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arXiv:cond-mat/0302324v1 cond-mat.stat-mech 17 Feb 2003Parrondos games as a discrete ratchetRaul Toral1, Pau Amengual1 and and Sergio Mangioni21 Instituto Mediterraneo de Estudios Avanzados, IMEDEA (CSIC-UIB),ed. Mateu Orfila, Campus UIB, E-07122 Palma de Mallorca, Spain2 Departament de Fsica, Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata, Dean Funes 3350, 7600 Mar del Plata, ArgentinaWe write the master equation describing the Parrondos games as a consistent discretizationof the FokkerPlanck equation for an overdamped Brownian particle describing a ratchet. Ourexpressions, besides giving further insight on the relation between ratchets and Parrondos games,allow us to precisely relate the games probabilities and the ratchet potential such that periodicpotentials correspond to fair games and winning games produce a tilted potential.PACS numbers: 05.10.Gg, 05.40.Jc, 02.50.LeThe Parrondos paradox1, 2 shows that the alternation of two losing games can lead to a winning game. Thissurprising result is nothing but the translation into the framework of very simple gambling games of the ratcheteffect3. In particular, the flashing ratchet4, 5 can sustain a particle flux by alternating two relaxational potentialdynamics, none of which produces any net flux. Despite that this qualitative relation between the Parrondo paradoxand the flashing ratchet has been recognized from the very beginning (and, in fact, it constituted the source ofinspiration for deriving the paradoxical games), only very recently there has been some interest in deriving exactrelations between both6, 7.In this paper, we rewrite the master equation describing the evolution of the probabilities of the different outcomesof the games in a way that shows clearly its relation with the FokkerPlanck equation for the flashing ratchet. In thisway, we are able to give an expression for the dynamical potentials in terms of the probabilities defining the games,as well as an expression for the current. Similarly, given a ratchet potential we are able to construct the games thatcorrespond to that potential.The Parrondos paradox considers a player that tosses different coins such that a unit of “capital” is won (lost)if heads (tails) show up. Although several possibilities have been proposed8, 9, 10, 11, 12, 13, 14 , in this paperwe consider the original and easiest version in which the probability of winning, pi, depends on the actual value ofthe capital, i, modulus a given number L. A game is then completely specified by giving the set or probabilitiesp0,p1,.,pL1 from which any other value pk can be derived as pk = pkmodL. A fair game, one in which gainsand losses average out, is obtained if producttextL1i=0 pi = producttextL1i=0 (1pi). The paradox shows that the alternation (eitherrandom or periodic) of two fair games can yield a winning game. For instance, the alternation of game A defined bypi p = 1/2,i, and game B defined by L = 3 and p0 = 1/10,p1 = p2 = 3/4 produces a winning game althoughboth A and B are fair games.A discrete time can be introduced by considering that every coin toss increases by one. If we denote by Pi()the probability that at time the capital is equal to i, we can write the general master equationPi( + 1) = ai1Pi1() +ai0Pi() +ai1Pi+1() (1)where ai1 is the probability of winning when the capital is i1, ai1 is the probability of losing when the capitalis i + 1, and, for completeness, we have introduced ai0 as the probability that the capital i remains unchanged (apossibility not considered in the original Parrondo games). Note that, in accordance with the rules described before,we have taken that the probabilitiesai1,ai0,ai1do not depend on time. It is clear that they satisfy:ai+11 +ai0 +ai11 = 1 (2)which ensures the conservation of probability: summationtexti Pi( + 1) = summationtexti Pi().It is a matter of straightforward algebra to write the master equation in the form of a continuity equation:Pi( + 1)Pi() =Ji+1(t)Ji(t) (3)where the current Ji() is given by:Ji() = 12 FiPi() +Fi1Pi1()DiPi()Di1Pi1() (4)andFi = ai+11 ai11 , Di = 12(ai+11 +ai11 ) (5)2This form is a consistent discretization of the FokkerPlank equation15 for a probability P(x,t)P(x,t)t =J(x,t)x (6)with a currentJ(x,t) = F(x)P(x,t)D(x)P(x,t)x (7)with general drift, F(x), and diffusion, D(x). If t and x are, respectively, the time and space discretization steps,such that x = ix and t = t, it is clear the identificationFi txF(ix), Di t(x)2D(ix) (8)The discrete and continuum probabilities are related by Pi() P(ix,t)x and the continuum limit canbe taken by considering that M = limt0,x0(x)2t is a finite number. In this case Fi M1xF(ix) andDiM1D(ix).From now on, we consider the case ai0 = 0. Since pi = ai+11 we haveDiD = 1/2 Fi =1+ 2pi (9)and the current Ji() =(1pi)Pi() +pi1Pi1() is nothing but the probability flux from i1 to i.The stationary solutions Psti can be found solving the recurrence relation derived from (4) for a constant currentJi = J with the boundary condition Psti = Psti+L:Psti = NeVi/D12JNisummationdisplayj=1eVj/D1Fj (10)with a currentJ = N eVL/D12summationtextLj=1 eVj/D1Fj(11)N is the normalization constant obtained fromsummationtextL1i=0 Psti = 1. In these expressions we have introduced the potentialVi in terms of the probabilities of the games16Vi =Disummationdisplayj=1lnbracketleftbigg1 +Fj11Fjbracketrightbigg=Disummationdisplayj=1lnbracketleftbigg pj11pjbracketrightbigg(12)The case of zero current J = 0, implies a periodic potential VL = V0 = 0. This reproduces again the conditionproducttextL1i=0 pi =producttextL1i=0 (1pi) for a fair game. In this case, the stationary solution can be written as the exponentialof the potential Psti = NeVi/D. Note that Eq.(12) reduces in the limit x 0 to V (x) = M1integraltext F(x)dxor F(x) = M V(x)x , which is the usual relation between the drift F(x) and the potential V (x) with a mobilitycoefficient M.The inverse problem of obtaining the game probabilities in terms of the potential requires solving Eq. (12) withthe boundary condition F0 = FL17:Fi = (1)ieVi/DsummationtextLj=1(1)jeVj/DeVj1/D(1)Le(V0VL)/D1 +isummationdisplayj=1(1)jeVj/DeVj1/D (13)These results allow us to obtain the stochastic potential Vi (and hence the current J) for a given set of probabilitiesp0,.,pL1, using (12); as well as the inverse: obtain the probabilities of the games given a stochastic potential,using (13). Note that the game resulting from the alternation, with probability , of a game A with pi = 1/2,i and3-1.5-1-0.500.511.5-20 -15 -10 -5 0 5 10 15 20V(x)x-1.5-1-0.500.511.5-20 -15 -10 -5 0 5 10 15 20V(x)xFIG. 1: Left panel: potential Vi obtained from (12) for the fair game B defined by p0 = 1/10, p1 = p2 = 3/4. Right panel:potential for game B, with p0 = 3/10, p1 = p2 = 5/8 resulting from the random alternation of game B with a game A withconstant probabilities pi = p = 1/2, i.a game B defined by the setp0,.,pL1has a set of probabilitiesp0,.,pL1with pi = (1)12 +pi. For theFis variables, this relation yields:Fi = Fi, (14)and the related potential V follows from (12).We give now two examples of the application of the above formalism. In the first one we compute the stochasticpotentials of the fair game B and the winning game B, the random combination with probability = 1/2 of game Band a game A with constant probabilities, in the original version of the paradox1. The resulting potentials are shownin figure 1. Notice how the potential of the combined game clearly displays the asymmetry under space translationthat gives rise to the winning game.-1.5-1-0.500.511.5-40 -30 -20 -10 0 10 20 30 40V(x)x-1.5-1-0.500.511.5-40 -30 -20 -10 0 10 20 30 40V(x)xFIG. 2: Left panel: Ratchet potential (15) in the case L = 9, A = 1.3. The dots are the discrete values Vi = V (i) used inthe definition of game B. Right panel: discrete values for the potential V i for the combined game B obtained by alternatingwith probability = 1/2 games A and B. The line is a fit to the empirical form V (x) = x + V (x) with = 0.009525, = 0.4718.The second application considers as input the potentialV (x) = Abracketleftbiggsinparenleftbigg2pixLparenrightbigg+ 14 sinparenleftbigg4pixLparenrightbiggbracketrightbigg(15)which has been widely used as a prototype for ratchets3. Using (13) we obtain a set of probabilitiesp0,.,pL1by discretizing this potential with x = 1, i.e. setting Vi = V (i). Since the potential V (x) is periodic, the resulting4game B defined by these probabilities is a fair one and the current J is zero. Game A, as always is defined bypi = p = 1/2,i. We plot in figure 2 the potentials for game B and for the game B, the random combination withprobability = 1/2 of games A and B. Note again that the potential Vi is tilted as corresponding to a winning gameB. As shown in figure 3, the current J depends on the probability for the alternation of games A and B.05e-061e-051.5e-052e-050 0.2 0.4 0.6 0.8 1JFIG. 3: Current J resulting from equation (11) for the game B as a function of the probability of alternation of games Aand B. Game B is defined as the discretization of the ratchet potential (15) in the case A = 0.4, L = 9. The maximum gaincorresponds to = 0.57.In summary, we have written the master equation describing the Parrondos games as a consistent discretization ofthe FokkerPlanck equation for an overdamped Brownian particle. In this way we can relate the probabilities of thegamesp0,.,pL1to the dynamical potential V (x). Our approach yields a periodic potential for a fair game anda tilted potential for a winning game. The resulting expressions, in the limit x 0 could be used to obtain theeffective potential for a flashing ratchet as well as its current.The work is supported by MCyT (Spain) and FEDER, projects BFM2001-0341-C02-01, BMF2000-1108. P.A. issupported by a grant from the Govern Balear.1 G.P. Harmer and D. Abbott, Nature 402, 864 (1999).2 G.P. Harmer and D. Abbott, Fluctuations and Noise Letters 2, R71 (2002).3 P. Reimann, Phys. Rep. 361, 57 (2002).4 R.D. Astumian and M. Bier, Phys. Rev. Lett. 72, 1766 (1994).5 J. Prost, J.F. Chauwin, L. Peliti and A. Ajdari, Phys. Rev. Lett. 72, 2652 (1994).6 A. Allison and D. Abbott, The Physical Bas