arXiv:cond-mat/0302324v1 cond-mat.stat-mech 17 Feb 2003 Parrondos games as a discrete ratchet Ra ul Toral1, Pau Amengual1and and Sergio Mangioni2 1 Instituto Mediterr aneo de Estudios Avanzados, IMEDEA (CSIC-UIB), ed. Mateu Orfi la, Campus UIB, E-07122 Palma de Mallorca, Spain 2 Departament de F sica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, De an Funes 3350, 7600 Mar del Plata, Argentina We write the master equation describing the Parrondos games as a consistent discretization of the FokkerPlanck equation for an overdamped Brownian particle describing a ratchet.Our expressions, 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 periodic potentials correspond to fair games and winning games produce a tilted potential. PACS numbers: 05.10.Gg, 05.40.Jc, 02.50.Le The Parrondos paradox1, 2 shows that the alternation of two losing games can lead to a winning game. This surprising result is nothing but the translation into the framework of very simple gambling games of the ratchet eff ect3. In particular, the fl ashing ratchet4, 5 can sustain a particle fl ux by alternating two relaxational potential dynamics, none of which produces any net fl ux. Despite that this qualitative relation between the Parrondo paradox and the fl ashing ratchet has been recognized from the very beginning (and, in fact, it constituted the source of inspiration for deriving the paradoxical games), only very recently there has been some interest in deriving exact relations between both6, 7. In this paper, we rewrite the master equation describing the evolution of the probabilities of the diff erent outcomes of the games in a way that shows clearly its relation with the FokkerPlanck equation for the fl ashing ratchet. In this way, we are able to give an expression for the dynamical potentials in terms of the probabilities defi ning the games, as well as an expression for the current. Similarly, given a ratchet potential we are able to construct the games that correspond to that potential. The Parrondos paradox considers a player that tosses diff erent 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 paper we consider the original and easiest version in which the probability of winning, pi, depends on the actual value of the capital, i, modulus a given number L. A game is then completely specifi ed by giving the set or probabilities p0,p1,.,pL1 from which any other value pkcan be derived as pk= pk modL. A fair game, one in which gains and losses average out, is obtained if QL1 i=0 pi= QL1 i=0 (1 pi).The paradox shows that the alternation (either random or periodic) of two fair games can yield a winning game. For instance, the alternation of game A defi ned by pi p = 1/2, i, and game B defi ned by L = 3 and p0= 1/10,p1= p2= 3/4 produces a winning game although both 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 equation Pi( + 1) = ai 1Pi1() + a i 0Pi() + a i 1Pi+1() (1) where ai 1 is the probability of winning when the capital is i 1, ai 1 is the probability of losing when the capital is i + 1, and, for completeness, we have introduced ai 0 as the probability that the capital i remains unchanged (a possibility not considered in the original Parrondo games). Note that, in accordance with the rules described before, we have taken that the probabilities ai 1,ai0,ai1 do not depend on time. It is clear that they satisfy: ai+1 1 + ai 0+ a i1 1 = 1(2) which ensures the conservation of probability: P iPi( + 1) = P iPi(). 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() = 1 2 FiPi() + Fi1Pi1() DiPi() Di1Pi1()(4) and Fi= ai+1 1 ai1 1 ,Di= 1 2(a i+1 1 + ai1 1 )(5) 2 This 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 current J(x,t) = F(x)P(x,t) D(x)P(x,t) x (7) with general drift, F(x), and diff usion, 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 identifi cation Fi t xF(ix), Di t (x)2 D(ix)(8) The discrete and continuum probabilities are related by Pi() P(ix,t)x and the continuum limit can be taken by considering that M =lim t0,x0 (x)2 t is a fi nite number. In this case Fi M1xF(ix) and Di M1D(ix). From now on, we consider the case ai 0= 0. Since pi= a i+1 1 we have Di D = 1/2Fi= 1 + 2pi(9) and the current Ji() = (1 pi)Pi() + pi1Pi1 () is nothing but the probability fl ux from i 1 to i. The stationary solutions Pst i can be found solving the recurrence relation derived from (4) for a constant current Ji= J with the boundary condition Pst i = Pst i+L: Pst i = NeVi/D 1 2J N i X j=1 eVj/D 1 Fj (10) with a current J = N eVL/D 1 2 PL j=1 eVj/D 1Fj (11) N is the normalization constant obtained from PL1 i=0 Pst i = 1. In these expressions we have introduced the potential Viin terms of the probabilities of the games16 Vi= D i X j=1 ln ?1 + F j1 1 Fj ? = D i X j=1 ln ? pj1 1 pj ? (12) The case of zero current J = 0, implies a periodic potential VL= V0= 0. This reproduces again the condition QL1 i=0 pi= QL1 i=0 (1 pi) for a fair game. In this case, the stationary solution can be written as the exponential of the potential Pst i = NeVi/D.Note that Eq.(12) reduces in the limit x 0 to V (x) = M1 R F(x)dx or F(x) = M V (x) x , which is the usual relation between the drift F(x) and the potential V (x) with a mobility coeffi cient M. The inverse problem of obtaining the game probabilities in terms of the potential requires solving Eq. (12) with the boundary condition F0= FL17: Fi= (1)ieVi/D PL j=1(1) jeVj/D eVj1/D (1)Le(V0VL)/D 1 + i X j=1 (1)jeVj/D eVj1/D (13) These results allow us to obtain the stochastic potential Vi(and hence the current J) for a given set of probabilities p0,.,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 and 3 -1.5 -1 -0.5 0 0.5 1 1.5 -20 -15 -10-505101520 V(x) x -1.5 -1 -0.5 0 0.5 1 1.5 -20 -15 -10-505101520 V(x) x FIG. 1:Left panel: potential Vi obtained from (12) for the fair game B defi ned 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 with constant probabilities pi= p = 1/2, i. a game B defi ned by the set p0,.,pL1 has a set of probabilities p0,.,p L1 with p i= (1) 1 2 +pi. For the Fis variables, this relation yields: F i = Fi,(14) and the related potential V follows from (12). We give now two examples of the application of the above formalism. In the fi rst one we compute the stochastic potentials of the fair game B and the winning game B, the random combination with probability = 1/2 of game B and a game A with constant probabilities, in the original version of the paradox1. The resulting potentials are shown in fi gure 1. Notice how the potential of the combined game clearly displays the asymmetry under space translation that gives rise to the winning game. -1.5 -1 -0.5 0 0.5 1 1.5 -40 -30 -20 -10010203040 V(x) x -1.5 -1 -0.5 0 0.5 1 1.5 -40 -30 -20 -10010203040 V(x) x FIG. 2:Left panel: Ratchet potential (15) in the case L = 9, A = 1.3. The dots are the discrete values Vi= V (i) used in the defi nition of game B. Right panel: discrete values for the potential V i for the combined game Bobtained by alternating with probability = 1/2 games A and B. The line is a fi t to the empirical form V (x) = x + V (x) with = 0.009525, = 0.4718. The second application considers as input the potential V (x) = A ? sin ?2x L ? + 1 4 sin ?4x L ? (15) which has been widely used as a prototype for ratchets3. Using (13) we obtain a set of probabilities p0,.,pL1 by discretizing this potential with x = 1, i.e. setting Vi= V (i). Since the potential V (x) is periodic, the resulting 4 game B defi ned by these probabilities is a fair one and the current J is zero. Game A, as always is defi ned by pi = p = 1/2, i. We plot in fi gure 2 the potentials for game B and for the game B, the random combination with probability = 1/2 of games A and B. Note again that the potential V i is tilted as corresponding to a winning game B . As shown in fi gure 3, the current J depends on the probability for the alternation of games A and B. 0 5e-06 1e-05 1.5e-05 2e-05 00.20.40.60.81 J FIG. 3:Current J resulting from equation (11) for the game Bas a function of the probability of alternation of games A and B. Game B is defi ned as the discretization of the ratchet potential (15) in the case A = 0.4, L = 9. The maximum gain corresponds to = 0.57. In summary, we have written the master equation describing the Parrondos games as a consistent discretization of the FokkerPlanck equation for an overdamped Brownian particle. In this way we can relate the probabilities of the games p0,.,pL1 to the dynamical potential V (x). Our approach yields a periodic potential for a fair game and a tilted potential for a winning game. The resulting expressions, in the limit x 0 could be used to obtain the eff ective potential for a fl ashing ratchet as well as its current. The work is supported by MCyT (Spain) and FEDER, projects BFM2001-0341-C02-01, BMF2000-1108. P.A. is supported 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 Basis fo