Computer simulation of Feynman s ratchet and pawl system

Transcription

1 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. [ ] ghc@everest.hku.hk 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 T 1. In the other chamber, there is a paddle wheel which also immersed in a bath of molecules with temperature T 2. 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 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 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 K φ is placed in one end of pawl. In the equilibrium position, the pawl angle is φ 0. 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 L = 1 2 N m gas vi I φ pawl K φ(φ φ 0 ) 2 V int (1) i=1 where v i is the velocity magnitude of the ith gas particle, φ (φ 0 ) is the pawl angle (the equilibrium pawl angle), φ is angular velocity, K φ is the elastic constant of the pawl spring, m gas and I pawl are the mass of the gaseous particles and the momentum of inertia of the pawl, respectively, and V int 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 φ(t) = φ 0 + [φ(t i ) φ 0 ] cos + φ(t i ) I pawl K φ sin [ [ K φ I pawl (t t i ) K φ I pawl (t t i ) ] ] (2) for t t i, where t i denotes the starting time. As showed in Figure 24, where θ is the rotation of the ratchet, φ (φ 0 ) is the angle of pawl (the equilibrium angle). K φ 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 N P = 6, the radius of each chamber R = 3.5, the equilibrium ratchet angle φ 0 = 0.8, the mass of the gas molecules in lower chamber m gas1 = 20, the mass of the gas molecules in upper chamber m gas2 = m gas1 T 2 /T 1 (mass of particle in pawl chamber should be scaled down to have similar velocities as particles in ratchet chamber), the length of the ratchet R B = 2.0, the length of the pawl R L = 0.8, the mass of the ratchet m B = 20, the mass of the pawl m L = 10, the distance between the fixed point of pawl and the center of chamber R P = 2.4, the temperature of ratchet chamber T 1 = 1000, and the force constant K φ = 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 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 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 t 0 to the end of simulation with a time step t = 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 t = 10 3 dt. After the restoration, the program performs 10 3 steps using t. It means one regular step is divided into 10 3 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 t 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 (T 2 ). 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 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 T 1 /T 2. The angular velocity become larger when higher temperature difference applied. After testing several combinations of T 1 /T 2, as showed in Figure 6, when T 1 /T 2 goes to 1, the ratchet has no net rotation. Angular velocities of the ratchet increased when T 1 /T 2 become larger. When T 1 /T 2 reaches 10 3, 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 T 1 /T 2 = 10 3 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 When the load exceeds , 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 φ 0. There are uncorrelated Gaussian type random force on both the ratchet and the pawl. The related Langevin equation can be written as, 6 ω θ = γω θ f θ (θ, φ, θ, φ) + Γ θ (t) θ = ω θ ω φ = γω φ f φ (θ, φ, θ, φ) + Γ φ (t) φ = ω φ (4) FIG. 9: An illustration of the simplified Feynman s ratchet and pawl system for Fokker Planck equation 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 (T 1 /T 2 < 10 3 ), higher temperature difference will result in higher rotating speed, and when temperature difference which is larger than 10 3, 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 External load which exceeds 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 dt = γv + F (x) + Γ(t) (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(t ) >= qδ(t t ). 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, m θ and m φ. The ratchet can rotate freely around the center as well as the pawl can Here, the random forces acted on θ and φ are uncorrelated and both are distributed in Gaussian type, we have < Γ θ (t)γ θ (t ) > = 2γ θkt θ I θ δ(t t ) < Γ φ (t)γ φ (t ) > = 2γ φkt φ I φ δ(t t ) < Γ θ (t)γ φ (t ) > = 0 (5) k is Boltzmann constant, T θ and T φ are the temperature of the ratchet and pawl chamber, γ θ and γ φ are friction coefficients, and I θ and I φ are the momentum inertia of the ratchet and pawl respectively. The related Fokker-Planck Equation can be written as follows[38], t W (θ, φ, θ, φ, t) = { θ θ + θ (f kt θ + γ θ θ) 2 + γθ I θ θ 2 φ φ } + φ (f kt φ + γ φ φ) 2 + γφ I θ θ W (θ, φ, θ, φ, t) (6) 2 Here, f θ = f θ (θ, φ, θ, φ) and f φ = f φ (θ, φ, θ, φ) are forces acted on ratchet and pawl. There are sharp values in these functions when ratchet collides to pawl. ( Here, f θ = δ θi θ and f φ = δ φi φ k φ (φ φ 0 ).) Detailed definition of δ θ and δ φ 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 7 FIG. 10: Net rotation in different T 1 and T 2. FIG. 11: Distribution of θ 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 N P = 6, the radius of each chamber R = 3.5, the equilibrium ratchet angle φ 0 = 0.8, R B = 2.0, the length of the pawl R L = 0.8, the mass of the ratchet m B = 0.2, the mass of the pawl m L = 0.1, the temperature of ratchet chamber T 1 = 10 9, the distance between the fixed point of pawl and the center of chamber R P = 2.4, the force constant K φ = 3500, and the friction constant of the ratchet and pawl are γ θ = 0.5, γ φ = 5. In order to have comparable results, the integrating time step and total simulation time is set to ( dt = 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 T 1 = T 2 = 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 T 1 = 1000, T 2 = 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 ( θ). Figure 11 is the distribution of θ. As it shows, the central point of the distribution is 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 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 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 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 T 1 /T 2 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 T 1 /T 2 = 10 3, The curve goes flat. Different friction coefficients of ratchet (γ θ ) 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 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 [ θ = m LR L R B I θ + I 2 sin φ 2I L φ = m LR L R B sin φ 2I L I θ sin φ φ θ + sin φ 2 θ2 + cos φ K φ (φ φ 0 ) 2I L ( I 1 I L I θ K φ (φ φ 0 ). Here, the moments of inertia are 9 φ 2 (8) ] I 1 θ2 + I 2 φ2 + 2I 2 φ θ + K φ (φ φ 0 ) I θ ) (9) FIG. 17: An illustration of the ratchet and pawl system. Left: Top view. Right: Side view. I B I L = 1 3 m BR 2 B = 1 3 m LR 2 L 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 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 K φ is placed in the rod between the branch and the leg. In equilibrium state, the branch-leg angle is φ 0. The potential energy of the ratchet is P E(ratchet) = 1 2 K φ(φ φ e ) 2. (7) I 1 I 2 = I B + I L + m L R 2 B + m L R L R B cosφ = I L m LR L R B cosφ I θ = I 1I L I 2 2 I L. (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 φ 0 = 0.8, the length of the branch R B = 2.0, the length of the leg R L = 0.8 the length between the pawl and centre point R P = 2.6, the mass of the branch m B = 20, the mass of the leg m L = 30, the mass of gas particle in branch chamber m gas1 = 50, the mass of gas particle in leg chamber m gas2 = m gas1 T 2 /T 1 (mass of particle in leg chamber should be scaled down to have similar velocities as particles in branch chamber), the size of gas particle r gas = 0.1, time step dt = , temperature of branch chamber T 1 = 1000, and the force constant K φ = 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 T 1 /T 2 = 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 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 setting. When T 1 /T 2 > 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 ( ) when the load is equals to This maximum efficiency is much lower than the one in Feynman s setting. The related Langevin equation can be written as, ω θ = γω θ f θ (θ, φ, θ, φ) + Γ θ (t) θ = ω θ ω φ = γω φ f φ (θ, φ, θ, φ) + Γ φ (t) φ = ω φ (11) Where the random forces acted on θ and φ. Here, f θ = f θ (θ, φ, θ, φ) and f φ = f φ (θ, φ, θ, φ) are interacting FIG. 21: Rotation of ratchet on different temperature applied 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 ( m B = 0.2, m L = 0.3 ) temperature of branch chamber is set to T 1 = 10 7, 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 T 1 /T 2 = 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 T 1 /T 2 > 1000, the ratchet rotates

11 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. SUMMARY AND DISCUSSION 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 , there is a maximum efficiency ( ). 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 T 2 is lower than T 1. 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 T 1 to T 2 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 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 12 APPENDIX A: DETAIL OF COLLISIONS FOR FEYNMAN S RATCHET AND PAWL SYSTEM 1. Without Collision The potential energy of the Ratchet and Pawl is due to the spring attached to the pawl (leg). P E(RP) = 1 2 K φ(φ φ e ) 2, (A1) where φ e 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 where P is P = 2 X Y X = AI B θ + BIL φ + CmG ẋ + Dm G ẏ Y = A 2 I B + B 2 I L + C 2 m G + D 2 m G A = θh I B B = φh I L C = xh m G D = yh m G. (A6) Here h represents the distance between the particle and the stick. I L = 1 3 m LR 2 L 3. Ratchet hits Pawl I B = 1 3 m BR 2 B, (A2) respectively. The equation of motion (EOM) for φ is as follows, I L φ = Kφ (φ φ e ). The exact solution for the EOM is (A3) h = R P sin φ + R B sin(θ φ) θ h = R B cos(θ φ) φ h = R P cos φ R B cos(θ φ) C = 0 D = Gas hits Ratchet (A7) ) ( ) ( ) Kφ φ(t) = φ e + (φ(t 0 ) φ e cos t + I φ(t I L Kφ 0 ) sin t, L K φ I L (A4) where φ(t 0 ) and φ(t 0 ) are the initial displacement and velocity for φ, respectively. h = x sin θ y cos θ θ h = x cos θ + y sin θ x h = sin θ y h = cos θ B = 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 θ, δ φ and δẋ(δẏ), respectively. Upon a single collision event, they are determined via the following relations, δ θ = A P δ φ = B P δẋ = C P δẏ = D P, (A5) 5. Gas hits the end of the Ratchet This happens when the gas molecule has a finite radius R. h = (x R B cos θ) 2 + (y R B sin θ) 2 R θ h = R B(x sin θ y cos θ) R x h = x R B cos θ R y h = y R B sin θ R B = 0, (A9)

13 13 where x and y are Cartesian coordinates in the system with OP as the positive x axis. 6. Gas hits Leg h = x sin φ y cos φ φ h = x cos φ + y sin φ x h = sin φ y h = cos φ A = 0, (A10) FIG. 25: schematic representation of the ratchet and pawl. 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 This happens when the gas molecule has a finite radius R. h = (x R L cos φ) 2 + (y R L sin φ) 2 R φ h = R L(x sin φ y cos φ) R x h = x R L cos φ R y h = y R L sin φ R A = 0, (A11) where x and y are Cartesian coordinates in the system with P O as the positive x axis. APPENDIX B: DETAIL OF COLLISIONS FOR SECOND RATCHET AND PAWL SYSTEM 1. Coordinate Transformation Transformation between XY and X Y coordinate systems: x = x cosθ y sinθ + R B cosθ y = y cosθ + x sinθ + R B sinθ x = x cosθ + y sinθ R B y = y cosθ x sinθ (B1) Velocity transformation between XY and X Y coordinate systems: v x v y = v x cosθ + v y sinθ x sinθ θ + y cosθ θ = v x sinθ + v y cosθ x cosθ θ y sinθ θ (B2) 8. 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, y OP ) and (x P O, y P O ), respectively. (x, y) can be transformed into (x OP, y OP ) via 2. Without Collision The potential energy of the ratchet is P E(ratchet) = 1 2 K φ(φ φ e ) 2. (B3) x OP y OP = x cos θ P + y sin θ P = y cos θ P x sin θ P Moments of inertia: ẋ OP = ẋ cos θ P + ẏ sin θ P ẏ OP = ẏ cos θ P ẋ sin θ P. (A12) I B = 1 3 m BR 2 B (x, y) can be transformed into (x P O, y P O ) via x P O y P O ẋ P O = R P x cos θ P y sin θ P = y cos θ P + x sin θ P = ẋ cos θ P ẏ sin θ P ẏ P O = ẏ cos θ P + ẋ sin θ P. (A13) I L I 1 I 2 = 1 3 m LR 2 L = I B + I L + m L R 2 B + m L R L R B cosφ = I L m LR L R B cosφ I θ = I 1I L I 2 2 I L. (B4)

14 14 EOMs for θ and φ: [ θ = m LR L R B I θ + I 2 sin φ 2I L φ = m LR L R B sin φ 2I L I θ sin φ φ θ + sin φ 2 φ 2 θ2 + cos φ K φ (φ φ 0 ) 2I L ( I 1 I L I θ K φ (φ φ 0 ). ] I 1 θ2 + I 2 φ2 + 2I 2 φ θ (B5) + K φ (φ φ 0 ) I θ ) 3. When a gas molecule hits the Branch (B6) Before collision the velocities are v n, v t, θ and φ, respectively. After collision the velocities are ṽ n, ṽ t, θ and φ, respectively. Define R G = x 2 G + y2 G v G = vn 2 + vt 2 C θ C φ = m GR G I θ = m GR G I L I 2 2 I 1I L = m GI 2 R G = m GR G I 2 I θ I L I 1 I L I2 2. (B7) The change of velocity of the gas molecule perpendicular to the Branch is δv n = ṽ n v n = 2I θ(v n R G θ) I θ + m G RG 2. (B8) Therefore the velocities after the collision are ṽ n ṽ t = δv n + v n = v t θ = C θ δv n + θ φ = C φ δv n + φ. Transformation between v n, v t and v x, v y is v n v t ṽ n ṽ t v x v y ṽ x = v y cos θ v x sin θ = v y sin θ + v x cos θ = ṽ y cos θ ṽ x sin θ = ṽ y sin θ + ṽ x cos θ = v n sin θ + v t cos θ = v n cos θ + v t sin θ (B9) = ṽ n sin θ + ṽ t cos θ ṽ y = ṽ n cos θ + ṽ t sin θ. (B10) 4. When a gas molecule hits the Leg Before collision the velocities are v x, v y, θ and φ, respectively. After collision the velocities are ṽ x, ṽ y, θ and φ, respectively. Define R G = x 2 G + y 2 G D xθ = m GR B R G cos φ (2I L m L R L R G ) 2I L I θ (y G R B sin θ) D xφ = m GR G (I 1R G I 2R G I 2R B cos φ) I L I θ (y G R B sin θ) D xy = x G R B cos θ y G R B sin θ. (B11) The change of velocity of the gas molecule along x direction is δv x = ṽ x v x 2m G = I 1 Dxθ 2 + I LDxφ 2 + 2I 2D xθ D xφ + m G Dxy 2 v x + m G 2 I 1D xθ θ + IL D xφ φ + I2 D xθ φ + I2 D xφ θ + mg D xy v y I 1 D 2 xθ + I LD 2 xφ + 2I 2D xθ D xφ + m G D 2 xy + m G. Therefore the velocities after the collision are ṽ x ṽ y = δv x + v x = D xy δv x + v y θ = D xθ δv x + θ φ = D xφ δv x + φ. 5. When the Leg hits the Pawl (B13) Since the pawl does not move, before and after the collision the velocities are θ, φ and θ, φ, respectively. Define D θφ = D xφ D xθ (B12) = 2I 2R B cos φ 2I B R G 2m LRB 2 R G m LR L R B R G cos φ m L R L R B R G cos φ 2I. LR B cos φ (B14) The velocities after the collision are When φ = π 2 collision are θ = (I LD 2 θφ I 1) θ 2(I L D θφ + I 2 ) φ I 1 + I L D 2 θφ + 2I 2D θφ φ = (I 1 I L Dθφ 2 ) φ 2(I 1 D θφ + I 2 Dθφ 2 ) θ I 1 + I L Dθφ 2 + 2I. (B15) 2D θφ or 3π 2, cos φ = 0, the velocities after the θ = θ φ = φ 2 θ. (B16)

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