Aplicaciones de la Paradoja de Parrondo Miguel Arizmendi Fac. Ingeniería Universidad Nacional de...
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Aplicaciones de laParadoja de Parrondo
Miguel ArizmendiFac. IngenieríaUniversidad Nacional de Mar del PlataArgentina
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Losing in order to win
Chess sacrifice bishop
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N/2 Wrongs Make a Right
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The Truel
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Motores Moleculares
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Parrondo’sParadoxical Games
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Biology is wet and dynamic. Molecules, subcellular organelles and cells,
inmersed in an aqueous environment, are in continous riotous motion.
H.C. Berg – Random Walks in Biology
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Games
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The small size of molecular machines means that their physics is dominatedby thermal fluctuations – macroscopic intuition is of limited use
Length scales
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Energy Scales
Thermal Energy
For proteins in water this energy is taken from collisions with water molecules
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Life at Low Reynolds Number
Reynold’s number: Re = vL/
v speed of the object – L characteristic length -
ρ liquid density and η viscosity
Example: fish vs. bacterium
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•fish of density approximately that of water ( = 1 gm/cc), length of 10 cm (L), moving at a velocity of 100 cm/sec (v) in
water ( = 0.01 g/cm sec), we calculate Re to be about 105.
•bacterium of the same density, length of 1 micron (L = 10-4 cm), moving at a velocity of 10-3 cm/sec through water,
we calculate Re to be 10-5.
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What about Proteins?
• fish we calculate Re to be about 105.
• bacterium we calculate Re to be 10-5.
• protein: size ~ 6 nm, speed 8m/s in water
Re ~ 0.05
Overdamped
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Quenched disorder effects on deterministic inertia ratchets
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Games
•Processes like this with no memory are called Markov Processes
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Random Walks and Diffusion
For short times and distances, diffusion is very fastK+ ion in water goes 1 micron in 0.25 ms, 0.1 mm in 0.25 sFor long time and distances, diffusion is very slow,K+ ion goes 1m in 8 years.
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Why bother moving?Rickettsia (tifus)
~ 100 years for mitochondrion synthesized in spinal chord to get to foot synapse. .
Active Transport is necessary: molecular motors
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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
Molecular Motor Model
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Thermodynamics Second Law?Thermodynamics Second Law?
Molecular Motor Model
Can a Net Current J be obtained from Noise?
Thermal Ratchet Model
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Feynman Lectures: Ratchet and Pawl
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Maxwell’s Demon
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i=1 i=2
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Flashing Ratchet Current
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- MA, JR Sanchez and F. Family: - MA, JR Sanchez and F. Family: PLAPLA 249,281 (’98)249,281 (’98) Physica A327Physica A327, 111 (2003), 111 (2003)
Flashing Ratchet Current / Entropy
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Thermal Ratchet:Not a very good molecular motor
model
• Force against viscous loads ~ 2kT/l ~ 1pN << 4-5pN (measured value)
•Diffuses in the right direction half of the time 2 molecules ATP hydrolized in average/step.• 1 step/ATP hydrolized for kinesin
(J. Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer, 2001)
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Highly diffusive, several ATP molecules hydrolyzed/step
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What about Games?
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win lose
¿Is X(t) a multiple of 3 ?
win lose win lose
Game A Game B
No Yes
: Player’s capital at -th run
(Fair games)
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No Yes
Is X(t) a multiple of 3?
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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.
-Amengual y Toral: -Amengual y Toral: 'Transfer of information in Parrondo's games‘'Transfer of information in Parrondo's games‘, , Fluctuation and Noise LettersFluctuation and Noise Letters 5, L63 (2005)
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The 2-girl paradox
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Leunberger´s volatility pumping
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How Often does the Parrondo EffectAppear?
G.C. Crisan, E. Nechita, M. Talmaciu, FNL 7, C19 (2007)
Game B (Capital dependent) bpwin :
cpwin :
Capital multiple of M ?
YES
NO
Notation: B: G(M,b,c), Original Parrondo: B: G(3,1/10-ε,3/4-ε)
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How Often does the Parrondo EffectAppear?
Probability that two randomly-chosen losing games A=G(3,a,a), and B=G(3,b,c) generate the triplet (A,B,1/2A+1/2B) that completesParrondo’s Paradox : 0.0306%.Parrondo effect quite unusual!Highest probability: 0.0537% when the mixing parameter α=0.173 andM=4
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Ensemble of interacting players. They chose either game A or game B randomly, i.e., with probability .
Cooperative games
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Reversals of Chance
Ensemble of N interacting players. They choose either game A or game B randomly, i.e., with probability .
2
1winpGame A :
Game B:
winners number
w
Winning Probabilities
w > [2N/3] p1
[N/3]< w ≤[2N/3] p2
w ≤ [N/3] p3
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Reversals of Chance
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Juegos con Memoria
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Matching Models
Consumers with specific wishes – Producers Employers – Job seekers Ph.D. Students - Supervisors
N men N women
Dating Game
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Das and Kamenica IJCAI 2005, 947Two Sided Bandits and the Dating Market
N men N women
Statistical decision model of an agent trying to optimizehis decisions while improving his information at the same time.
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Can Losers do Better?
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Game A:
Random man j Best valued woman i
mod(n)1jj withmatch otherwise
man valued best notj ifotherwise
matchingP
)(
2/1)(
Woman chooses greedily best valued man
Fair game for every man
(sparkling personality)
New Rules for Dating Game
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Game B:
Random man j Best valued woman i
)(N1)mod(j withmatch otherwise
man valued best notj if otherwise
Woman chooses greedily best valued man to match with
/ p and p ,p 32 1
Fair game for every man
Previous averagesparkling personality Trend follower?
New Rules for Dating Game
winners number
w
Winning Probabilities
w > [2N/3] p1
[N/3]< w ≤[2N/3] p2
w ≤ [N/3] p3
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N=4
Results:Parrondo Effect in Total Matches
2/1
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Results:Expected Payoff in Loser Matches
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N=3N=3 – Mixing Probability of games A and B
2/1
No change for losers
N=3N=3 – Mixing Probability of games A and B
5/1
....ABABAB
....ABBBBA
Losers do worse!
Not losers also do worse!
Results: How Many Players?
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