parrondo’s games as a discrete ratchet
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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 .
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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 .
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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 as a discrete ratchet. Leipzig. May 2005
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
Average capital per player
Time evolution of the variance of the single player capital distribution
i iii tC
NtC
Nt
222 11
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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 ,
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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.
Parrondo’s games as a discrete ratchet. Leipzig. May 2005
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).