Discrete–time ratchets, the Fokker–Planck equation and Parrondo's paradox

Amengual, Pau; Allison, Andrew; Toral, Raúl; Abbott, Derek

doi:10.1098/rspa.2004.1283

Cited by 38 publications

(37 citation statements)

References 45 publications

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“…A useful picture for understanding how this works, is to notice that the states in one of the games are set up such that there is detailed balance when the game is played in isolation, but when it is randomly mixed with another losing game the symmetry of the balance is broken resulting in different equilibrium probabilities that can be designed to then result in a winning game. This interaction between symmetry breaking and random behavior is a ratchet effect and, in the case of Parrondo's games, this connection to ratchets has been formally established via discretization of the Fokker-Planck equation [35][36][37].…”

Section: The Two-envelope Switching Process As a Ratchetmentioning

confidence: 99%

The Two-Envelope Problem Revisited

2010

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The two-envelope problem has intrigued mathematicians for decades, and is a question of choice between two states in the presence of uncertainty. The problem so far, is considered open and there has been no agreed approach or framework for its analysis. In this paper we outline an elementary approach based on Cover's switching function that, in essence, makes a biased random choice where the bias is conditioned on the observed value of one of the states. We argue that the resulting symmetry breaking introduced by this process results in a gain counter to naive expectation. Finally, we discuss a number of open questions and new lines of enquiry that this discovery opens up.

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Section: The Two-envelope Switching Process As a Ratchetmentioning

confidence: 99%

The Two-Envelope Problem Revisited

2010

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“…The goal is to minimize the inverse current [Eq. (11)] for the piecewise constant temperature [Eq. (28)].…”

Section: A Numerical Solution For a Piecewise Constant Temperaturementioning

confidence: 99%

“…Paradoxical Parrondo games [10] which can be interpreted as discrete analogues of Brownian ratchets [11], have also been optimized recently [12].…”

Section: Introductionmentioning

confidence: 99%

Optimal potentials for temperature ratchets

2009

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In a spatially periodic temperature profile, directed transport of an overdamped Brownian particle can be induced along a periodic potential. With a load force applied to the particle, this setup can perform as a heat engine. For a given load, the optimal potential maximizes the current and thus the power output of the heat engine. We calculate the optimal potential for different temperature profiles and show that in the limit of a periodic piecewise constant temperature profile alternating between two temperatures, the optimal potential leads to a divergent current. This divergence, being an effect of both the overdamped limit and the infinite temperature gradient at the interface, would be cut off in any real experiment.

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“…This sort of game, first devised by J. M. R. Parrondo, has played a very relevant role in understanding the intriguing behavior shown by many physical systems, wherein the addition of disorder can lead to the emergence of some kind of order. This is the case of Brownian-ratchet-related problems [36,37], but Parrondo's games may have further implications in very diverse fields such as genetics [38] or finance [38,39].…”

Section: Introductionmentioning

confidence: 99%

Parrondo-like behavior in continuous-time random walks with memory

Montero

2011

Phys. Rev. E

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The continuous-time random walk (CTRW) formalism can be adapted to encompass stochastic processes with memory. In this paper we will show how the random combination of two different unbiased CTRWs can give rise to a process with clear drift, if one of them is a CTRW with memory. If one identifies the other one as noise, the effect can be thought of as a kind of stochastic resonance. The ultimate origin of this phenomenon is the same as that of the Parrondo paradox in game theory.

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Discrete–time ratchets, the Fokker–Planck equation and Parrondo's paradox

Cited by 38 publications

References 45 publications

The Two-Envelope Problem Revisited

The Two-Envelope Problem Revisited

Optimal potentials for temperature ratchets

Parrondo-like behavior in continuous-time random walks with memory

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