parrondo’s games as a discrete ratchet

Parrondo’s games as a discrete ratchet. Leipzig. May 2005

Parrondo’s games as a Parrondo’s games as a discrete ratchetdiscrete ratchet

Pau AmengualRaúl Toral

Instituto Mediterráneo de Estudios Avanzados - IMEDEAUniversitat de les Illes Balears – UIB

SPAIN


Outline

1. Introduction: flashing ratchet.2. Original Parrondo’s games.3. Other classes of games. Cooperative

games4. Relation between Parrondo’s games and

ratchets.5. Coupled ratchets and coupled games.6. Conclusions


1.Introduction

Brownian MotorsBrownian MotorsTransport phenomena in small-scale systems

System subjectedTo thermal noise

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


DiffusionDiffusion

x

Net Motion

Ratchet Potential OnRatchet Potential On

Ratchet Potential OnRatchet Potential On

x

x

VB(x)

P(x)

P(x)

x

x

x

P(x)

Flashing ratchet : Flashing ratchet : Potential switched on and off periodically or stochastically with a flip rate .


2. Original Parrondo’s games

Game A :

2

1)(winp

Game B (Capital dependent)

10

1)(winp

4

3)(winp

Capital multiple of three ?

YES

NO

2

1)(losep

Both games are losing when played separatedly. Either periodic or random alternation between both games gives as a result a winning game .


Average gain of a single player versus time with a value of The simulations were averaged over 50000 ensembles.

300

1

The player, with probability

)1(

Plays game A

Plays game B

Random case

Periodic case The player alternates between game A and B following a given Sequence of plays.


Parrondo’s games with self-transitionParrondo’s games with self-transition

3.Other classes of games

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


History dependent gamesHistory dependent games

Time step t-2 Time step t-1Winning

ProbabilitiesLosing

Probabilities

Loss Loss p1 1-p1

Loss Win p2 1-p2

Win Loss p3 1-p3

Win Win p4 1-p4

Parrondo et al. PRL 85, 24 2000

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

Game A :

Game B:

2

1)(winp


Cooperative gamesCooperative gamesCapital redistribution between players

New versions for game A are presented :

Games B and B’ :Games B and B’ :

Game A’ :Game A’ : A player chosen randomly gives away one unit of capital to a randomly selected player

Game A’’ :Game A’’ : A player chosen randomly gives away one unit of capital to ny of its nearest neighbours. Probability proportional to the capital difference.

1max1 ii CCiip

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

i iii tC

NtC

Nt

222 11


Ensemble of interacting players. They chose either game A or game B randomly, i.e., with probability .

2

1winp

Rules of chosing between probabilities for game B depend on the state of the neighbour’s player

Player site

i-1Player site i+1

Winning Probabilities

Losing Probabilities

Loser Loser p1 1-p1

Loser Winner p2 1-p2

Winner Loser p3 1-p3

Winner Winner p4 1-p4

Cooperative games

Game A :

Game B:

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 :

110111 ii

ii

ii

i PaPaPaP

We have the following Master Equation for the evolution of the capital i of the player at the th coin tossed

iiii JJPP 11

11112

1 iiiiiiiii PDPDPFPFJ

11

11

ii

i aaF 11

112

1 ii

i aaD

It can be rewritten as a continuity equation :

where

and


For the original games we have (ri=0)

111 iiiii PpPpJ

The current is then

We can define a potential in the following way :

i

k k

ki

k k

ki p

p

F

FV

1

1

1

1

1ln

2

1

1

1ln

2

1

This potential assures the periodicity of the potential when the condition of fairness is fulfilled by the set of probabilities {pi,qi}, that is

1

1

1

1

L

i

L

iii qp

11 i

i ap

2

1DDi

12 ii pF

00 ia


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

i

j j

VVst

i F

e

N

JeNP

ji

1

22

1

21

L

j j

V

V

Fe

eNJ

j

L

1

2

2

12

1

Plot of the current J vs the alternating probability between games


The solutions obtained for Pn and J are equivalent to the continuous solution of the Langevin equation with additive noise

'0

0

'

dxeD

JPexP

x

D

xV

D

xV

'

10

0

'

dxe

eDP

J L

D

xV

D

LV

As

i

j

FD

FF

F

i

j

FF

Fi

j j

D

V

j

i

k

jk

j

j

k k

kj

eeF

e

1

22

1

1ln1

1ln

1

1

0

1

1

1

is the numerical approx. to the integral using Simpson’s rule.

x

D

xV

dxe0

'

'

txDttxFx ,


Inverse problem

Solving the equation for the potential with the boundary condition gives

i

j

VVjVVL

L

j

VVj

Vii

jj

L

jj

i eee

ee

eF1

22

21

22

2 1

0

1

111

1

1

Last equation together with

2

1 ii

Fp

can be used to obtain the

probabilities pi in terms of the potential Vi

LFF 0


Multiplicative noise :

Now , which corresponds to the case of non-null self-transition probability.

We can define an effective potentialwith the same properties than the previous one as

For this case 12 iii rpF and ii rD 12

1

i

j

j

jj

j

j

i

r

rp

r

p

V1

1

1

1

1

1ln

For the stationary case we obtain for the probability and the current :

n

j D

F

n

v

n

stVst

n

j

j

j

n

D

eJ

D

PDeP

1 21

00

1

L

j D

F

V

VL

st

j

j

j

L

e

eDDPJ

1 21

00

1

00 ia


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

tttxDttxFx ,2,

x dxxdxx

st dxeJNxD

exP

x

x

'

'

''''

L dxx

dxx

dxe

eLDDP

J x

L

0

''''

'

00

'

0

0

xDxF

x where


5. Coupled ratchets and coupled games

Master Equation for the joint PDF is :

N

jnj

cn

cnj

c

N

N

j

N

jjj

njjNNn

tcccPatccPatcccPa

tccccPtccP

jjj

1111011

1

1''''

1'''''11

11

;,..,1,..;,..,;,..,1,..

;,..,1,..,1,..1;,..,

Due to the constant transition probabilities, we can obtain the ME governing the evolution of one player performing the following sum

Ncc

nj tccPtcP...

1

1

;,...;

Set of N players, we choose a random player and then :

Plays game B

Gives 1 coin to a random player

)1( With probability


tP

tPtPtPatPatPatP

iNN

NN

iiNii

ii

ii

Ni

211

111101111

The result is

Solving the latter equation for the stationary case gives

n

jk k

kn

j j

n

k k

kn q

p

q

JNP

q

pP

1

1

10

1

1

1

1

11

1

With a current

L

j

L

jkqp

q

L

kqp

N

k

k

j

k

kP

J

1 11

11

1

11

10

1

1

1 1


Comparison between the current obtained theoretically and numerically. The probabilities for game B are those of the original game. N = 50 players.

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

iiii xfx 1

Capital redistribution between neigbours


Conclusions

• Our relations work both in the cases of additive noise or multiplicative noise

• 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.

• 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.

• Inversely, we can also obtain the probabilities corresponding to a given physical potential

• Our next goal is to ellaborate a relation between the collective games and theirassociated 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).

parrondo’s games as a discrete ratchet

Documents