Exact microscopic analysis of a thermal Brownian motor C. Van den Broeck,1 R. Kawai,2 and P. Meurs1 1Limburgs Universitair Centrum, B-3590 Diepenbeek, Belgium 2Department of Physics, University of Alabama at Birmingham, Birmingham, AL 35294 (Dated: February 2, 2008) 4 0 WestudyagenuineBrownian motorbyhard diskmolecular dynamicsand calculate analytically 0 its properties, including its drift speed and thermal conductivity,from microscopic theory. 2 n PACSnumbers: 05.20.Dd,05.40.Jc,05.60.Cd,05.70.Ln a J It is (believed to be) impossible to systematically rec- ization of this construction. First, we can dispose of the 1 tifythermalfluctuationsinasystematequilibrium. Such pawl and spring in the ratchet and consider any rigid ] a perpetuum mobile of the second kind, also referred to but asymmetric object. An example with a cone-shaped h as a Maxwell demon [1], would violate the second law of object in one compartment and a flat “blade” or “sail” c e thermodynamics and would, from the point of view of located in the other one is illustrated in Fig. 1b. Sec- m statistical mechanics, be in contradiction with the prop- ond, we replace the single rotational degree of freedom - erty of detailed balance. Yet, it may require quite subtle with a single translational degree of freedom (Figs. 1c at arguments to explain in detail on specific models why and 1d). Third, we restrict ourselvesto two-dimensional t rectification fails. Apart from the academic and peda- systems. Finally, the substrate particles in the various s . gogical interest of the question, the study of small scale compartments aremodeled by harddisks whichundergo t a systemsismotivatedbyrapidlyincreasingcapabilitiesin perfectlyelasticcollisionswitheachotherwhiletheircen- m nanotechnology and by the huge interest in small scale ter collides elastically with the edges of the motor [5]. - biological systems. Furthermore, when operating under We firstreporton the results obtainedfrommolecular d n nonequilibrium conditions, as is the case in living organ- dynamicsfortwodifferentrealizationsofourmotor. The o isms, the rectification of thermal fluctuations becomes first one, referred to as arrow/bar or AB is inspired by c possible. This mechanism, also referred to as a Brown- theabovediscussion. Itconsistsofonetriangular-shaped [ ian motor [2], could furnish the engine that drives and arrowinthefirstcompartmentandaflatbarintheother 2 controls the activity on a small scale. In this letter we v proposea breakthroughin the theoreticalandnumerical 1 studyofasmallscalethermalengine. Ourstartingpoint a b 0 is the observation that one of the basic and most popu- 7 2 lar models, namely the Smoluchowski-Feynman ratchet T2 T2 1 [2, 3, 4], is needlessly complicated, and can be replaced 3 by a simplified construction involving exclusively hard 0 core interactions. Its properties, including speed, diffu- t/ sion coefficient and heat conductivity, can be measured T1 T1 a veryaccuratelybyharddiskmoleculardynamicsandcan c d m be calculated exactly from microscopic theory. - d T2 S 2q 0 T2 In Fig. 1a, we have schematically depicted the con- n o structionoriginallyintroducedbySmoluchowski[4]inhis c discussionofMaxwelldemonsandre-introducedwithtwo T S T v: compartmentsatdifferenttemperaturesbyFeynman[3]. 1 1 i One compartment contains a ratchet with a pawl and X a spring, mimicking the rectifier device that is used in FIG. 1: (a) Schematic representation of the Smoluchowski- r a clockworks of all kinds. The macroscopic mode of op- Feynman ratchet. (b) Similar construction without a pawl eration of such an object generates the impression that and spring. (c) Two dimensional analogue referred to as the only clockwise rotations can take place, suggesting that AB motor. Themotorisconstrainedtomovealongthehori- this construction can be used as a rectifier of the im- zontalx-direction(withoutrotationorverticaldisplacement). Thehostgasconsistsofharddiskswhosecenterscollideelas- pulses generated by the impacts of the particles in the tically with the engine parts. The shape of the arrow is de- othercompartmentonthe bladeswithwhichthe ratchet terminedbytheapexangle2θ0 andtheverticalcrosssection is rigidly linked. As Feynman has argued, such a recti- S. Periodic boundary conditions are used in the computer fication is only possible when the temperature in both simulations. (d) A symmetric construction referred to as the compartments is different. We now introduce a model AA motor. which is at the same time a simplification and general- 2 1.0 60] 2V> 4000 P(x,t) 2·<x(t)> [ 1024 0 t [ 1· 105] 2 t<V(0)V()>/<00..050 t [ ·1 103] 2 <x(t)> 20000 -2000 0 0.5 1.0 1.5 2.0 -1000 -500 x0 500 1000 t [ · 105] 0.20 FIG. 2: Probability density P(x,t) for the position x of mo- tor AB at times t = 1000 (shaded) and t = 4000 (open). model AA Insetleft: meansquaredisplacementversustime. Insetright: 0.15 0.02 model AB velocity correlation function. > V0.01 < > V0.10 < 0.00 (Fig. 1c). The other motor, called arrow/arrow or AA, -0.01 0.05 -2 -1 0 1 2 consists of an identical triangular-shaped arrow in both T-T 1 2 compartments (Fig. 1d). Both units of the motor are constrained to move as a rigid whole along the horizon- 0.00 0 50 100 150 200 tal x-direction as a result of their collisionwith the hard mass ratio M/m disks in the two compartments. The initial state of the hard disk gases corresponds to uniform (number) densi- FIG. 3: Upper panel: Position of the motor as a function of ties ρ1 and ρ2 and Maxwellian speeds at temperatures time. The thin solid curve shows a typical trajectory. All T1 and T2, in the compartments 1 and 2 respectively, [6] othercurvesrepresenttheaveragehx(t)i. Thethicksolidline (k =1bychoiceofunits). Theboundaryconditionsare is the equilibrium case (T1 = T2 = 1) for motor AB. The B dotted and dashed curves correspond to the nonequilibrium periodic both left and right, and top and bottom. Un- situation for respectively motor AA and AB (T1 =1.9, T2 = lessmentionedotherwise,the followingparametervalues 0.1). ThesituationwithswitchedtemperaturesforAB isthe are used: Each compartment is a 1200 by 300 rectangle dashed curvewith anegative velocity. Lower panel: Average and contains 800 hard disks (mass m = 1, diameter 1), velocity of motor AB as a function of its mass M. Inset: i.e., particle densities ρ1 =ρ2 =0.00222. Initial temper- averagespeed(avg. over2000runs)ofmotorsAAandABas atures were set to T1 = 1.9 and T2 = 0.1. The motor afunction of theinitial temperaturedifferenceT1−T2 (T1+ has a mass M =20, apex angle 2θ0 =π/18, and vertical T2 =2fixed). Thetheoreticalresults(7)and(8)predictlower speeds than the simulations. However, when the magnitude cross section S = 1. The averages are taken over 1000 isscaled,thetheoreticalcurves(dottedlines)fitwellwiththe runs. simulations. When the temperatures are the same in both com- partments, T1 = T2, no rectification takes place. In fact, Fig. 2 shows that the motor undergoes plain Brow- nian motion, with average zero speed, exponentially de- culiar behavior, resulting from the fact that both units cayingvelocity correlationsand linearlyincreasing mean are identical. Whereas equilibrium is usually a point of squaredisplacement. Thecorrespondingfrictionanddif- flux reversal,the velocity now displays a parabolic curve fusioncoefficientobeytheEinsteinrelation. Ontheother as a function of T1 T2 with a minimum equal to zero − hand, as soon as the temperatures are no longer equal, at the equilibrium state T1 = T2. It is clear from its the motor spontaneously developsan averagesystematic symmetric construction that, at least when ρ1 = ρ2, an drift along the x-axis. The amplitude and direction of interchange of T1 with T2 can not modify the speed so the speed depend in an intricate way on the parame- that the latter has to be an even function of T1 T2, cf. − ters of the problem. In particular, the average speed in- Fig. 3. creaseswiththetemperaturedifferenceandthedegreeof We finally note that the observed systematic speed asymmetry(decreasingθ0)anddecreaseswithincreasing doesnotpersistforever. Indeed,themotionofthemotor mass of the motor roughly as 1/M (see the lower panel along the x direction is a single degree of freedom that in Fig. 3). Note furthermore that the observed average allows for (microscopic) energy transfers hence thermal speed can be very large, i.e. comparable to the thermal contactbetweenthecompartments,afactthatwasover- speed k T/M of the motor. The AA motor has a pe- looked by Feynman in his analysis and first pointed out B p 3 2.0 To obtain analytic results from microscopic theory, T erature 11..05 T12 winhwichhicahretahseycmopmtpoatirctamlleynetxsaacrte,winefifnoicteulsyolnartgheewsihtuilaettiohne p m densities of the hard disk gases are extremely low (more e 0.5 t precisely, the so-called high Knudsen number regime re- 0.0 4000 quires that the mean free path is much larger than the linear dimensions of the motor units). In this limit, each > x(t)2000 compartment,characterizedbyitsparticledensityρiand < temperature T , acts as an ideal thermal reservoir. We i will furthermore assume that all the constituting units 0 0.0 0.5 1.0 1.5 2.0 of the motor are closed and convex. Under these cir- time t [ · 105 ] cumstances, the motor never undergoes recollisions and the assumption of molecular chaos becomes exact [10]. FIG.4: Exponentialdecayofthetemperaturestoafinalcom- The probability density P(V,t) for the speed V~ =(V,0) mon value (Tfinal =(T1+T2)/2=1.) in motor AB and con- comitantdisappearanceoftheaveragedriftspeed. Toenlarge ofthe motorthus obeysthe followingBoltzmann-Master the conductivity, a small mass M = 1, a large vertical cross equation: section S =10, and a large apex angle θ0 =π/9 are used. ∂ P(V,t)= t dr[W(V r,r)P(V r,t) W(V,r)P(V,t)](1) in [7, 8]. As a result one observes that the tempera- Z − − − tures in both compartments converge exponentially to a commonfinaltemperaturewithaconcomitantreduction W(V,r) is the transition probability per unit time for and eventual disappearance of the systematic motion as the motor to change speed from V to V +r due to the shown in Fig. 4. While this feature has already been collisionswiththegasparticlesinvariouscompartments. documented in detail in other constructions [9], we have The explicit expressionfor W(V,r) follows from elemen- focused here on conditions in which this conductivity is taryargumentsfamiliarfromkinetictheoryofgases,tak- smallandthecompartmentssufficientlylargesothatone inginto accountthe statisticsofthe perfectly elasticcol- reachesaquasi-steadystatewithawelldefinedandmea- lisions of the motor, constrained to move along the x- surable averagedrift velocity. direction, with the impinging particles: 2π ∞ ∞ m W(V,r)= dθ dv dv ρ φ (v ,v )L F (θ)(V~ ~v)~e(θ)H[(V~ ~v)~e(θ)]δ r+ B(θ)(V v +v cotθ) x y i i x y i i x y Xi Z0 Z−∞ Z−∞ − · − · h M − i (2) Here, the sum over i runs over all the different lowing a procedure similar to the one used in one- compartments, L is the total circumference of the dimensional problems such as the Rayleigh piston [11] i 2 ith unit of the motor, B(θ) = 2Msin θ/(M + or the adiabatic piston [12]. The details are somewhat 2 msin θ), H[x] is the Heaviside function, and ~e(θ) = involved and will be given elsewhere [13]. To lowest or- (sinθ, cosθ) is the unit vector normal to a surface der in the perturbation, the Master equation(1) reduces − at angle θ, θ [0,2π], the angles being measured to a Fokker-Planck equation equivalent to the following ∈ counter-clockwise from the horizontal axis. φ (v ,v ) = linear Langevin equation: i x y mexp m(v2+v2)/2k T /2πk T is the Maxwellian velocit(cid:2)y−distrxibutiyon inBcoim(cid:3)partmBenit i. The shape of MV˙ = γiV + 2γikBTi ηi (3) − any closed convex unit of the motor is defined by the Xi Xi p (normalized) probability density F(θ) such that F(θ)dθ with η independent Gaussian white noises of unit i isthefractionofitsoutersurfacethathasanorientation strength and between θ and θ+dθ. Note that sinθ = cosθ = 0, where the average is with respecthto Fi(θ),ha proiperty k T m 2π B i 2 γ =4ρ L dθF (θ)sin θ (4) resulting from the requirement that the object is closed. i i ir 2π Z0 i The Boltzmann-Masterequations (1) and (2) cannow the friction coefficient experienced by the motor due to be solved by a perturbation expansion in m/M, fol- its presence in compartment i. We conclude that at p 4 this order of the perturbation the contributions from m πk 2 B the separate compartments add up and are each - taken hViAB = ρ1ρ2(1−sin θ0)rMr2M separately - of the linear equilibrium form. In par- (T1 T2)√T1 ticular the motor has no (steady state) drift velocity, − (8) × [2ρ1√T1+ρ2√T2(1+sinθ0)]2 V = 0. It does however conduct heat. In the case h i of two compartments 1 and 2, the heat flow per unit Inagreementwithpreviousarguments,theAAmotor,cf. time between them is, as anticipated in Ref. 7, given (7), always moves in the same direction, namely the di- by a Fourier law: Q˙1→2 = κ(T1−T2) with conductivity rectionofthearrow. Furthermoreitisanexamplewhere κ = kBγ1γ2/[M(γ1+γ2)]. One also concludes from (3) one canincrease the asymmetry to generate a maximum and(4)thatthe(steadystate)velocitydistributionofthe 3 2 driftspeed. The limit sin θ = sin θ is reachedhere motor is Maxwellian, but at the effective temperature |h i| h i when θ0 0, which corresponds to an infinitely elon- → gated and sharp arrow in both compartments. Due to Teff = γiTi γi (5) strong finite size effects (e.g. sound waves among oth- Xi .Xi ers), the agreement between the theoretical results (7) and(8)andthecomputersimulationsisonlyqualitative: At the next order of perturbation in m/M, the cor- the theory predicts speeds which are typically 20 40% responding Langevin equation becomespnonlinear in V − lower. However,Eqs.(7)and(8)canbe fittedtothesim- while at the same time the Gaussiannature of the white ulation results by appropriately rescaling the magnitude noise is lost, a feature well-known from the Van Kam- of the velocity (see Fig. 3), indicating that their depen- pen 1/Ω expansion [11]. The most relevant observation dencies on the parameters, M, T1 and T2, are in good istheappearance,atthesteadystate,ofanon-zerodrift agreement with the simulations. velocity: In conclusion, we have provided a detailed analytic and numerical study of a simplified version of the πkBTeff m V = Smoluchowski-Feynman ratchet, including an exorcism h i r 8M rM - based on microscopic theory - of its operation as a ρiLiTiT−eTffeff 02πdθFi(θ)sin3θ Maxwell demon. Pi R (6) This work was supported in part by the National Sci- × Pi ρiLiqTTeiff R02πdθFi(θ)sin2(θ) ence Foundation under Grant No. DMS-0079478. This speed is of the order of the thermal speed of the motor, times the expansion parameter m/M, and fur- ther multiplied by a factor that deppends on the de- [1] H. S. Leff and A. F. Rex, Maxwell’s Demon (Adam tails of the construction. Note that the Brownian mo- Hilger, Bristol 1990). tor ceases to function in the absence of a temperature [2] P. Reimann, Phys.Rep. 361, 57 (2002). difference (Ti Teff, i) and in the macroscopic limit [3] R.P.Feynman,R.B.Leighton,andM.Sands,TheFeyn- ≡ ∀ M ( V 1/M). Notealsothatthe speedis scale- man Lectures on Physics I (Addison-Wesley, Reading, →∞ h i∼ independent, i.e., independent of the actual size of the MA, 1963), Chapter 46. motor units: V is invariant under the rescaling L to [4] M. v.Smoluchowski, Physik.Zeitschr. 13, 1069 (1912). i h i [5] Thistrickavoidscollisionsofthedisk’ssurfaceswiththe CL . To isolatemoreclearlythe effectofthe asymmetry i sharpcornersofthemotor;formoredetails,see: C.Van of the motor on its speed, we focus on the case where denBroeck,R.KawaiandP.Meurs,ProceedingsofSPIE the units have the same shape in all compartments, i.e. Volume: 5114,NoiseinComplexSystemsandStochastic Fi(θ) = F(θ). In this case Teff is independent of F(θ) Dynamics, p. 1 (2003). 3 2 and the drift velocity is proportional to sin θ / sin θ , [6] Note that oursystem as awhole is finiteand isolated so h i h i withtheaveragedefinedwithrespecttoF(θ). Thelatter that strictly speaking the equilibrium state should refer ratio is in absolute value always smaller than 1, a value to a micro-canonical ensemble. [7] J. M. R. Parrondo and P. Espagnol 64, 1125 (1996). thatcanbereachedfor“strongly”asymmetricobjectsas [8] K. Sekimoto, J. Phys.Soc. Jap. 66, 1234 (1997). will be shown below on a particular example. [9] C. Van den Broeck, E. Kestemont, and M. Malek Man- Wenowturntoacomparisonbetweentheoryandsim- sour, Europhys.Lett. 56, 771 (2001). ulations. Fromthegeneralresult(6),oneobtainsthefol- [10] J. R. Dorfman, H. Van Beijeren, and C. F. McClure, lowing expressions for the speed of the two motors that Archives of Mechanics 28, 333 (1976). were studied: [11] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland,Amsterdam, 1981). V AA = ρ1ρ2(1 sinθ0) [12] Ch. Gruber and J. Piasecki, Physica A268, 412 (1999); h i − E. Kestemont, C. Van den Broeck, and M. Malek Man- m πkB (T1 T2)(√T1 √T2) sour, Europhys.Lett. 49, 143 (2000). − − (7) × rMr8M [ρ1√T1+ρ2√T2]2 [13] P. Meurs, C. Van den Broeck, and A.Garcia, preprint.