force driven system. The results from both
systems are consistent. The ratchet rotates to designed
direction when T2 is lower than T1 .
Efficiency of the system and different temperature configurations are tested. We found out that the efficiency
in this system is much lower then the one in Feynman’s
system. The reason is, in Feynman’s setting, heat transfers from T1 to T2 only when ratchet is colliding with
pawl. However, in this new design, when the rotor is
rotating, the branch drags the leg around and particles
SUMMARY AND DISCUSSION
In this work, we have presented two designs of ratchet
and pawl systems, and have performed molecular dynamics simulations for both particle driven and Langevin
force driven systems to test the detailed behavior of these
systems. We find out that the ratchet rotates as designed
when the temperature of the pawl chamber is lower than
the ratchet side. Several parameters and configurations
have been tested. The results show that there is a maximum efficiency when optimum load applied. The correlation between ratchet and environment of particle-driven
system is much lower than that of Langevin force-driven
system. After comparing results between different designs of ratchet, we find out that the efficiency of the
ratchet is greatly depended on the design of the system.
That may help us to design new nano devices in this kind.
Acknowledgments
This work was supported by the Hong Kong Research
Grants Council (HKU 7012/04P and HKU 7127/05E)
and the Committee on Research and Conference Grants
(CRCG) of University of Hong Kong.
12
APPENDIX A: DETAIL OF COLLISIONS FOR
FEYNMAN’S RATCHET AND PAWL SYSTEM
where ∆P is
X
Y
AIB θ̇ + BIL φ̇ + CmG ẋ + DmG ẏ
A2 IB + B 2 IL + C 2 mG + D2 mG
∂θ h
IB
∂φ h
IL
∂x h
mG
∂y h
.
mG
∆P = −2
1.
X =
Y =
Without Collision
The potential energy of the Ratchet and Pawl is due
to the spring attached to the pawl (leg).
A =
B =
1
P E(RP) = Kφ (φ − φe )2 ,
2
(A1)
where φe stands for the minimum position of the above
parabolic potential. The moments of inertia of the leg
with respect to point P and of the branch with respect
to point O are
1
2
mL RL
3
1
2
=
mB RB
,
3
C =
D =
Here h represents the distance between the particle and
the stick.
IL =
IB
h
∂θ h
∂φ h
C
D
(A3)
s
!
Ãr
!
Kφ
IL
Kφ
φ(t) = φe + φ(t0 )−φe cos
t +φ̇(t0 )
sin
t ,
IL
Kφ
IL
(A4)
where φ(t0 ) and φ̇(t0 ) are the initial displacement and
velocity for φ, respectively.
´
Ratchet hits Pawl
=
=
=
=
=
RP sin φ + RB sin(θ − φ)
RB cos(θ − φ)
RP cos φ − RB cos(θ − φ)
0
0.
4.
The exact solution for the EOM is
³
3.
(A2)
respectively. The equation of motion (EOM) for φ is as
follows,
IL φ̈ = −Kφ (φ − φe ).
(A6)
Ãr
h
∂θ h
∂x h
∂y h
B
(A7)
Gas hits Ratchet
=
=
=
=
=
x sin θ − y cos θ
x cos θ + y sin θ
sin θ
− cos θ
0,
(A8)
where x and y are Cartesian coordinates in the system
−−→
with OP as the positive x axis.
2.
General formulation for collision events between
particles and sticks
The change of velocities for the branch, the leg and the
gas molecules are denoted by δ θ̇, δ φ̇ and δ ẋ(δ ẏ), respectively. Upon a single collision event, they are determined
via the following relations,
δ θ̇
δ φ̇
δ ẋ
δ ẏ
=
=
=
=
A ∆P
B ∆P
C ∆P
D ∆P,
(A5)
5.
Gas hits the end of the Ratchet
This happens when the gas molecule has a finite radius
R.
p
h =
(x − RB cos θ)2 + (y − RB sin θ)2 − R
RB (x sin θ − y cos θ)
∂θ h =
R
x − RB cos θ
∂x h =
R
y − RB sin θ
∂y h =
R
B = 0,
(A9)
13
where x and y are Cartesian coordinates in the system
−−→
with OP as the positive x axis.
6.
h
∂φ h
∂x h
∂y h
A
Gas hits Leg
=
=
=
=
=
x sin φ − y cos φ
x cos φ + y sin φ
sin φ
− cos φ
0,
(A10)
where x and y are Cartesian coordinates in the system
−−→
with P O as the positive x axis.
7.
Gas hits the end of the Pawl
APPENDIX B: DETAIL OF COLLISIONS FOR
SECOND RATCHET AND PAWL SYSTEM
1.
This happens when the gas molecule has a finite radius
R.
p
(x − RL cos φ)2 + (y − RL sin φ)2 − R
h =
RL (x sin φ − y cos φ)
∂φ h =
R
x − RL cos φ
∂x h =
R
y − RL sin φ
∂y h =
R
A = 0,
(A11)
where x and y are Cartesian coordinates in the system
−−→
with P O as the positive x axis.
8.
FIG. 25: schematic representation of the ratchet and pawl.
Coordinate Transformation
Transformation between XY and X 0 Y 0 coordinate systems:
x
y
x0
y0
=
=
=
=
x0 cosθ − y 0 sinθ + RB cosθ
y 0 cosθ + x0 sinθ + RB sinθ
x cosθ + y sinθ − RB
y cosθ − x sinθ
(B1)
Velocity transformation between XY and X 0 Y 0 coordinate systems:
vx0 = vx cosθ + vy sinθ − x sinθ θ̇ + y cosθ θ̇
vy0 = −vx sinθ + vy cosθ − x cosθ θ̇ − y sinθ θ̇ (B2)
Coordinate transformation
Assume the geometric center of a gas molecule is at
−−→
(x, y) in the OX coordinate system, its coordinates in
−−→
−−→
OP and P O systems are (xOP , yOP ) and (xP O , yP O ),
respectively. (x, y) can be transformed into (xOP , yOP )
via
xOP
yOP
ẋOP
ẏOP
=
=
=
=
x cos θP
y cos θP
ẋ cos θP
ẏ cos θP
+ y sin θP
− x sin θP
+ ẏ sin θP
− ẋ sin θP .
=
=
=
=
RP − x cos θP − y sin θP
−y cos θP + x sin θP
−ẋ cos θP − ẏ sin θP
−ẏ cos θP + ẋ sin θP .
Without Collision
The potential energy of the ratchet is
P E(ratchet) =
1
Kφ (φ − φe )2 .
2
(B3)
Moments of inertia:
(A12)
IB =
IL =
(x, y) can be transformed into (xP O , yP O ) via
xP O
yP O
ẋP O
ẏP O
2.
I1 =
I2 =
(A13)
Iθ =
1
2
mB RB
3
1
2
mL RL
3
2
IB + IL + mL RB
+ mL RL RB cosφ
1
IL + mL RL RB cosφ
2
I1 IL − I22
.
IL
(B4)
14
EOMs for θ and φ:
"
mL RL RB
sin φ 2
θ̈ =
sin φ φ̇ θ̇ +
φ̇
Iθ
2
(B5)
˜
φ̇, respectively. Define
q
0
=
x0 2G + y 0 2G
RG
#
I2 sin φ 2 cos φ Kφ
Kφ
θ̇ +
(φ − φ0 ) +
(φ − φ0 )
2IL
2IL
Iθ
Ã
!
mL RL RB sin φ
2
2
φ̈ = −
I1 θ̇ + I2 φ̇ + 2I2 φ̇ θ̇
(B6)
2IL Iθ
+
I1
−
Kφ (φ − φ0 ).
IL Iθ
3.
0
0
cos φ (2IL − mL RL RG
)
mG RB RG
2IL Iθ (yG − RB sin θ)
0
0
0
mG RG
(I1 RG
− I2 RG
− I2 RB cos φ)
=
IL Iθ (yG − RB sin θ)
xG − RB cos θ
= −
.
(B11)
yG − RB sin θ
Dxθ =
Dxφ
Dxy
The change of velocity of the gas molecule along x direction is
When a gas molecule hits the Branch
δvx = ṽx − vx
Before collision the velocities are vn , vt , θ̇ and φ̇, re˜
spectively. After collision the velocities are ṽn , ṽt , θ̇ and
˜
φ̇, respectively. Define
q
2
RG =
x2G + yG
q
vG =
vn2 + vt2
mG RG
mG RG IL
= 2
Iθ
I2 − I1 IL
m G I2 R G
mG RG I2
=
.
=
Iθ IL
I1 IL − I22
=
−2mG
vx −
2 + I D 2 + 2I D D
2
I1 Dxθ
L xφ
2 xθ xφ + mG Dxy + mG
2 ×
I1 Dxθ θ̇ + IL Dxφ φ̇ + I2 Dxθ φ̇ + I2 Dxφ θ̇ + mG Dxy vy
.
2 + I D 2 + 2I D D
2
I1 Dxθ
L xφ
2 xθ xφ + mG Dxy + mG
(B12)
Therefore the velocities after the collision are
ṽx
ṽy
˜
θ̇
˜
φ̇
Cθ = −
Cφ
(B7)
The change of velocity of the gas molecule perpendicular
to the Branch is
δvn = ṽn − vn =
−2Iθ (vn − RG θ̇)
.
2
Iθ + mG RG
= δvn + vn
= vt
(B9)
Transformation between vn , vt and vx , vy is
vn
vt
ṽn
ṽt
vx
vy
ṽx
ṽy
4.
=
=
=
=
=
=
=
=
vy cos θ − vx sin θ
vy sin θ + vx cos θ
ṽy cos θ − ṽx sin θ
ṽy sin θ + ṽx cos θ
−vn sin θ + vt cos θ
vn cos θ + vt sin θ
−ṽn sin θ + ṽt cos θ
ṽn cos θ + ṽt sin θ.
= Dxφ δvx + φ̇.
(B13)
When the Leg hits the Pawl
Dxφ
Dxθ
0
2 0
0
2I2 RB cos φ − 2IB RG
− 2mL RB
RG − mL RL RB RG
cos φ
.
0
mL RL RB RG cos φ − 2IL RB cos φ
(B14)
Dθφ =
= Cθ δvn + θ̇
= Cφ δvn + φ̇.
= Dxθ δvx + θ̇
Since the pawl does not move, before and after the col˜ ˜
lision the velocities are θ̇, φ̇ and θ̇, φ̇, respectively. Define
Therefore the velocities after the collision are
ṽn
ṽt
˜
θ̇
˜
φ̇
5.
(B8)
= δvx + vx
= Dxy δvx + vy
=
The velocities after the collision are
2
(IL Dθφ
− I1 )θ̇ − 2(IL Dθφ + I2 )φ̇
˜
θ̇ =
2 + 2I D
I1 + IL Dθφ
2 θφ
2
2
)θ̇
)φ̇ − 2(I1 Dθφ + I2 Dθφ
(I1 − IL Dθφ
˜
. (B15)
φ̇ =
2
I1 + IL Dθφ + 2I2 Dθφ
(B10)
When a gas molecule hits the Leg
Before collision the velocities are vx , vy , θ̇ and φ̇, re˜
spectively. After collision the velocities are ṽx , ṽy , θ̇ and
When φ = π2 or
collision are
3π
2 ,
cos φ = 0, the velocities after the
˜
θ̇ = θ̇
˜
φ̇ = −φ̇ − 2θ̇.
(B16)
15
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