We develop an exactly-solvable three-state discrete-time minimal Brownian
ratchet (MBR), where the transition probabilities between states are
asymmetric. By solving the master equations we obtain the steady-state
probabilities. Generally the steady-state solution does not display detailed
balance, giving rise to an induced directional motion in the MBR. For a reduced
two-dimensional parameter space we find the null-curve on which the net current
vanishes and detailed balance holds. A system on this curve is said to be
balanced. On the null-curve, an additional source of external random noise is
introduced to show that a directional motion can be induced under the zero
overall driving force. We also indicate the off-balance behavior with biased
random noise.Comment: 4 pages, 4 figures, RevTex source, General solution added. To be
  appeared in Phys. Rev. Let

A. Allison

A. P. Flitney

Andrew Allison

C. E. M. Pearce

C. R. Doering

D. Abbott

D. Heath

D. Kinderlehrer

Derek Abbott

G. P. Harmer

H. Eugene Stanley

H. V. Westerhoff

J. Buceta

J. M. R. Parrondo

M. von Smoluchowski

P. C. W. Davies

R. P. Feynman

Youngki Lee

English

arXiv

We develop an analytically solvable three-state discrete-time minimal Brownian ratchet (MBR), where the transition probabilities between states are asymmetric. By solving the master equations, we obtain the steady-state probabilities. Generally, the steady-state solution does not display detailed balance, giving rise to an induced directional motion in the MBR. For a reduced two-dimensional parameter space, we find the null curve on which the net current vanishes and detailed balance holds. A system on this curve is said to be balanced. On the null curve, an additional source of external random noise is introduced to show that a directional motion can be induced under the zero overall driving force.Youngki Lee, Andrew Allison, Derek Abbott, and H. Eugene Stanle

Lee, Y.

Allison, A.

Abbott, D.

Stanley, H.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION  Lee, Youngki; Allison, Andrew Gordon; Abbott, Derek; Stanley, H. Eugene  Minimal Brownian ratchet: an exactly solvable model Physical Review Letters, 2003; 91(22):220601   ©2003 American Physical Society  http://link.aps.org/doi/10.1103/PhysRevLett.91.220601             http://link.aps.org/doi/10.1103/PhysRevD.62.093023                        http://hdl.handle.net/2440/2468         PERMISSIONS http://publish.aps.org/authors/transfer-of-copyright-agreement   “The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright Act, 17 U.S.C. §101, the employer named [below], shall have the following rights (the “Author Rights”): [...] 3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.”    8th May 2013  P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22Minimal Brownian Ratchet: An Exactly Solvable ModelYoungki Lee,1,2 Andrew Allison,3 Derek Abbott,3 and H. Eugene Stanley21Yanbian University of Science & Technology, Beishan St. Yanji, Jilin 133000, China2Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA3Centre for Biomedical Engineering (CBME) and School of Electrical and Electronic Engineering,The University of Adelaide, SA 5005, Australia(Received 5 August 2002; published 24 November 2003)220601-1We develop an analytically solvable three-state discrete-time minimal Brownian ratchet (MBR),where the transition probabilities between states are asymmetric. By solving the master equations, weobtain the steady-state probabilities. Generally, the steady-state solution does not display detailedbalance, giving rise to an induced directional motion in the MBR. For a reduced two-dimensionalparameter space, we find the null curve on which the net current vanishes and detailed balance holds. Asystem on this curve is said to be balanced. On the null curve, an additional source of external randomnoise is introduced to show that a directional motion can be induced under the zero overall driv-ing force.DOI: 10.1103/PhysRevLett.91.220601 PACS numbers: 05.40.CaFIG. 1. State-transition diagram of a three-state discrete-timeBrownian ratchet with asymmetric transition probabilities p0,p1, and p2 in the positive direction (counterclockwise) and1 p0, 1 p1, and 1 p2 in the negative direction(clockwise). Each transition has two numbers associated withit; fpk; Rkg. The first number in the brackets, pk, is the condi-tional probability of that transition (given the initial state). Thesecond number, Rk, is the reward associated with that transi-tion. Note that we have a skip-free process, which means thereward structure is 1 for ‘‘winning’’ transitions and 1 forthe null surface, in the parameter space, of the noisy and ‘‘losing’’ transitions.The Brownian ratchet and pawl system was first cor-rectly explained by Smoluchowski [1] and later revisitedby Feynman [2]—this has inspired much activity in thearea of Brownian ratchets, despite flaws in Feynman’sanalysis of the thermal efficiency of the ratchet engine [3]and detailed balance [4].Interest has revived because molecular motors [5] havebeen described in terms of Brownian ratchet [6,7] models.Another area of interest has been in Parrondo’s paradox[8], where losing strategies cooperate to win. This can beillustrated in terms of games that lose when played indi-vidually, but win when alternated—this has been shownto be a discrete-time Brownian ratchet [9], otherwiseknown as a ‘‘Parrondian game.’’ Parrondo’s games havesignificantly sparked recent interest in the areas of latticegas automata [10], spin models [11], random walks anddiffusions [12–14], biogenesis [15], molecular transport[16,17], noise induced patterns [18], stochastic control[19,20], stochastic resonance [21], and quantum gametheory [22,23]. Recently, Reimann [24] has performedan extensive review of the ratchet field.Jarzynski et al. [25] developed an exactly solvableBrownian ratchet that can be operated as a heating systemor refrigerator, depending on the parameters between twoheat reservoirs of different temperatures. However, this istreated as a six-state system and solution is via matrixinversion of coupled linear equations. The derivation issomewhat complex, so the physical picture and key in-gredients of the observed properties are obscured.Westerhoff et al. [26] have analyzed enzyme transportusing a four-state model. In this paper, for the first time,we develop a three-state discrete-time Brownian ratchetmodel that can be solved analytically. We call it theminimal Brownian ratchet (MBR) [27]. By setting upand solving the steady-state solution of the correspondingmaster equations, we obtain the probability current and0031-9007=03=91(22)=220601(4)$20.00 noise-free MBR. The obtained solution does not show anycritical behavior and can be suitably explained in terms ofnonsingular behaviors.The minimal ingredients of a Brownian ratchet are anasymmetric potential and random noise. In Fig. 1, weshow the state diagram of the MBR. The MBR hasthree states, fS0; S1; S2g, where the transition probabilitiesbetween states are asymmetric. The transition probabilitythat a random walker in state Sk steps in the positive2003 The American Physical Society 220601-1P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22direction is pk. The probability of a shift in the negativedirection is ~pk. This is true for k 2 f0; 1; 2g. We define thepositive direction as counterclockwise. The condition ofnormalization, pk  ~pk  1, is automatically enforced byour choice of symbols. These ingredients comprise athree-state random walk model with generalized asym-metric potential and we call it a noise-free MBR.It is straightforward to set up the following differenceequations for the probability distributions of the noise-free MBR model:Pkt 1  Pk1t~pk1  Pk2tpk2; (1)for all cyclic (modulo-3) state indices k. Pkt is theprobability for the random walker at time t to be on thestate of Sk. This can be written in matrix form as Pt1 PtB, where Pt is the time varying probability (row) vectorat time t and B is the transition probability matrix.We canwrite	Bi;j 2640 p0 ~p0~p1 0 p1p2 ~p2 0375:The steady-state probability, after a sufficiently longtime, limt!1Pt  P is simply given asP  PB; (2)which is a characteristic value problem. A partial proba-bility current, I, can be defined asI  Pkpk  Pk1 ~pk1: (3)If I  0, there is no net current and detailed balance [28]is satisfied, otherwise there exists a net current and thesystem will assume a nonequilibrium steady state.Solving Eq. (2) together with the normalization con-dition, P0  P1  P2  1, is again straightforward.Using the standard methods for characteristic value oreigenvalue problems, we obtainPk  ~pk1  pk1pk2=D; (4)for all k. The denominator D is given asD  2 p0p1p2  ~p0 ~p1 ~p2: (5)These expressions are consistent with the results ofPearce [29]. It is easy to check that they are the solutionto Eq. (2) by direct substitution.We can substitute the results from Eq. (4) into Eq. (3) tosolve for the net current, I:I  p0p1p2  ~p0 ~p1 ~p2=D: (6)The condition for detailed balance I  0 is thenp0p1p2  ~p0 ~p1 ~p2; (7)which is the equation of a two-dimensional surface in thethree-dimensional parameter space, fp0; p1; p2g. Note220601-2that Eq. (6) is independent of state index k as is requiredby its definition given in Eq. (3).The second part of the MBR is to introduce additionalrandom noise to the system. To the noise-free MBR, weadd more noise, controlled by the parameter , to theMBR as follows.With a probability of ~, a random walkerfollows the dynamic rule of the noise-free MBR other-wise, with the probability of  (1 ~), the walkerrandomly takes a right or left step with the equal proba-bility of a half. For   0, the model is exactly the samewith the noise-free MBR. In the other limit, for   1,the randomizing process dominates and the system re-duces to a simple unbiased random walk where the netcurrent remains zero. It is important to note that  influ-ences the level of noise in the ratchet but is not identicalwith the noise itself. We refer to  as a ‘‘noise parameter.’’With this modification the transition probability matrix Bchanges as	Bi;j 2640 ~p0  =2 ~~p0  =2~~p1  =2 0 ~p1  =2~p2  =2 ~~p2  =2 0375:From the transition matrix, we know that adding therandom noise with parameter  effectively changes theexisting parameters aspk ! ~pk  =2; (8)and the same holds for ~pk. The steady-state solution andnet current for noisy MBR can be obtained by exchangingall pk in Eqs. (4) and (6) according to Eq. (8). Theexpression for current is given asI  	~3A  ~=21 =2B=D; (9)whereD  2 ~2A  ~=21 =2;A  p0p1p2  ~p0 ~p1 ~p2;B  p0  p1  p2  ~p0  ~p1  ~p2:(10)It is possible to further restrict the choices offp0; p1; p2g without losing the important properties ofthe ratchet. Parrondo’s original definition imposed thefurther constraints p0  q and p1  p2  p. This re-duced the parameter space to a two-dimensional spacewith parameters fp; qg. In two-dimensional fp; qg pa-rameter space, the condition of detailed balance, i.e.,Eq. (7), gives the equation for a curve that we call thenull curve:q  ~p2p2  ~p2 1 p2p2  1 p2 : (11)The null curve is a special case of the more general nullsurface or null hypersurface, in higher dimensions.Figure 2 shows the ‘‘positive’’ and ‘‘negative’’ net current220601-20.0 0.2 0.4 0.6 0.8 1.0p0.00.20.40.60.81.0q null surface, I=0postive region, I>0negative region, I<0FIG. 2. The null surface of a three-state discrete-time Brown-ian ratchet. On the null surface, q  1 p2=	p2  1 p2,the current vanishes. Above the curve, the system has a posi-tive net current. Below the curve, the system has a negative netcurrent.0.0 0.2 0.4 0.6 0.8 1.0noise, γ−0.4−0.20.00.20.4current, I γp=0.9p=0.7p=0.5p=0.3p=0.1FIG. 3. The probability current, I, versus noise parameter,, on the null surface. For values of p  0:5, additionalnoise induces a net current that increases, in magnitude, withincreasing  and then decreases, in magnitude, to zero after exceeds an optimum value. The bottom curve correspondsto p  0:1. All the other curves represent increments ofp  0:1. The middle curve corresponds to p  0:5. The topcurve corresponds to p  0:9. Parrondo’s original games hadp  0:75.0.0 0.2 0.4 0.6 0.8 1.0noise, γ−0.2−0.10.00.10.2current, I γp77,q12p76,q11p75,q10p74,q09p73,q08FIG. 4. The probability current I versus noise parameter off the null surface. The bottom curve corresponds to p  0:73and q  0:08 and the top to p  0:77 and q  0:12.P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22regions of the noise-free MBR. Note that as expectedfrom the symmetry of the system the curve is invariantunder the transformations q! 1 q and p! 1 p.This is also apparent from a consideration of Eq. (7).On the null surface, we introduce additional randomnoise to the system by controlling the value of . For  0, the model is exactly the same as the noise-freeMBR, and the net current remains zero since we are onthe null surface. In the other limit, for   1, the random-izing process dominates and the system reduces to asimple unbiased random walk where the net current isalso zero. However, counterintuitively, for 0< < 1 non-zero current is induced by introducing random noisecontrolled by .In Fig. 3, we show the current versus noise parameter,, for different values of parameters p and q  ~p2=p2  ~p2. As  is increased from zero, the currentincreases to a maximum and then falls off, which hasthe form of stochastic resonance [21]. The position ofthis extremum can be obtained from the condition,@I=@  0.  varies a little from 0.408 for p  0:1 to0.423 for p  0:5.Figure 4 shows the net current, I, versus noise  whenthe system is not on the null surface any more. In theoff-balance region, q  ~p2=p2  ~p2, the net current isnot zero for   0 but still should be zero for   1 andthe intermediate behavior is qualitatively the same as thebalanced behavior. The actual values of p and q for thevarious curves in Fig. 4 are in linear increments of0.01 for p and q. The top curve has parameters p 0:77 and q  0:12. The bottom curve has parameters p 0:73 and q  0:08.We can generalize the MBR by introducing a bias intothe added noise. The walker takes a right step with220601-3probability of 0:5  and a left step with probability of0:5 . For   0 this noise introduces nonzero netcurrent. The new parameter, , is essentially a measureof the bias in the added noise. This generalized modelqualitatively shows the same behavior as the noisy MBR.We can generalize this model to a system of size Nby repeating the unit cell of modulo-3 N times with aperiodic boundary condition. In this case, the periodicpotential ensures pkt  pk3nt 8 n  0;1;2; . . . .Because of the normalization condition,PNk1 Pkt  1,the current will be reduced by a factor of N. Otherwise,220601-3P H Y S I C A L R E V I E W L E T T E R S week ending28 NOVEMBER 2003VOLUME 91, NUMBER 22the corresponding master equations and solutions areexactly the same as the minimal model.For different moduli, in principle, we can also set upthe master equations and solve them exactly by matrix in-version for the set of linear equations. It can be shownthat these results have qualitatively the same statisticalbehavior as the three-state MBR. Note that for even num-ber moduli there are oscillatory nonstationary solutions.The transformation in Eq. (8) tells us effectively thatthe MBR gives the same results as a biased random walk,where the transition probability is not symmetric butbiased. Although this analogy can be used to investigatethe characteristics of MBR, it is absolutely impossible todetermine whether the system is itself a biased randomwalk or an MBR by analyzing the result of measure-ments, without prior knowledge that the model is a com-bination of a balanced unbiased random walk and addedrandom external noise. This makes the MBR valuable forunderstanding the minimal features of the discrete-timeBrownian ratchet. The MBR has applicability in discrete-time processes where the transition probabilities do notfluctuate in time, such as in game and computation theorywhere transitions occur at precisely defined times.Funding from GTECH Australasia, Sir Ross and SirKeith Smith Fund, and the Australian Research Council(ARC) is gratefully acknowledged.22060[1] M. von Smoluchowski, Phys. Z. XIII, 1069 (1912).[2] R. P. Feynman, R. B. Leighton, and M. Sands, TheFeynman Lectures on Physics (Addison-Wesley,Reading, MA, 1963), Vol. 1, p. 46.1.[3] J. M. R. Parrondo and P. Espan˜ol, Am. J. Phys. 64, 1125(1996).[4] D. Abbott, B. R. Davis, and J. M. R. Parrondo, inProceedings of the 2nd International Conference on Un-solved Problems of Noise and Fluctuations (UPoN’99),Adelaide, Australia, 1999 (American Institute ofPhysics, New York, 2000), Vol. 511, p. 213.[5] R. D. Astumian and M. Bier, Phys. Rev. Lett., 72, 1766(1994).[6] C. R. Doering, Nuovo Cimento Soc. Ital. Fis. 17D, 685(1995).[7] M. O. Magnasco, Phys. Rev. Lett., 71, 1477 (1993).[8] J. M. R. Parrondo, in EEC HC&M Network onComplexity and Chaos (ERBCHRX-CT940546), ISI,Torino, Italy (1996); G. P. Harmer and D. Abbott,Nature (London) 402, 846 (1999); J. M. R. Parrondo,G. P. Harmer, and D. Abbott, Phys. Rev. Lett. 85, 5226(2000); G. P. Harmer and D. Abbott, Fluct. Noise Lett. 2,R71 (2002); G. P. Harmer, D. Abbott, P. G. Taylor, andJ. M. R. Parrondo, in Proceedings of the 2nd Inter-national Conference on Unsolved Problems of Noise andFluctuations (UPoN’99), Adelaide, Australia, 1999(Ref. [4], p. 189).1-4[9] G. P. Harmer and D. Abbott, Stat. Sci. 14, 206 (1999).[10] D. A. Meyer and H. Blumer, J. Stat. Phys. 107, 225(2002).[11] H. Moraal, J. Phys. A 33, L203 (2000).[12] E. S. Key, M. M. Kłosek, and D. Abbott, math.PR/0206151.[13] R. Pyke, math.PR/0206150.[14] B. Cleuren and C. Van den Broeck, Phys. Rev. E 65,030101 (2002).[15] P. C.W. Davies, Lecture in honor of Abdus Salam, inProceedings of the Sixth Trieste Conference on ChemicalEvolution, Trieste, Italy, 2000, edited by J. Chela-Flores,T. Owen, and F. Raulin (Kluwer, Dordrecht, 2001),pp. 13–20.[16] D. Kinderlehrer and M. Kowalczyk, Arch. Ration. Mech.Anal. 161, 299 (2002).[17] D. Heath, D. Kinderlehrer, and M. Kowalczyk, DiscreteContinuous Dynam. Syst. B 2, 153 (2002).[18] J. Buceta, K. Lindenberg, and J. M. R. Parrondo, Fluct.Noise Lett. 2, L21 (2002); J. Buceta, K. Lindenberg, andJ. M. R. Parrondo, Phys. Rev. Lett. 88, 024103 (2002).[19] A. Allison and D. Abbott, Chaos 11, 715 (2001).[20] L. Kocarev and Z. Tasev, Phys. Rev. E 65, 046215 (2002).[21] A. Allison and D. Abbott, Fluct. Noise Lett. 1, L239(2001).[22] A. P. Flitney, J. Ng, and D. Abbott, Physica (Amsterdam)314A, 35 (2002).[23] C. F. Lee and N. F. Johnson, quant-ph/0203043/.[24] P. Reimann, Phys. Rep. 361, 57 (2002).[25] C. Jarzynski and O. Mazonka, Phys. Rev. E 59, 6448(1999).[26] H.V. Westerhoff, T.Y. Tsong, P. B. Chock, Y. Chen, andR. D. Astumian, Proc. Natl. Acad. Sci. U.S.A. 83, 4734(1986).[27] Although a two-state system is possible, it cannot modelthe class of skip-free discrete-time ratchet systems de-scribed by Parrondo’s games. This is because two statesdo not provide the necessary asymmetry for theParrondo effect. We can obtain a winning game fromtwo losing games in a two-state system, if we violatethe skip-free condition. A skip-free process impliespayoffs in steps of 1 or 1. The significance is thatparticles in a continuous-time Brownian ratchet gener-ally do not skip over multiple teeth at once — ourdiscrete-time model reflects this by imposing the skip-free condition. This is why our model can be regarded asa minimal discrete-time ratchet.[28] Notice that Eq. (7) is a sufficient condition for detailedbalance only when the transition probabilities do notdepend on time, which is our case. Equation (7) isanalogous to a corollary of the Onsager condition forcircular chemical reactions at equilibrium: L. Onsager,Phys. Rev. 37, 405 (1931). Also see R. Dean Astumianet al., Phys. Rev. A, 39, 6416 (1989).[29] C. E. M. Pearce, in Proceedings of the 2nd InternationalConference on Unsolved Problems of Noise and Fluctua-tions (UPoN’99), Adelaide, Australia, 1999 (Ref. [4],p. 207).220601-4

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