BBallistic Transport at Uniform Temperature
Nawaf Bou-Rabee ∗ Houman Owhadi ∗
21 November 2018
Abstract
A paradigm for isothermal, mechanical rectification of stochastic fluctuations is in-troduced in this paper. The central idea is to transform energy injected by randomperturbations into rigid-body rotational kinetic energy. The prototype considered inthis paper is a mechanical system consisting of a set of rigid bodies in interactionthrough magnetic fields. The system is stochastically forced by white noise and dis-sipative through mechanical friction. The Gibbs-Boltzmann distribution at a specifictemperature defines the unique invariant measure under the flow of this stochasticprocess and allows us to define “the temperature” of the system. This measure is alsoergodic and strongly mixing. Although the system does not exhibit global directed mo-tion, it is shown that global ballistic motion is possible (the mean-squared displacementgrows like t ). More precisely, although work cannot be extracted from thermal energyby the second law of thermodynamics, it is shown that ballistic transport from thermalenergy is possible. In particular, the dynamics is characterized by a meta-stable state inwhich the system exhibits directed motion over random time scales. This phenomenonis caused by interaction of three attributes of the system: a non flat (yet bounded)potential energy landscape, a rigid body effect (coupling translational momentum andangular momentum through friction) and the degeneracy of the noise/friction tensoron the momentums (the fact that noise is not applied to all degrees of freedom). Mechanical rectification was introduced to describe devices that convert small-amplitudemechanical vibrations into directed rotary or rectilinear mechanical motion [6]. The pur-pose of this paper is to extend this concept to mechanical systems in isothermal environmentssubjected to stochastic fluctuations. Although the second law of thermodynamics preventsdirected mechanical motion from thermal fluctuations in an isothermal environment, vio-lations of the second law of thermodynamics on certain time-scales in microscopic systemshave been theoretically predicted [10; 12] and experimentally observed [7; 20].In this paper we exhibit a system characterized by ballistic non-directed motion atuniform temperature. Set µ to be the unique invariant measure of the system and x ( t ) thedisplacement of the system at time t then µ a.s. lim t →∞ ( x ( t ) − x (0)) /t → E µ [ { x ( t ) − E ( x ( t )) } ] ∼ t . Moreover the system is characterized by “dynamic” meta-stable states inwhich it experiences ballistic directed motion over random time scales (flights) in additionto “static” meta-stable states where the system is “captured” in potential wells.The mechanism introduced in this paper is different from the one associated to theGallavotti-Cohen fluctuation theorem [10]. It is based on the fact that one can obtainanomalous diffusion by introducing degenerate noise and friction in the momentums. This ∗ Applied & Computational Mathematics (ACM), Caltech, Pasadena, CA 91125 ( [email protected] , [email protected]). a r X i v : . [ m a t h . P R ] O c t Preview of the Paper 2 anomalous diffusion manifests itself in the just mentioned flights over time-scales which arerandom and the probability distributions of the flight-durations have a heavy tail. Moreoverwe expect that it will be of relevance to Brownian motors [2; 19] since it shows how thermalnoise can be used to get ballistic transport, in other words how to obtain fast and efficientintracellular transport without using chemical energy. Although global ballistic transportis achieved it is not directed and hence work is not produced from thermal fluctuations(this mechanism is not in violation of the second law of thermodynamics). However thismechanism would be sufficient for intracellular transport of a large fraction of the materialbeing carried to target areas with ballistic speed without draining cellular energy reserves.Figure 1.1:
Picture of mechanical system.
The physical system consists of a magnetized axisymmetrictop on a flat surface and a magnetized ring as shown above. The dipole moments in the ring are orientedradially.
The prototype analyzed in this paper consists of a magnetized top (ball) on a surfaceinteracting with a suspended magnetized ring as shown in Fig. 1.1. The following dynamicsis observed: when one lowers the magnetic ring to within a certain range of heights andthen tilts the ring, one observes the top transition from a state of no spin about its axis ofsymmetry to a state of nonzero spin. The reader is referred to the following urls for moviesand simulations of the phenomenon:
This phenomenon seems counterintuitive and non-Hamiltonian, since one would expect theangular momentum of the top to be conserved (since without friction the Hamiltonian ofthe system is invariant under an S rotation of the ball). The origin of this mechanicaldevice can be traced back to the work of David Hamel on magnetic motors in the unofficialsub-scene of physics [15]. The mechanism presented in this paper when the device is notat uniform temperature could in principle be used as a method of extracting energy frommacroscopic fluctuations. In section 4 Hamel’s device is analyzed through an idealized model based on magnetostaticsand the spinning of the top is caused by the introduction of surface frictional forces. Sim-ulations (figure 1.2) are done using variational integrators and concur with experimentalobservations.In section 5 the magnetic ring is allowed to be dynamic, and a fixed outer ring of a finitenumber of magnetic dipoles is introduced to stabilize it (see figure 5.1). The inner magneticring is excited through white noise applied as a torque. The steel ball and inner ring arecoupled through a magnetostatic potential and the motion of the steel ball is dissipativethrough slip friction.
Preview of the Paper 3
Figure 1.2:
Snapshots of Nonconservative Tilted Simulation.
The above are snapshots of thesimulation described in Fig. 4.9. The top is initially set with its axis of symmetry pointing nearly vertical.The axis of symmetry then aligns with the ambient magnetic field. However, the state is not a minimum of V e , and hence, the state is unstable. The top moves towards a state that minimizes its magnetic potentialenergy, and acquires spin in this process. The mathematical description of the fluctuation driven motor is obtained by generalizingLangevin processes from a system of particles on a linear configuration space to rigidifiedparticles whose configuration space is the Lie group SE(3). The question of random pertur-bations of a rigid body was treated by previous investigators who added perturbations tothe Lie-Poisson equations without potential or dissipative torques [13; 14]. Hence, they donot consider generalizing Langevin processes to the Lie-Poisson setting.We refer to figure 2.1 for a plot of the angular position of the ball versus time for differentvalues of noise amplitude α . The simulation of the system exhibits two distinct kinds ofmetastable states. In the first kind the ball is “stuck” in a magnetic potential well whosedepth depends on the position of the inner ring, and moves into another potential well whenthe energy barrier between the two wells is close to minimum (a phenomenon known asstochastic resonance) or transitions to a metastable state of the second kind. In the secondkind the ball spins in circles clockwise or counter-clockwise in a directed way for a randomamount of time (that numerically and heuristically observe to be exponential in law) untilit gets stuck in a potential well. −100−80−60−40−20020 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −30−25−20−15−10−5051015 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −200−150−100−50050 (a) α = 0 . α = 0 . α = 0 . Angular-position of magnetic ball (non-uniform temperature).
The angular com-ponent of the center of mass is plotted for three different realizations. The plots show the magnetic balltransition between two meta-stable states: noise-driven and inertia-driven motion.
Since the system associated to figure 2.1 is described by a generalized Langevin processit is possible to introduce a notion of temperature defined as (noise amplitude) friction constant . However,since noise but no friction is applied at the level of the inner ring and friction but no noise Preview of the Paper 4 is applied at the level of the ball, the system is not at uniform temperature, i.e., the innerring is at infinite temperature and the ball is at zero temperature. Nevertheless, one couldin principle use such a mechanism as a way to extract energy from macroscopic fluctuations.In a second step frictional torque is introduced to the inner ring and thermal torque(white noise) to the ball, so that the generator of the process is characterized by a uniqueGibbs-Boltzmann invariant distribution. Throughout this paper we interpret this propertyas placing the mechanical system at uniform temperature. We refer to figure 2.2 for a plotof the angular position of the ball versus time for different values of noise amplitude α (andtemperature). The system is still characterized by ballistic directed motion meta-stablestates. −60−50−40−30−20−100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −30−25−20−15−10−505 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −90−80−70−60−50−40−30−20−100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −2.5−2−1.5−1−0.500.5 (a) α = 0 .
001 (b) α = 0 .
001 (c) α = 0 . α = 0 . Angular-position of magnetic ball (uniform temperature).
Four different realizationsof the angular component of the center of mass are plotted. The plots for α = 0 . α is increased to α = 0 . Figure 2.3:
Sliding Disk.
Consider a sliding disk of radius r that is free to translate and rotate on asurface. We assume the disk is in sliding frictional contact with the surface. The configuration space of thesystem is SE(2), but with the surface constraint the configuration space is just R × SO(2).
To understand the behavior of the fluctuation driven motor prototype, the paper con-siders in section 3 a sliding disk (figure 2.3) that has the same essential behavior, but whoseconfiguration space is SE(2). The solution of this simplified system is a Langevin processwith degenerate noise and friction matrices in the momentums. The disk is free to slide androtate. Assume that one rescales position by r and time by some characteristic frequencyof rotation or other time-scale. The dimensionless Lagrangian is given by the difference ofkinetic and potential energy L ( x, v, θ, ω ) = 12 v + σ ω − U ( x ) Preview of the Paper 5
Figure 2.4:
Ballistic vs. Normal Diffusion. If U is non-constant theorem 3.1 implies that the meansquared displacement with respect to the invariant law is ballistic. Numerically, flights are observed inthe x -displacement as shown in the diagram. However, if U is constant this diffusion is normal, i.e., the x -displacement behaves like Brownian motion. where U : R → R is assumed to be smooth and periodic. If U = cos( x ), figure 2.5 showsthat the sliding disk is a modified one-dimensional pendulum. The contact with the surfaceis modelled using a sliding friction law. For this purpose we introduce a symmetric matrix C defined as, C = (cid:20) /σ /σ /σ (cid:21) . C is degenerate since the frictional force is actually applied to only a single degree of freedom,and hence, one of its eigenvalues is zero. In addition to friction the system is excited bywhite noise so that the governing equations become dx = vdtdθ = ωdt (cid:34) dvdω (cid:35) = (cid:34) − ∂ x U (cid:35) dt − c C (cid:34) vσω (cid:35) dt + α C / (cid:34) dB v dB ω (cid:35) (2.1)where C / is the matrix square root of C . Let E denote the energy of the mechanicalsystem given by, E = 12 v + 12 σω + U ( x ), (2.2)and let β = 2 c/α .In [4], we show that the Gibbs-Boltzmann distribution µ = exp ( − βE ) ,defines the unique invariant measure under the flow of the sliding disk. This measure is alsoergodic and strongly mixing. Using this result we prove in §
3, that if U is non-constant thenthe x -displacement of the sliding disk is µ a.s. not ballistic (cf. proposition 3.2). However, themean-squared displacement with respect to the invariant law is ballistic (cf. theorem 3.1).More precisely, we show that the squared standard deviation of the x -displacement withrespect to its noise-average grows like t . This implies that the process exhibits not onlyballistic transport but also ballistic diffusion. If U is constant then the squared standarddeviation of the x -displacement is diffusive (grows like t ).In the numerics the sliding disk is initially at rest and averages are computed withrespect to realizations. Numerically one observes the following consequences of this ballistic Preview of the Paper 6 behavior. Figure 2.6 shows that when U is non-constant then the motion is characterizedby meta-stable directed motion states. Figure 2.7 shows that the mean square displacement E [ x ( t ) ] grows like t (with respect to time) when U is constant or in the case of a onedimensional standard Langevin process (control) whereas it grows like t as soon as U isnon-constant (the motion becomes ballistic). A diagram comparing the solution behaviorin the U constant and non-constant cases is provided in Figure 2.4. Figure 2.8 correspondsto the plot of Cov( x ( t + 2 s ) − x ( t + s ) , x ( t + s ) − x ( t ))(Var( x ( t + 2 s ) − x ( t + s ))) (Var( x ( t + s ) − x ( t ))) (2.3)for t large as a function of s . It clearly shows that the system is characterized by long timememory/correlation when U is non-constant whereas it has almost no memory when U isconstant or in in the case of a one dimensional standard Langevin process.Figure 2.5: Ballistic Pendulum.
If the dimensionless potential is U = cos( x ), then the sliding disk issimply a pendulum in which the bob in the pendulum is replaced by a disk and the pendulum is placedwithin a cylinder as shown. Preview of the Paper 7 −20000200040006000800010000 Time M ean o f X UsymmetricUasymmetricUflatControl -2000 0 2000 4000 6000 8000 100000.0e+00 1.0e+04 2.0e+04 3.0e+04 4.0e+04 5.0e+04 6.0e+04 7.0e+04 8.0e+04 9.0e+04 1.0e+05timeMean of X, U-symmetric -2000-1000 0 1000 2000 3000 4000 50000.0e+00 1.0e+04 2.0e+04 3.0e+04 4.0e+04 5.0e+04 6.0e+04 7.0e+04 8.0e+04 9.0e+04 1.0e+05timeMean of X, U-asymmetric -50 0 50 100 150 200 2500.0e+00 1.0e+04 2.0e+04 3.0e+04 4.0e+04 5.0e+04 6.0e+04 7.0e+04 8.0e+04 9.0e+04 1.0e+05timeMean of X, U-flat -300-200-100 0 100 200 300 4000.0e+00 1.0e+04 2.0e+04 3.0e+04 4.0e+04 5.0e+04 6.0e+04 7.0e+04 8.0e+04 9.0e+04 1.0e+05timeMean of X, Ornstein-Uhlenbeck (a) U-symmetric (b) U-asymmetric (c) U-flat (d) ControlFigure 2.6:
Sliding Disk at Uniform Temperature, h = 0 . , α = 5 . , c = 0 . . The mean of the x -displacement of the disk for U symmetric, asymmetric, flat and control case. Figures (a) and (b) show thatwhen U is non-constant, directed motion as a meta-stable state is possible. On the other hand, (c) and (d)do not show such behavior. The figure on the top superposes these graphs in a single plot for comparison. Time M ean o f X UsymmetricUasymmetricUflatControl
Figure 2.7:
Sliding Disk at Uniform temperature, h = 0 . , α = 5 . , c = 0 . . A log-log plot of themean squared displacement of the ball. It clearly shows that the x-position exhibits anomalous diffusionwhen U is symmetric or asymmetric. In the control and flat U cases the diffusion is normal. Sliding Disk: Simplified Model 8 −0.200.20.40.60.81 Time C o rr e l a t i on o f X UsymmetricUasymmetricUflatControl -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.13.0e+04 3.5e+04 4.0e+04 4.5e+04 5.0e+04 5.5e+04 6.0e+04 6.5e+04 7.0e+04timeCorrelation of X (U-flat) -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.083.0e+04 3.5e+04 4.0e+04 4.5e+04 5.0e+04 5.5e+04 6.0e+04 6.5e+04 7.0e+04timeCorrelation of X (Ornstein-Uhlenbeck) (a) U-symmetric (b) U-asymmetric (c) U-flat (d) ControlFigure 2.8:
Sliding Disk at Uniform Temperature, h = 0 . , α = 5 . , c = 0 . . The correlation of the x displacement of the disk for U symmetric, asymmetric, flat, and the control case. When U is symmetricor asymmetric the correlation in the x displacement is nonzero for a certain time-scale which is larger thanthe characteristic time-scales associated with U and with the friction factor. However, in the other casesthe correlation is negligible. The figure on the top superposes these graphs in a single plot for comparison. To understand the behavior of the prototype stochastic mechanical rectifier which will betreated in detail in subsequent sections, we designed a simplified model whose configurationspace is SE(2). The system consists of a disk sliding on a surface as shown in Figure 2.3.The effect of the outer ring is modelled as a periodic, one-dimensional potential; while theeffect of the inner ring is incorporated into white noise. The SDE for the isothermal, slidingdisk is a Langevin process, but the noise and friction matrices are degenerate. A statisticalnumerical analysis discussed below shows that this process has two interesting statisticalproperties which are both linked to rigid-body inertial effects: 1) a non-trivial correlationon certain time-scales and 2) ballistic diffusion.
Ballistic Transport at Uniform Temperature
The proof that the sliding disk is atuniform temperature is based on finding the generator for (2.1) and showing that the Gibbsmeasure is the unique, invariant measure under the flow of this generator.
Theorem 3.1.
Let χ := T × R × T × R denote the phase space of the sliding disk ( T standingfor the torus of dimension one). Set ξ := ( x, v, θ, ω ) to be the solution of (2.1). Let E denotethe energy of the sliding disk given by, E = v + σω + U ( x ) . Set β = 2 c/α and let µ bethe Gibbs probability measure defined by µ ( dξ ) := e − βE Z dξ (3.1) where Z := (cid:82) χ e − βE dξ . If U is non-constant, then the Gibbs measure µ is ergodic andstrongly mixing with respect to the stochastic process ξ . Furthermore, it is the unique,invariant probability measure for the stochastic process ξ . This result is a special case of the proof provided in [4]. To determine the measure isinvariant, one computes the infinitesimal generator L associated to the stochastic process ξ Sliding Disk: Simplified Model 9 and proves that if f is µ - measureable then, (cid:90) χ L f µ ( dξ ) = 0Using the fact that the measure is ergodic, one can readily prove that the x -displacementitself is not ballistic. Proposition 3.2.
Provided that U is non-constant, then µ a.s. lim t →∞ x ( t ) − x (0) t → . Proof.
Since µ is ergodic with respect to ξ ,lim t →∞ x ( t ) − x (0) t = lim t →∞ t (cid:90) t vdt = (cid:90) χ vµ ( dξ ) = 0. (cid:4) Moreover, as stated in the following proposition, the squared standard deviation of the x + θ -degree of freedom of the sliding disk grows like t . Let E denote the expectation withrespect to the Brownian noise and E µ the expectation with respect to the Brownian noiseand the initial configuration (sampled from the invariant measure µ ). Set x t = x ( t ) and θ t = θ ( t ). Proposition 3.3.
The squared standard deviation of the x t + θ t -degree of freedom is diffu-sive, i.e., lim t →∞ E µ [( x t + θ t − E [ x t + θ t ]) ] t = 2 α σ c ( σ + 1) . (3.2) Proof.
First, diagonalize the diffusion and friction matrices in (2.1) using the followinginvertible matrix: V = 11 + σ (cid:20) − σ (cid:21) as follows, V (cid:20) dvdω (cid:21) = V (cid:20) − ∂ x U (cid:21) dt − c (cid:20) σ +1 σ (cid:21) V (cid:20) vω (cid:21) dt + α ( σ + 1) √ σ + 1 (cid:20) (cid:21) V (cid:20) dB v dB ω (cid:21) .Simplifying this expression yields the following pair of equations: d ( − v + σω ) = ∂ x U dt , (3.3) d ( v + ω ) = − ∂ x U dt − cγ ( v + ω ) + ¯ α ( dB v + dB ω ). (3.4)where γ = ( σ + 1) σ , ¯ α = α ( σ + 1) √ σ + 1 .Set B s := ( B v + B ω ) / √
2. Integrating (3.3) and (3.4) gives − v + σω = − v + σω + (cid:90) t ∂ x U ds , (3.5) v + ω = ( v + ω ) e − cγt − (cid:90) t e − cγ ( t − s ) ∂ x U ds + √ α (cid:90) t e − cγ ( t − s ) dB s . (3.6) Sliding Disk: Simplified Model 10
Integrating (3.6) gives( x t + θ t ) − ( x + θ ) = v + ω cγ (1 − e − cγt ) − (cid:90) t − e − cγ ( t − s ) cγ ∂ x U ds + √ α (cid:90) t − e − cγ ( t − s ) cγ dB s .The result follows by observing that (cid:90) t ∂ x U ds = ( − v t + σω t ) − ( − v + σω ) (3.7) (cid:4) However, one can prove that the squared standard deviation of the − x t + σθ t -degree offreedom grows like t . This implies that the process exhibits not only ballistic transport butalso ballistic diffusion along this degree of freedom. Proposition 3.4.
Assume that U is non constant, then lim sup t →∞ E µ (cid:104)(cid:0) − x t + σθ t − E [ − x t + σθ t ] (cid:1) (cid:105) t ≤ σβ (3.8) and lim inf t →∞ E µ (cid:104)(cid:0) − x t + σθ t − E [ − x t + σθ t ] (cid:1) (cid:105) t ≥
14 1 + σβ (3.9) Proof.
From Cauchy-Schwartz inequality one obtains that (cid:18) t (cid:90) t ( − v s + σω s ) ds (cid:19) ≤ t (cid:90) t ( − v s + σω s ) ds. Hence lim sup t →∞ E µ (cid:104)(cid:0) − x t + σθ t − ( − x + σθ ) (cid:1) (cid:105) t ≤ µ (cid:2) ( − v + σω ) (cid:3) . (3.10)We obtain the first inequality of the proposition by observing that µ (cid:2) ( − v + σω ) (cid:3) = 1 + σβ .and E µ (cid:104)(cid:0) E [ − x t + σθ t ] − ( − x + σθ ) (cid:1) (cid:105) ≤ E µ (cid:104)(cid:0) − x t + σθ t − ( − x + σθ ) (cid:1) (cid:105) (3.11)Let us now prove the lower bound. Integrating equation (3.5) gives − x t + σθ t − E [ − x t + σθ t ] = t (cid:90) t (1 − st )( ∂ x U − E [ ∂ x U ]) ds (3.12)Write A t = (cid:90) t (1 − st )( ∂ x U − E [ ∂ x U ]) ds (3.13) Sliding Disk: Simplified Model 11 and B t = (cid:90) t st ( ∂ x U − E [ ∂ x U ]) ds (3.14)Observe that A t + B t = ( − v t + σω t ) − E [ − v t + σω t ]. (3.15)Since µ is strongly mixing when U is non constant, it follows that [18]lim inf t →∞ E µ [( A t + B t ) ] = µ [( − v t + σω t ) ] (3.16)Furthermore E µ [( A t + B t ) ] ≤ (cid:0) E µ [ A t ] + E µ [ B t ] (cid:1) (3.17)and since the law of the process ( x t , θ t , v t , ω t ) remains invariant under P µ by reversing timeand flipping the velocities v t , ω t we deduce that E µ [ A t ] = E µ [ B t ] andlim inf t →∞ E µ [ A t ] = 14 µ [( − v t + σω t ) ] (3.18)We conclude by the taking the expectation of square of (3.12) with respect to E µ . (cid:4) The following theorem is a straightforward consequence of the previous propositions.
Theorem 3.1.
We have • If U is constant then lim t →∞ E µ (cid:2) ( x t − E [ x t ]) (cid:3) t = 2 α σ c ( σ + 1)( σ + 1) (3.19) • If U is non constant then lim sup t →∞ E µ (cid:2) ( x t − E [ x t ]) (cid:3) t ≤ β (1 + σ ) (3.20) and lim inf t →∞ E µ (cid:2) ( x t − E [ x t ]) (cid:3) t ≥ β (1 + σ ) (3.21) Remark . Using Ito’s formula and (2.2) one obtains that, dE = − c ( v + ω ) dt + α (cid:18) σ (cid:19) dt + martingales.Integrating this expression gives E ( t ) − E (0) = − c (cid:90) t ( v + ω ) ds + α (cid:18) σ (cid:19) t + martingales.The first and second terms represent the energy loss due to friction and the energy injecteddue to the noise respectively. Remark . Setting v t = v ( t ), it follows from (3.5) and (3.6) that v t = σ ( v + ω ) e − cγt + v − σω σ + 1 − (cid:90) t σe − cγ ( t − s ) + 1 σ + 1 ∂ x U ds + √ ασ + 1 (cid:90) t e − cγ ( t − s ) dB s . (3.22)The long term memory effect exhibited in Figure 2.8 has its origin in the term (cid:82) t ∂ x U ( x ( s )) ds in (3.22). That term is equal to zero when U is constant, and itself has its origin in therigid body interaction between rotation and translation and the fact the friction matrix issingular. Sliding Disk: Simplified Model 12
Stochastic Variational Integrator
To simulate the dynamics of the sliding disk atconstant temperature, a stochastic variational Euler method is applied [5]. The discretescheme for the isothermal case is given explicitly by: x n +1 = x n + hv n +1 , θ n +1 = θ n + hω n +1 , (cid:34) v n +1 ω n +1 (cid:35) = (cid:34) v n ω n (cid:35) + h (cid:34) − ∂ x U ( x n )0 (cid:35) − hc C (cid:34) v n σω n (cid:35) + α C / (cid:34) dB v dB ω (cid:35) . (3.23)It is an explicit, first-order strongly convergent method. Simulation
We will consider four different systems to simulate. The first three are slidingdisks with a symmetric, asymmetric, and flat potentials: U ( x )sin( x ) symmetricsin( x ) + 0 . x ) asymmetric0 flatThe fourth case is a control consisting of a simulation of a 1-D Langevin process at the sametemperature and with a symmetric potential: dX = V dt , dV = − cV dt − cos( X ) dt + αdB V .For all of the simulations the sliding disk is initially at rest , r = 0 . m = 1 . J = mr / x displacement of the disk as shown in Fig. 2.6 are very small compared with the spread as,e.g., shown in the histogram of the final position of the ball as shown in Fig. 3.3. However,the mean squared displacement shows ballistic diffusion in the cases when U is symmetric orasymmetric and normal diffusion otherwise (see Figures 3.1-2.7). The time-scale associatedwith this ballistic diffusion is plotted in Fig. 2.8 which shows the correlation in the x-displacement when U is symmetric or asymmetric. This time-scale is much greater thanthe characteristic time-scale associated with the friction or the potential. Recall from theintegral expression of the velocity, that when U is zero a rigid-body term is neglected. Thisdemonstrates the important role of the rigid-body effect in the ballistic diffusion of the x -displacement when U is symmetric and asymmetric.Finally we also consider adding a non-degenerate, but anisotropic dissipation matrix tothe sliding disk. It is numerically observed that if the anisotropy is large enough, E ( x t ) isballistic for a long period of time. Sliding Disk: Simplified Model 13 Time M ean o f X UsymmetricUasymmetricUflatControl (a) U-symmetric (b) U-asymmetric (c) U-flat (d) ControlFigure 3.1:
Sliding Disk at Uniform Temperature, h = 0 . , α = 5 . , c = 0 . . From left: themean squared position of the disk for U symmetric, asymmetric, flat, and the control. The figure on the topsuperposes these graphs in a single plot for comparison. D i ff u s i on o f X UsymmetricUasymmetricUflatControl (a) U-symmetric (b) U-asymmetric (c) U-flat (d) ControlFigure 3.2: Sliding Disk at Uniform Temperature, h = 0 . , α = 5 . , c = 0 . . From left: “diffusion”of x -displacement of the disk for U symmetric, asymmetric, flat, and the control. For the cases when U isflat and the control, the diffusion is normal. Whereas in the other cases the diffusion is ballistic. The figureon the top superposes these graphs in a single plot for comparison. Hamel’s magnetic top 14 −3 −2 −1 0 1 2 3x 10 (a) U-symmetric (b) U-asymmetric (c) U-flat (d) ControlFigure 3.3: Sliding Disk at Uniform Temperature, h = 0 . , α = 5 . , c = 0 . . From left: histogramof the x -displacement of the disk at T = 50000 for U symmetric, asymmetric, and flat. We observe a widerspread in the cases when U is symmetric or asymmetric. In this section an observed magnetism induced spinning phenomenon is analyzed. Up toour knowledge, the first magnetism induced spinning device was “Tesla’s egg of colombus”exhibited in 1892 [11]. Using two phase AC energizing coils in quadrature, Tesla placed acopper plated ellipsoid on a wooden plate above a rotating magnetic field. The egg stood onits pointed end without cracking its shell and began to spin at high speed to the amazementof the scientists who witnessed the experiment. This effect was caused by induced eddycurrents on the surface of the ellipsoid. A related magnetism induced spinning phenomenonis the Einstein-de Hass effect in which the rotation of an object is caused by a change inmagnetization [9]. In this effect the magnetic field causes an alignment of electronic spinsand their angular momenta is transferred to the atomistic lattice. We will show belowthrough an idealized model based on magnetostatics that the spinning of the top in Hamel’sdevice is due to surface friction.
Mechanism behind curious rotation
The following observation was made on a simplemechanical system consisting of a magnetized top and ring as shown in Fig. 1.1. When thering is held above the top within a certain range of heights, and then tilted, one observes thetop transitions from a state of no spin to a state of nonzero spin about its axis of symmetry.The system is modeled as two rigid bodies in magnetostatic interaction: a ball with amagnetic dipole aligned to one of its axes, and a fixed magnetized ring of radially alignedmagnetic dipoles. The main tool used to analyze the observed curious rotation is a Lagrange-Dirichlet stability criterion [16]. It is shown that the fixed points of the system’s governingequations correspond to the magnetic top being at rest with its axis of symmetry alignedwith the local magnetic field and its translational position at a critical point of the magneticpotential energy. Stability of this point is determined by analyzing the nature of this criticalpoint. If the attitude of the ring is normal to the surface, the magnetic potential energyis very nearly axisymmetric. A cross-sectional sketch of this potential energy is shown inFig. 4.1. In this case there exists a ring of minima that are not individually stable. However,if the ring is tilted slightly, the potential has a unique local minimum opposite a saddle pointas shown in the same figure.If the ring is tilted and the ball’s initial position is unstable, the ball moves towards thestable fixed point which induces a resisting frictional force (see Fig. 4.2). The torque due tothe sliding friction is in directions orthogonal to the moment arm q . However, the torquedue to the magnetic field counters the torques about the axes perpendicular to ξ . Thismagnetic torque keeps ξ aligned with the local magnetic field. Thus, the torque due tofriction mainly causes a spin about the ξ axis.Thus far, the ring has been kept fixed. If the ring is perturbed slightly the position of Hamel’s magnetic top 15 the local minimum will change. Hence this system is unstable with respect to perturbationsof the ring. This instability will be utilized to design a prototype stochastic mechanicialrectifier. ring of minima global minimumsaddle point
No Tilt With Tilt
Figure 4.1:
Cross-Section of Potential Energy Surface.
Cross-sectional sketches of the magneticpotential energy with and without tilt. With no tilt in the ring, the potential energy surface is axisymmetricand has a ring of minima. If the ring is tilted slightly, a unique local minimum exists across from a saddlepoint. ! Fri c ti o nal F o r ce Magn e ti c T o rqu e Figure 4.2:
Mechanism behind spin.
A sketch of the ball, its axis of symmetry ξ , the ambient magneticfield (blue arrows), the restoring magnetic torques, and frictional force. The ball tends to a position thatminimizes its potential energy. If the ball is initially unstable, then as it moves towards the minimum africtional force resists this motion. The frictional force causes torques about axes orthogonal to the momentarm q . However, the restoring magnetic torque counters the torques about the axes orthogonal to ξ asshown in the sketch. This argument clarifies why the spin is primarily along ξ . ringring ballring 3 HO Or ! Figure 4.3:
Illustration of Magnetized Rigid Ball and Ring.
The figure depicts a rigid ball on aflat surface with a dipole fixed at the ball’s centroid O ball and in the direction ξ . We will assume that thecenter of mass of the ball is coincident with O ball . The ring is located at a height H ring above a referencepoint O on the surface, and its radius is r ring . The magnetic dipoles are placed symmetrically around thisring with dipole moments in the radial direction. Hamel’s magnetic top is modeled as an axisymmetric rigid ball of radius r and mass m . A magnetic dipole is attached to the center of mass, and in the direction of the axisof symmetry ξ ∈ R as shown in Fig. 4.3. For simplicity, the surface friction is modelledusing a sliding friction law. In what follows the continuous and discrete model are derivedusing the Hamilton-Pontryagin (HP) variational principle [3]. Magnetostatic Field
Let B : R → R be the ambient magnetostatic field. For simplic-ity, we assume the magnetostatic field is due to a magnetic ring consisting of N magnetic Hamel’s magnetic top 16 dipoles equally spaced around a ring of radius r ring centered at the point O ring and in theplane defined by the vector ζ ∈ R . The point O ring is at a height H ring above a refer-ence point O on the surface of the ball. To the reference point O , an orthonormal frame( e x , e y , e z ) is attached. For i = 1 , ..., N , let d i ∈ R denote the location of the ith-dipolewith respect to O ring and m i ∈ R denote its dipole moment. The dipole moments areassumed to be aligned in the radial direction of the ring. Let µ be the permeability of freespace. The magnetic field of the ith-dipole at a field point x ∈ R can be computed fromthe vector r i connecting the field point to the ith-dipole given by r i = x − r ring d i − H ring e z using the standard formula for the magnetostatic field due to a dipole: B ( r i , m i ) = µ π (cid:107) r i (cid:107) (cid:0) r T i m i ) r i − (cid:107) r i (cid:107) m i (cid:1) . (4.1)Using the principle of superposition, the magnetic field at a field point x ∈ R due to the N dipoles is determined as the vector sum of the magnetic fields due to each dipole: B ( x ) = N (cid:88) i =1 B ( r i , m i ). ! ! ! ! ! ! ! ! ! ! ! ! (a) axisymmetric φ R = 0 , θ R = 0 (b) tilted φ R = π/ , θ R = 0 (c) tilted φ R = π/ , θ R = π Figure 4.4:
Illustration of Magnetized Ring and its Magnetic Field.
The figures depict variousorientations of a magnetized ring consisting of a discrete number of dipoles equally spaced around a circlein the plane defined by ζ which is also depicted. The magnetic fields lines in the plane y = 0 are plotted.The angles φ R and θ R shown are the altitude and azimuth of the normal to the plane in which the ring isin. The height of the center of the ring, the radius of the ring, the number of dipoles, and the orientationof the ring are all parameters in this magnetostatic field model. Lagrangian of Magnetic Ball
The configuration space of the system is Q = R × SO(3) and its Lagrangian is denoted L : T Q → R . Let ( x ( t ) , ˙x ( t )) ∈ R × R denotethe translational position and velocity of the ball measured with respect to the referenceframe attached to O . Let ( e , e , e ) denote an inertial orthonormal frame attached to O ball and related to a body-fixed frame ( ξ , ξ , ξ ) via the rotation matrix R ( t ) ∈ SO(3). Let J = diag( I , I , I ) be the standard diagonal inertia matrix of the body and ξ the axisof symmetry of the ball. We assume the mass distribution of the body is symmetric withrespect to the axis of symmetry ξ . This assumption implies that the principal moments ofinertia about ξ and ξ are equal, i.e., I = I = I . Let ( x ( t ) , ˙x ( t )) ∈ R × R denote thetranslational position and velocity of the ball.We will use the isomorphism between R and the Lie algebra of SO(3), so (3), given by thehat map (See appendix.). In terms of this identification, we define the reduced Lagrangian Hamel’s magnetic top 17 (cid:96) : R × R × SO(3) × R → R as (cid:96) ( x , ˙x , R, ω ) = L ( x , ˙x , R, (cid:98) ω R ).For the free magnetic ball, i.e., magnetic ball without dissipation, it is given explicitly by, (cid:96) ( x , ˙x , R, ω ) = m ˙x T ˙x + 12 ω T RJR T ω − mg x T e + ξ T3 B ( x ). (4.2)From left the terms represent the translational and rotational kinetic energy of the ball, thegravitational potential energy and the dipole potential energy. The ball is also subject tothe surface constraint ϕ : R → R given by: ϕ ( x ) = − r + e T3 x . (4.3)This holonomic constraint restricts the translational motion of the ball to a plane. Governing Conservative Equations
The equations of motion will be determined usinga HP description [3]. The constrained HP action integral is given by, s = (cid:90) ba (cid:104) (cid:96) ( x , v , R, ω ) + (cid:104) p , ˙x − v (cid:105) + (cid:68)(cid:98) π , ˙ RR T − (cid:98) ω (cid:69) + λϕ ( x ) (cid:105) dt .The HP principle states that δs = 0where the variations are arbitrary except that the endpoints ( x ( a ) , R ( a )) and ( x ( b ) , R ( b ))are held fixed. The equations are given by, ˙x = v (reconstruction equation), (4.4) p = ∂(cid:96)∂ v (Legendre transform), (4.5) ddt p = ∂(cid:96)∂ x + λ ∂ϕ∂ x (Euler-Lagrange equations), (4.6) ϕ ( x ) = 0 (constraint equation), (4.7) ddt R = (cid:98) ω R (reconstruction equation), (4.8) π = ∂(cid:96)∂ ω (reduced Legendre transform), (4.9) ddt (cid:98) π = ∂(cid:96)∂R R T − (cid:100)(cid:98) πω (Lie-Poisson equations). (4.10)Evaluating these equations at (cid:96) as defined in (4.2) yields ˙x = v m ˙v = ( DB ( x )) T ξ + ( λ − mg ) e x T e = r ˙ R = (cid:98) ω R π = RJR T ω ˙ π = (cid:98) ξ B ( x ). (4.11) Hamel’s magnetic top 18
Remark . Since the system is axisymmetric the Legendre transform in (4.11) simplifies: π = I ω + ( I − I )( ω T ξ ) ξ = ⇒ ω = 1 I (cid:18) π + I − I I ( π T ξ ) ξ (cid:19) . (4.12)As a consequence one does not need to solve for the evolution of all three columns of R ( t )to integrate the ODE in π . Instead one just needs to solve for the evolution of the thirdcolumn, ξ , using: ˙ ξ = (cid:98) ωξ . (4.13)The following conservation law follows from axisymmetry. Proposition 4.1.
The following momentum map is conserved under the flow of (4.11), J = π T ξ . Proof.
This momentum map is due to an S symmetry of the Lagrangian about the axis ξ which can be computed by formula (12.2.1) of [16]. The group acts on Q by:Φ Qs ( x , R ) = ( x , exp( s (cid:98) ξ ) R ).The corresponding infinitesimal generator is given by: ψ Q ( x , R ) = dds Φ Qs ( x , R ) (cid:12)(cid:12) s =0 .Thus, one can invoke Noether’s theorem to conclude the following momentum map J : T Q → ( S ) ∗ is preserved J = (cid:28) ∂L∂ ˙ R , ψ Q ( x , R ) (cid:29) = ∂(cid:96)∂ ω T ξ = π T ξ . (cid:4) This conservation law indicates that if initially the top is not spinning about its axisof symmetry, then it will never spin about this axis. Thus, one cannot obtain the curiousrotation using magnetic and gravitational effects alone. This result suggests surface frictionwill also play a role in producing this phenomenon.
Governing Nonconservative Equations
Let q = − r e denote the vector connectingthe center of mass C to the contact point Q as shown in Fig. 4.3. We model the surfacefrictional force using a sliding friction law proportional to the slip velocity, i.e., the velocityof the contact point on the rigid body relative to the center of mass V Q : F f = − c V Q This law assumes zero static friction but, is nevertheless reasonable on very slippery surfaceswhere static friction is negligible. Moreover, it is an experimental fact that the magnetictop exhibits this curious rotation even on oily surfaces. In reality one must keep in mindthat the ball is not in point-contact with the surface; rather, a finite area of the ball is incontact with the surface and is moving relative to the surface to make spinning possible. Inthis case this sliding model of friction is quite reasonable. A more refined model of frictionwould include rotational torque and dry frictional effects (Coulomb friction).The slip velocity is given by: V Q = ˙ x + (cid:98) ω q . (4.14) Hamel’s magnetic top 19
The force of friction is therefore, F f = − c V Q ,and the torque due to friction is, τ f = (cid:98) qF f .The governing dynamical equations of the magnetic ball with friction are given by: ˙x = v , m ˙v = ( DB ( x )) T ξ + ( λ − mg ) e − c ( v + r (cid:98) e ω ), x T e = r ,˙ ξ = (cid:98) ωξ , ω = I (cid:16) π + I − I I ( π T ξ ) ξ (cid:17) ,˙ π = (cid:98) ξ B ( x ) + cr (cid:98) e ( v + r (cid:98) e ω ). (4.15) Remark . The fixed points of (4.15) satisfy:˙ x = 0 , ω = 0˙ π = 0 = ⇒ ξ = κ B ( x ) , κ ∈ R , constant,˙ v = 0 = ⇒ ( DB ( x )) T B ( x ) = 0.The Lagrange-Dirichlet criterion can be used to analyze the stability of these states sincethe kinetic energy vanishes at these fixed points and since the dissipation is proportionalto velocity [16]. In particular, one can invoke a classical theorem due to Thomson-Tait-Chetaev, to conclude that if the fixed point is potentially stable or unstable, then it remainsstable or unstable after introducing arbitrary dissipative forces proportional to velocities[17]. By this criterion, the fixed points are stable provided that the equilibrium is a strictlocal minimum of the potential energy. The potential energy evaluated at this equilibriumpoint is given by V e : R → R , V e ( x ) = − κ B ( x ) T B ( x ).In the limit as the number of dipoles along the ring is infinite, B is axisymmetric withrespect to the surface provided the attitude of the ring is normal to the surface on whichthe ball is on. In this case V e can be written as a function of the distance to the origin: f ( ρ ) = V e ( x ) where ρ = (cid:107) x (cid:107) with x T e = r . Assume that the ring is above the ball and thepotential energy is V e ( x ) ≤ x ∈ R . The origin is an unstable critical point, since f has a local maximum at that point. Moreover, f ( ρ ) tends to zero as ρ becomes large.One can pick a distance M (cid:29) r ring (the radius of the ring) sufficiently large to make f ( ρ )arbitrarily close to zero. In the interval ρ ∈ (0 , M ), V e ( ρ ) is continuous, and therefore, thefunction has a local minimum in this interval. However, since f is axisymmetric this criticalpoint is not a local minimum in the plane, but rather a circle of critical points. As shownin Fig. 4.5, the ball will be unstable with respect to circumferential perturbations. Thefigure also shows that when the axisymmetry of the magnetic field is broken by changingthe orientation of the ring, there is a critical point that is a local minimum of V e ( x ). Simulations
Four simulations are performed on the top to confirm the theory and explainthe curious rotation of the physical system. We use the HP integrator introduced in [3]. Let φ B and θ B denote the azimuth and altitude of the ball’s attitude ξ . In all of the simulations Hamel’s magnetic top 20 (a) axisymmetric φ R = 0 , θ R = 0 (b) tilted φ R = π/ , θ R = 0 (c) tilted φ R = π/ , θ R = π Figure 4.5:
Surface plot of V e . The potential energy surface is plotted as a function of x and y for ballradius r = 0 .
3, ring height H = 2, 25 dipoles on the ring, and ring radius r ring = 4. The angles φ and θ shown are the altitude and azimuth of the normal to the plane in which the ring is in. By tilting the ringone can obtain a critical point which is a local minimum of V e . the top is initialized as follows: c = 0 . · m / s , r = 1 . , m = 200 g , φ B = 0 , θ B = 0 , I = I = 2 / mr ,µ (cid:107) m i (cid:107) / (4 π ) = µ (cid:107) ξ (cid:107) / (4 π ) = 10 − Tesla meter ,H ring = 20 cm , N = 20 dipoles , r ring = 34 cmThe timestep size and number of timesteps are h = 0 .
025 and N = 20000 respectively. Inthe first four simulations the orientation of the ring is kept fixed. The first case simulatedis when the magnetostatic field is axisymmetric and friction is absent. In this case the topundergoes what appears to be chaotic motion involving a balance between the magneticpotential and translational kinetic energies as shown in the simulation and Fig. 4.6. In thesecond simulation the altitude of the attitude of the ring is slightly perturbed causing themagnetostatic field to be noticeably asymmetric, see Fig. 4.7. However, in both of thesecases the top does not acquire spin about the symmetry axis ξ as predicted by the theory.In the presence of friction c = 0 .
3, the motion changes. The third case considered isthe same configuration as the first case, but with friction. In this case the kinetic energyof the top is dissipated and the top moves towards a point where the magnetic potentialenergy is minimum, see Fig. 4.8. However, no spin develops. If the attitude of the ring isslightly perturbed, spin does develop as shown in Fig. 4.9. The origin of this spin in case 4is explained here.It is very clear from (4.15) that one can get spin about the axis of symmetry in thepresence of friction. However, the frictional forces may produce torques about the other axesas well. So what is not clear is why the ball spins primarily about ξ . This phenomenon isclarified in the following remark. Remark . If the initial position of the ball is not a local minimum of the magneticpotential energy (as in the simulation), the position will be unstable as predicted by aLagrange-Dirichlet criterion. The ball’s position then adjusts to minimize its magneticpotential energy which causes sliding friction (see Fig. 4.2). The torque due to the slidingfriction will introduce a torque in directions orthogonal to the moment arm q . However, thetorque due to the magnetic field will counter the torques about the axes perpendicular to ξ . Keep in mind that the magnetic torque keeps ξ aligned with the local magnetic field.Thus, the torque due to friction mainly causes a spin about the ξ axis as shown in thesimulation. Snapshots of the simulation are provided in Fig. 1.2. Hamel’s magnetic top 21 (a) potential surface V e (b) momentum map J ( t )Figure 4.6: Conservative Axisymmetric.
The figures plot x ( t ) , y ( t ) and J ( t ) of the top in the absenceof friction, and for φ R = 0 and θ R = 0. Superimposed on (a) is the potential energy level set which appearsaxisymmetric. (a) shows that the top bounces between the local maximum at the center and the circle ofminima (dark blue ring). (b) shows that J ( t ) is preserved as predicted by the theory. The accompanyingsimulation vividly illustrates this motion and demonstrates that one does not get curious rotation in thiscase. (a) potential surface V e (b) momentum map J ( t )Figure 4.7: Conservative Tilted.
The figures plot x ( t ) , y ( t ) and J ( t ) of the top in the absence offriction, and for φ R = 0 and θ R = π/
64. Superimposed on (a) is the potential energy level set which is nolonger axisymmetric. (a) shows that the top moves erratically within an annulus. (b) confirms that J ( t )is preserved. The accompanying simulation vividly illustrates this motion and demonstrates that one doesnot get curious rotation in this case. (a) potential surface V e (b) momentum map J ( t )Figure 4.8: Nonconservative Axisymmetric.
The figures plot x ( t ) , y ( t ) and J ( t ) of the top in thepresence of friction, and for φ R = 0 and θ R = 0. Superimposed on (a) is the potential energy level setwhich appears axisymmetric. (a) shows that the top moves from the local maximum at the center to apoint that minimizes the potential energy (dark blue ring). However, (b) shows that no spin develops. Theaccompanying simulation vividly illustrates this motion and demonstrates that one does not get curiousrotation in this case. Fluctuation Driven Magnetic Motor 22 (a) potential surface V e (b) momentum map J ( t )Figure 4.9: Nonconservative Tilted.
The figures plot x ( t ) , y ( t ) and J ( t ) of the top in the presence offriction, and for φ R = 0 and θ R = π/
64. Superimposed on (a) is the potential energy level set which is nolonger axisymmetric. (a) shows that the top moves along arcs. (b) confirms that one does get spin in thiscase. The accompanying simulation vividly illustrates this motion and clearly demonstrates that the ballgoes from a state of rest to a state of nonzero spin about its attitude.
The instability of the top with respect to perturbations of the ring is the key idea behindthe fluctuation driven magnetic motor. These random fluctuations are modelled as a whitenoise torque on the ring. However, if the ring is allowed to move freely, it could possiblyturn on its side. To stabilize the ring, a fixed magnetized outer ring is installed.By adjusting the radii and heights of the rings and the inertia of the inner ring, onecan obtain a configuration in which the attitude of the inner ring can be randomly torquedwithout undergoing large excursions from the vertical position. In this case numerical exper-iments reveal that one can adjust the amplitude of the white noise so that the ball undergoesdirected motion on certain time-scales. To be precise the numerics indicates that startingfrom a position of rest initially the top’s motion is dominated by the effect of white noise. Ifthe outer ring is close enough one can see the top oscillate between the wells in the magneticpotential caused by the dipoles in the outer ring. This behavior is reminiscent of stochasticresonance.After some time, the top accumulates enough kinetic energy that it displays directedmotion along a circle of certain radius. This motion is inertia-driven and the energy injectedinto the system by the white noise mainly adds to the speed of the top. Provided that theinner ring does not turn on its side, one of two things can happen: 1) the top reachesa critical velocity in which the amount of energy dissipated by the surface friction is onaverage equal to the amount of energy injected by the thermal noise or 2) the top gathersenough kinetic energy to escape from the potential well created by the inner and outer rings.Conducting this same experiment at uniform temperature, reveals that this phenomenonpersists. The isothermal, magnetic ball-ring system is a prototype fluctuation-driven motor.The fluctuation-driven motor considered in this paper is related to the granular, mag-netic balls in ferrofluidic thermal ratchets. In such ratchets a time-varying magnetostaticfield transfers angular momentum to magnetic spherical grains in a ferrofluid [8]. The ferro-magnetic grains are modelled in the same way as the rigid ball in the prototype. However,our work is different in that the magnetic potential in the prototype is autonomous. Infact, the use of a non-autonomous potential to design a fluctuation driven motor is wellunderstood [1].In this section the equations of motion for the dynamics of a magnetic ball interactingwith a dynamic, inner magnetized ring are derived. The magnetic effects of a fixed outer ringare also taken into account. As before the magnetic ball is assumed to be constrained to a
Fluctuation Driven Magnetic Motor 23 flat surface that resists the motion of the top via surface sliding friction. The center of massof the inner ring is kept at a fixed height, but otherwise it is free to rotate and subjectedto white noise torques. The governing stochastic differential equations are analyzed usingenergy arguments. Simulations validate this analysis and show that one can get sustaineddirected motion of the top from random perturbations of the inner ring. Moreover, it isshown that this phenomenon persists even when the system is at uniform temperature.This system is called a fluctuation driven magnetic motor.The mass and radius of the magnetic ball are m and r respectively. The mass of theinner ring is m inner . The heights and radii of the inner and outer rings are plotted anddefined in Fig. 5.1. outerballouterouter inner innerinner 3 rOOH H O rO ! Figure 5.1:
Illustration of Magnetized Rings and Magnetic Top.
The figure shows the magnetictop on the surface with two rings above it. The inner and outer rings have radii r outer and r inner andheights H outer and H inner respectively. The centroids of the ball, inner and outer rings are also labeled.The purpose of the outer ring is to prevent the trivial equilibrium in which the inner ring turns on its side. Lagrangian of Magnetic Ball & Inner Ring
The configuration space of the systemis Q = R × SO(3) × SO(3). Let O ball , O inner and O outer denote the centroids of theball, inner and outer rings respectively. Let ( e , e , e ), ( f , f , f ), and ( g , g , g ) denoteinertial orthonormal frames attached to O ball , O inner and O outer , and related to body-fixedframes ( ξ , ξ , ξ ), ( ζ , ζ , ζ ), and ( η , η , η ) via the rotation matrices R B ( t ), R R ( t ),and R O ∈ SO(3). Let I B = diag( I , I , I ) and I R = diag( J , J , J ) be the standarddiagonal inertia matrices of the body and inner ring, and let ξ and ζ be the attitudeof the ball and inner ring. The following assumptions are made: the outer ring is fixed,the mass distribution of the inner ring is symmetric with respect to its attitude (or axisof symmetry), and the ball’s mass distribution is spherically symmetric. This assumptionimplies that J B = I = I = I and J = J = J .The magnetostatic field at any field point is due to the dipole in the ball, and an innerand outer ring consisting of N and M magnetic dipoles respectively. On each ring thedipoles are equally spaced around a circle of radius r inner and r outer in the planes definedby the vectors η and ζ ∈ R . We assume that there is no self-interaction between thedipoles within each body. The heights of the rings above the surface are denoted by H inner and H outer .For i = 1 , · · · , N and j = 1 , · · · , M , let d inner i ( t ) and d outer j ∈ R denote the location ofthe ith and jth-dipoles on the inner and outer rings with respect to the points O inner and O outer and let m inner i ( t ) and m outer j ∈ R denote the orientation of their respective dipolemoments. The dipole moments are assumed to be in the radial direction of the ring. Themagnetic field of each dipole at a field point can be determined from the vector r connecting Fluctuation Driven Magnetic Motor 24 the field point to the dipole using (4.1) B inner i ( r ) = B ( r , m inner i ) (field of inner ring dipole), B outer j ( r ) = B ( r , m outer i ) (field of outer ring dipole), B ( r ) = B ( r , ξ ) (field of ball).Define the following vectors r ij ∈ R i = 0 , · · · , N and j = 0 , · · · , M which joins the innerand outer dipoles, the inner dipoles to the ball, and outer dipoles to the ball as follows, r ij = − d outer j − H outer e + d inner i + H inner e if i, j > x − d inner i − H inner e if j = 0 (jth outer to ball), x − d outer j − H outer e if i = 0 (ith inner to ball).The spatial representation of the reduced Lagrangian (cid:96) : T R × SO(3) × R × SO(3) × R → R of the free magnetic ball, i.e., magnetic ball without dissipation, is given by, (cid:96) ( x , ˙x , R B , ω B , R R , ω R ) = m ˙x T ˙x + 12 J B ω T B ω B + 12 ω T R R R I R R T R ω R + 2 N (cid:88) i =1 ξ T3 B inner i ( r i ) + M (cid:88) j =1 ξ T3 B outer j ( r j ) + N (cid:88) i =1 M (cid:88) j =1 ( m inner i ) T B outer j ( r ij ) (5.1)From left the terms represent the translational and rotational kinetic energy of the ball, therotational kinetic energy of the ring, the gravitational potential energy of the ball and themagnetic potential energy of the inner ring and ball dipoles. Governing Equations for Magnetic Ball & Ring
The equations of motion are de-termined from a Hamilton-Pontryagin (HP) principle [3]. The HP action integral is givenby, s = (cid:90) ba (cid:104) (cid:96) ( x , ˙x , R B , ω B , R R , ω R ) + (cid:68)(cid:98) π B , ˙ R B R T B − (cid:99) ω B (cid:69) + (cid:68)(cid:98) π R , ˙ R R R T R − (cid:99) ω R (cid:69) + λϕ ( x ) (cid:105) dt .The HP principle states that δs = 0where the variations are arbitrary except that the endpoints ( x ( a ) , R B ( a ) , R R ( a )) and( x ( b ) , R B ( b ) , R R ( b )) are held fixed. The equations are given by, ddt ∂(cid:96)∂ ˙x = ∂(cid:96)∂ x + λ ∂ϕ∂ x (Euler-Lagrange equations for ball), (5.2) ϕ ( x ) = 0 (constraint equation), (5.3) ddt R B = (cid:99) ω B R B (reconstruction equation for ball), (5.4) ddt R R = (cid:99) ω R R R (reconstruction equation for ring), (5.5) π B = ∂(cid:96)∂ ω B (reduced Legendre transform for ball), (5.6) π R = ∂(cid:96)∂ ω R (reduced Legendre transform for ring), (5.7) ddt (cid:99) π B = ∂(cid:96)∂R B R T B − (cid:92) (cid:99) π B ω B (Lie-Poisson equations for ball), (5.8) ddt (cid:99) π R = ∂(cid:96)∂R R R T R − (cid:92) (cid:99) π R ω R (Lie-Poisson equations for ring). (5.9) Fluctuation Driven Magnetic Motor 25
The nonconservative system is obtained by adding the frictional force and torque derivedearlier to the translational and rotational equations of the ball: (5.2) and (5.8).
Random Perturbations & Uniform Temperature
Consider driving the inner ring bythe following Wiener process. Let W ∈ R denote Brownian motion in R , and append thefollowing random torque: T R = αd W ∈ R to the Lie-Poisson equation for the ring (5.9).The work performed by T R is equal to the change in total energy of the system. On theother hand, the work done by the kicks over a time interval [ a, b ] is given by the followingformula: Work of kicks = α (cid:90) ba T T R ω R . (5.10)The work transferred from the inner ring to the top is given by:Work transferred to top = ∆Total Energy − ∆Ring Energy. (5.11)Thus a measure of the efficiency of the magnetic motor is given by the following ratio:Efficiency = Work transferred to topWork of kicks . (5.12)As a next step frictional torque is introduced to the inner ring and thermal torque (whitenoise) to the ball, so that the generator of the process is characterized by a unique Gibbs-Boltzmann invariant distribution. The reader is referred to the Appendix for the governingequations of the magnetic motor at uniform and non-uniform temperatures. Simulations
A stochastic variational integrator is used to carry out these simulations [5].Simulations are conducted at uniform and non-uniform temperature as described below.The initial conditions and parameters used are given by c = 0 .
15 kg · m / s , r = 4 . , m = 500 g , φ B = 0 , θ B = 0 , I = I = 2 / mr ,µ (cid:107) m inner, outer i (cid:107) / (4 π ) = µ (cid:107) ξ (cid:107) / (4 π ) = 2 × − T · m ,H inner = H outer = 48 cm , N = 20 dipoles , M = 5 dipoles ,r inner = 75 cm , r outer = 98 cm , m inner = 6 kg , J = m inner r , J = 1 / J . In all of the simulations the magnetic top is initially at rest with its attitude aligned tothe vertical. The difference in the simulations is the amplitude of the oscillations and thediscrete sample from the normal distribution. Figures 5.6-2.2 present data from the uniformtemperature simulations. The key point about the uniform temperature simulations is asseen in Figures 5.9 and 2.2 the magnetic top transitions from noise to ballistic motion. Asexpected the non-uniform temperature case also exhibits this behavior as shown in Figures5.2-2.1.
Fluctuation Driven Magnetic Motor 26 −2.4−2.3−2.2−2.1−2−1.9−1.8−1.7−1.6x 10 −3 −2.3−2.2−2.1−2−1.9−1.8−1.7−1.6−1.5x 10 −3 −2.4−2.2−2−1.8−1.6−1.4−1.2x 10 −3 (a) α = 0 . α = 0 . α = 0 . Total and inner ring energy (non-uniform temperature).
The total energy of thesystem is shown in green and the ring energy in cyan for different amplitudes of the noise and for twodifferent samples. Although there is dissipation, the thermal fluctuations inject energy into the system. −6−4−202468101214x 10 −4 −1−0.500.511.5x 10 −3 −1.5−1−0.500.511.52x 10 −3 (a) α = 0 . α = 0 . α = 0 . Energy injected and dissipated (non-uniform temperature).
A plot of the energyinjected by the white noise as computed analytically and dissipated by friction computed numerically. −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.6−0.4−0.200.20.40.60.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.8−0.6−0.4−0.200.20.40.6 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8−0.6−0.4−0.200.20.40.60.8 (a) α = 0 . α = 0 . α = 0 . xy-position of magnetic ball (non-uniform temperature). An aerial view of the pathof the center of mass of the ball in the xy-plane. −0.8−0.6−0.4−0.200.20.40.60.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.8−0.6−0.4−0.200.20.40.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.8−0.6−0.4−0.200.20.40.60.8 (a) α = 0 . α = 0 . α = 0 . xy-position of magnetic ball planar view (non-uniform temperature). The x and y components of the center of mass are plotted in green and blue respectively. It is very clear from this plotthat the motion transitions from noise driven to being inertia driven. Fluctuation Driven Magnetic Motor 27 −2.35−2.3−2.25−2.2−2.15−2.1−2.05x 10 −3 −2.35−2.3−2.25−2.2−2.15−2.1−2.05x 10 −3 −2.35−2.3−2.25−2.2−2.15−2.1−2.05x 10 −3 −2.35−2.3−2.25−2.2−2.15−2.1−2.05x 10 −3 (a) α = 0 .
001 (b) α = 0 .
001 (c) α = 0 . α = 0 . Total and inner ring energy (uniform temperature).
The total energy of the systemis shown in green and the ring energy in cyan for different amplitudes of the noise. Although there isdissipation, the thermal fluctuations inject energy into the system. −0.4−0.3−0.2−0.100.10.20.30.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.4−0.3−0.2−0.100.10.20.30.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25 (a) α = 0 .
001 (b) α = 0 .
001 (c) α = 0 . α = 0 . Energy injected and dissipated (uniform temperature).
A plot of the energy injectedby the white noise as computed analytically and dissipated by friction computed numerically. In the constanttemperature case we see an approximate balance between these energies. −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8−0.6−0.4−0.200.20.40.60.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8−0.6−0.4−0.200.20.40.60.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8−0.6−0.4−0.200.20.40.60.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.100.10.20.3 (a) α = 0 .
001 (b) α = 0 .
001 (c) α = 0 . α = 0 . xy-position of magnetic ball (uniform temperature). An aerial view of the path of thecenter of mass of the ball in the xy-plane. −0.8−0.6−0.4−0.200.20.40.60.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.8−0.6−0.4−0.200.20.40.60.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.8−0.6−0.4−0.200.20.40.60.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −0.8−0.6−0.4−0.200.20.40.60.8 (a) α = 0 .
001 (b) α = 0 .
001 (c) α = 0 . α = 0 . xy-position of magnetic ball planar view (uniform temperature). The x and y components of the center of mass are plotted in green and blue respectively. It is very clear from this plotthat the motion transitions from noise driven to being inertia driven. Conclusion 28
This paper introduces a novel mechanism to rectify random perturbations to achieve directedmotion as a meta-stable state and ballistic mean-squared displacement with respect to theinvariant law. The basic idea behind the mechanism is explained using the simple slidingdisk model. With this model one can prove that if the potential energy is non-constant,then the invariant Gibbs measure of the system, µ , is ergodic and strongly mixing. Asa consequence µ a.s. it is shown that the translational displacement of the sliding disk isnot ballistic. However, it is shown that the mean-squared displacement with respect to theinvariant law is ballistic.This anomalous diffusion manifests itself numerically. That is, starting from a positionat rest and when U is non-constant, one observes flights in the x -displacement over time-scales which are random and the probability distribution of the flight-durations have a heavytail. Moreover, from a numerical statistical analysis, it is observed that the mean-squaredtranslational displacement of the sliding disk is ballistic and its translational displacementis characterized by long time memory/correlation. The paper proceeds to show that thisbasic phenomenon arises in a more complex, rigid-body system consisting of two rigid bodiesinteracting via magnetostatic effects. Along the way we explain the observed dynamics ofHamel’s magnetic device. SO(3)
Preliminaries
In the paper we use the following isomorphism between an elementof the Lie algebra of SO(3), T e SO(3) = so (3), and R . Recall that elements of so (3) are skew-symmetric matrices with Lie bracket given by the matrix commutator. Let ω = ( ω , ω , ω ).Then one can relate R with a skew-symmetric matrix via the hat map (cid:98) : R → so (3), (cid:98) ω = − ω ω ω − ω − ω ω .Let g ( t ) be a curve in SO(3). With this identification of so (3) to R , the right-trivializationof a tangent vector ˙ g to this curve, given by ξ = T R g − · ˙ g ∈ so (3), can be written in termsof the spatial angular velocity vector ω ∈ R , i.e., as ξ = (cid:98) ω ∈ so (3). Magnetic motor at non-uniform temperature governing equations
Below the gov-erning equations of the ring-ball system with nonconservative effects due to surface fric-tion and white noise are written in Ito form. We introduce the potential energy function U : R × SO(3) × SO(3) → R which represents the total potential energy of the ring-ball Appendix 29 system. In terms of U the governing equations can be written as, d x = v dtd v = 1 m (cid:18) − ∂U∂ x + ( λ − mg ) e − c V Q (cid:19) dt x T e = rdR B = (cid:99) ω B R B dtdR R = (cid:99) ω R R R dt π B = J B ω B π R = R R I R R T R ω R d (cid:99) π B = (cid:18) − ∂U∂R B R T B − c (cid:91) (cid:98) qV Q (cid:19) dtd (cid:99) π R = (cid:18) − ∂U∂R R R T R (cid:19) dt + α (cid:100) d W Evaluating these governing equations at (cid:96) defined in (5.1) yields d x = v dt (7.1) d v = 1 m N (cid:88) i =1 DB inner i ( r i ) T ξ + M (cid:88) j =1 DB outer j ( r j ) T ξ dt + 1 m ( − c V Q + ( λ − mg ) e ) dt (7.2) x T e = r (7.3) dR B = (cid:99) ω B R B dt (7.4) dR R = (cid:99) ω R R R dt (7.5) π B = J B ω B (7.6) π R = R R I R R T R ω R (7.7) d π B = (cid:98) ξ N (cid:88) i =1 B inner i ( r i ) + M (cid:88) j =1 B outer j ( r j ) dt − c (cid:98) qV Q dt , (7.8) d π R =2 N (cid:88) i =1 (cid:104) (cid:92) m inner i B ( r i ) − (cid:92) d inner i (cid:0) DB inner i ( r i ) T ξ (cid:1)(cid:105) dt + N (cid:88) i =1 M (cid:88) j =1 (cid:104) (cid:92) m inner i B outer j ( r ij ) + (cid:92) d inner i (cid:0) DB outer j ( r ij ) T m inner i (cid:1)(cid:105) dt + αd W . (7.9)Since the ring is axisymmetric its Legendre transform simplifies: π R = J ω R + ( J − J )( ω T R ζ ) ζ = ⇒ ω R = 1 J (cid:18) π R + J − J J ( π T R ζ ) ζ (cid:19) . (7.10) Remark . The energy of the magnetic motor is given by E ( x , v , R B , π B , R R , π R ) = m v T v + 12 J − B π T B π B + 12 π T R R R I − R R T R π R + U ( x , R B , R R ). Appendix 30
Using Ito’s formula one can calculate the stochastic differential of the energy in the casewhen the magnetic motor is at non-uniform temperature dE = (cid:18) − c V T Q V Q + 12 α trace (cid:2) I − R (cid:3)(cid:19) dt + α ω T R d W .Integrating yields, E ( t ) − E (0) = − c (cid:90) t V T Q V Q ds + 12 α trace (cid:2) I − R (cid:3) t + martingales.The first term represents the energy dissipated by friction. The next terms represent thework done by the white noise torque as computed by Ito’s integral. Isothermal, magnetic motor governing equations
Similar to the sliding disk, to putthe magnetic motor at constant temperature we define the following dissipation matrix, C = /m − r/ ( mJ )0 1 /m r/ ( mJ ) 00 r/ ( mJ ) r /J − r/ ( mJ ) 0 0 r /J The translational position of the ball is written in coordinates as x = ( x , x , r ). Thetranslational and angular velocities of the ball are given by v = ( v , v ,
0) and ω B =( ω (1) B , ω (2) B , ω (3) B ). The dynamical equations for the constant temperature magnetic motorare given by: dx = v dt (7.11) dx = v dt (7.12) dR B = (cid:99) ω B R B dt (7.13) dR R = (cid:99) ω R R R dt (7.14) dv dv dω (1) B dω (2) B = − U T x e /mU T x e /mU T B e /JU T B e /J dt − c B C mv mv Jω (1) B Jω (2) B + α B C / dB v dB v dB ω (1) B dB ω (2) B (7.15) dω (3) B = − U T B e /J (7.16) π R = R R I R R T R ω R (7.17) d π R = − U R − c R ω R + α R d B R (7.18)The terms U x , U B , and U R are defined in terms of the inner product on R as, U T x y = (cid:28) ∂U∂x , y (cid:29) = ∂ x U ( x, R B , R R ) · yU T B y = (cid:28) ∂U∂R B R T B , (cid:98) y (cid:29) = ∂ R B U ( x, R B , R R ) · (cid:98) yR B U T R y = (cid:28) ∂U∂R R R T R , (cid:98) y (cid:29) = ∂ R R U ( x, R B , R R ) · (cid:98) yR R and ∂ R B U, ∂ R R U : SO(3) → T ∗ R SO(3), and ∂ x U : R → T ∗ x R . Similar to the sliding disk atuniform temperature, one can prove the following. Appendix 31
Theorem 7.1.
Let χ denote the phase space of the magnetic motor. Set ξ to be the solutionof (7.11)-(7.18). Suppose that c R /α R = c B /α B . Let E denote the energy of the magneticmotor given by, E ( x , v , R B , π B , R R , π R ) = m v T v + 12 J − B π T B π B + 12 π T R R R I − R R T R π R + U ( x , R B , R R ) .Let β = 2 c B /α B . The following measure, µ ( dξ ) := e − βE Z dξ (7.19) is the unique Gibbs invariant measure of the stochastic process ξ .Remark . By using Ito’s formula one can show that: dE = (cid:18) − c B V T Q V Q − c R ω T R ω R + α B (cid:18) m + r J (cid:19) dt + 12 α R trace (cid:2) I − R (cid:3)(cid:19) dt + martingales.Integrating yields, E ( t ) − E (0) = − c B (cid:90) t V T Q V Q ds − c R (cid:90) t ω T R ω R ds + α B (cid:18) m + r J (cid:19) t + 12 α R trace (cid:2) I − R (cid:3) t + martingales. EFERENCES 32
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