Parrondo’s games as a discrete ratchet Pau Amengual Raúl Toral Instituto Mediterráneo de Estudios Avanzados - IMEDEA Universitat de les Illes Balears – UIB SPAIN

Outline Introduction: flashing ratchet. Original Parrondo’s games. Other classes of games. Cooperative games Relation between Parrondo’s games and ratchets. Coupled ratchets and coupled games. Conclusions

1.Introduction Brownian Motors The system must be out System subjected To thermal noise Transport phenomena in small-scale systems Two basic features are needed for the existence of directed transport : The system must be out of its equilibrium state Breaking of thermal equilibrium: Accomplished either through stochastic or periodic forcing : F(t) Breaking of spatial inversion symmetry Ratchet potential : it consists of a periodic and asymmetric potential

Flashing ratchet : Potential switched on and off periodically or stochastically with a flip rate . Ratchet Potential On P(x) x x Diffusion P(x) x x Ratchet Potential On Net Motion P(x) VB(x) x x

2. Original Parrondo’s games Game A : Game B (Capital dependent) YES Capital multiple of three ? NO Both games are losing when played separatedly. Either periodic or random alternation between both games gives as a result a winning game .

Random case Periodic case Plays game A The player, with probability Plays game B Periodic case The player alternates between game A and B following a given Sequence of plays. Average gain of a single player versus time with a value of The simulations were averaged over 50000 ensembles.

3.Other classes of games Parrondo’s games with self-transition In this class of games a new probability is introduced : self-transition probability. The player has a probability distinct from zero of remaining with the same capital after a round played p =9/20 - ε, r=1/10, p1=9/100 – ε, r1=1/10, p2=3/5 – ε, r2=1/5 and ε=1/500

Winning Probabilities History dependent games Game A : Parrondo et al. PRL 85, 24 2000 Game B: Time step t-2 Time step t-1 Winning Probabilities Losing Probabilities Loss p1 1-p1 Win p2 1-p2 p3 1-p3 p4 1-p4 Simulations are carried out with ε = 0.003 and averaging over 500 000 ensembles. The probabilities are p1=9/10 - ε, p2 = p3 = 1/4 - ε, p4 = 7/10 - ε

Cooperative games Capital redistribution between players New versions for game A are presented : Game A’ : A player chosen randomly gives away one unit of capital to a randomly selected player Game A’’ : A player chosen randomly gives away one unit of capital to ny of its nearest neighbours. Probability proportional to the capital difference. Games B and B’ : We will use either original game B, or the history dependent game B’ with probabilities p1 = 0.9 - ε p2 = p3 = 0.25 - ε p4 = 0.7 - ε ε = 0.01

Average capital per player Time evolution of the variance of the single player capital distribution

Cooperative games Game A : Game B: Ensemble of interacting players. They chose either game A or game B randomly, i.e., with probability  . Game A : Rules of chosing between probabilities for game B depend on the state of the neighbour’s player Game B: Player site i-1 Player site i+1 Winning Probabilities Losing Probabilities Loser p1 1-p1 Winner p2 1-p2 p3 1-p3 p4 1-p4 The probabilities used for the simulation are p = 0.5, p1 = 1, p2 = p3 = 0.16, p4 = 0.7

4. Relation between Parrondo’s games and Brownian ratchet Additive noise : We have the following Master Equation for the evolution of the capital i of the player at the  th coin tossed It can be rewritten as a continuity equation : where and

of fairness is fulfilled by the set of probabilities {pi,qi}, that is For the original games we have (ri=0) The current is then We can define a potential in the following way : This potential assures the periodicity of the potential when the condition of fairness is fulfilled by the set of probabilities {pi,qi}, that is

Some examples of potentials : Potential V(x) obtained from the probabilities defining game B Potential V’(x) obtained from the probabilities of the combined game A and B

For the stationary case ( Ji = ctant and Pi() = Pi) we obtain Plot of the current J vs the alternating probability between games 

is the numerical approx. to the integral using Simpson’s rule. The solutions obtained for Pn and J are equivalent to the continuous solution of the Langevin equation with additive noise As is the numerical approx. to the integral using Simpson’s rule.

Inverse problem Solving the equation for the potential with the boundary condition gives Last equation together with can be used to obtain the probabilities pi in terms of the potential Vi

Multiplicative noise : Now , which corresponds to the case of non-null self-transition probability. For this case and We can define an effective potential with the same properties than the previous one as For the stationary case we obtain for the probability and the current :

These expressions can be compared with the continuous solutions of the Langevin equation with multiplicative noise where

5. Coupled ratchets and coupled games Set of N players, we choose a random player and then : Gives 1 coin to a random player With probability Plays game B Master Equation for the joint PDF is : Due to the constant transition probabilities, we can obtain the ME governing the evolution of one player performing the following sum

The result is Solving the latter equation for the stationary case gives With a current

Capital redistribution between neigbours Comparison between the current obtained theoretically and numerically. The probabilities for game B are those of the original game. N = 50 players. Capital redistribution between neigbours Although the joint PDF function might be slightly different, the ME we obtain for a single player is the same as in the previous case, and so are the results. From the discrete solution for the stationary Pn that we have obtained, we can derive its corresponding Fokker-Planck Equation and then the Langevin equation, giving

Conclusions We have presented a consistent way of relating the master equation for the Parrondo games with the formalism of the Fokker–Planck equation describing Brownian ratchets. This relation works in two ways: we can obtain the physical potential corresponding to a set of probabilities defining a Parrondo game, as well as the current and its stationary probability distribution. Inversely, we can also obtain the probabilities corresponding to a given physical potential Our relations work both in the cases of additive noise or multiplicative noise Our next goal is to ellaborate a relation between the collective games and their associated collective ratchets.

Bibliography R. Toral, P. Amengual and S. Mangioni, Parrondo’s games as a discrete ratchet, Physica A 327 (2003). R. Toral, P. Amengual and S. Mangioni, A Fokker-Planck description for Parrondo’s games, Proc. SPIE Noise in complex systems and stochastic dynamics eds. L. Schimansky-Geier, D. Abbott, A. Neiman and C. Van den Broeck), Santa Fe, 5114 (2003). P. Amengual, A. Allison, R. Toral and D. Abbott, Discrete-time ratchets, the Fokker-Planck equation and Parrondo’s paradox, Proc. Roy. Soc. London A 460 (2004). R. Toral, Cooperative Parrondo’s games, Fluctuation and Noise Letters 1 (2001). R. Toral, Capital redistribution brings wealth by Parrondoís paradox, Fluctuation and Noise Letters 2 (2002). G. P. Harmer and D. Abbott, Losing strategies can win by Parrondo’s paradox, Nature 402 (1999) 864. G. P. Harmer and D. Abbott, A review of Parrondo’s paradox, Fluctuation and Noise Letters 2 (2002).