New paradoxical games based on brownian ratchets

Based on Brownian ratchets, a counterintuitive phenomenon has recently emerged-namely, that two losing games can yield, when combined, a paradoxical tendency to win. A restriction of this phenomenon is that the rules depend on the current capital of the player. Here we present new games where all the rules depend only on the history of the game and not on the capital. This new history-dependent structure significantly increases the parameter space for which the effect operates.     

Quantum Parrondo game based on a quantum ratchet effect

A Parrondo game is a counterintuitive game where two losing games can be combined to form a winning game. We construct a quantum version of a Parrondo game based on a quantum ratchet effect for a delta-kicked model, which can be realized in optical lattices. A game set is presented and a quantum anti-Parrondo game is also investigated.     

Parrondo Games as Lattice Gas Automata

Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.     

Maximal Parrondo’s Paradox for Classical and Quantum Markov Chains

Parrondo’s paradox refers to the situation where two, multi-round games with a fixed winning criteria, both with probability greater than one-half for one player to win, are combined. Using a possibly biased coin to determine the rule to employ for each round, paradoxically, the previously losing player now wins the combined game with probability greater than one-half. In this paper, we will analyze classical observed, classical hidden, and quantum versions of a game that displays this paradox. The game we have utilized is simpler than games for which this behavior has been previously noted in the classical and quantum cases. We will show that in certain situations the paradox can occur to a greater degree in the quantum version than is possible in the classical versions.     

Books received

Two losing gambling games, when alternated in a periodic or random fashion, can produce a winning game. This paradox has been inspired by certain physical systems capable of rectifying fluctuations: the so-called Brownian ratchets. In this paper we review this paradox, from Brownian ratchets to the most recent studies on collective games, providing some intuitive explanations of the unexpected phenomena that we will find along the way.     

The paradox of group behaviors based on Parrondo’s games

We assume a multi-agent model based on Parrondo’s games. The model consists of game A between individuals and game B. In game A, two behavioral patterns are defined: competition and inaction. A controlled alternation strategy of behavioral pattern that gives a single player the highest return is proposed when game A+B is played randomly. Interesting phenomena can be found in collective games where a large number of individuals choose the behavioral pattern by voting. When game B is the capital-dependent version, the outcome can be better for the players to vote randomly than to vote according to their own capital. An explanation of such counter-intuitive phenomena is given by noting that selfish voting prevents the competition behavior of game A that is essential for the total capital to grow. However, if game B is the history-dependent version, this counter-intuitive phenomenon will not happen. The reason is selfish voting results in the competition behavior of game A, and finally it produces the winning results.     

Quantum Parrondo’s Games Under Decoherence

We study the effect of quantum noise on history dependent quantum Parrondo’s games by taking into account different noise channels. Our calculations show that entanglement can play a crucial role in quantum Parrondo’s games. It is seen that for the maximally entangled initial state in the presence of decoherence, the quantum phases strongly influence the payoffs for various sequences of the game. The effect of amplitude damping channel leads to winning payoffs. Whereas the depolarizing and phase damping channels lead to the losing payoffs. In case of amplitude damping channel, the payoffs are enhanced in the presence of decoherence for the sequence AAB. This is because the quantum phases interfere constructively which leads to the quantum enhancement of the payoffs in comparison to the undecohered case. It is also seen that the quantum phase angles damp the payoffs significantly in the presence of decoherence. Furthermore, it is seen that for multiple games of sequence AAB, under the influence of amplitude damping channel, the game still remains a winning game. However, the quantum enhancement reduces in comparison to the single game of sequence AAB because of the destructive interference of phase dependent terms. In case of depolarizing channel, the game becomes a loosing game. It is seen that for the game sequence B the game is loosing one and the behavior of sequences B and BB is similar for amplitude damping and depolarizing channels. In addition, the repeated games of A are only influenced by the amplitude damping channel and the game remains a losing game. Furthermore, it is also seen that for any sequence when played in series, the phase damping channel does not influence the game.     

Randomly chosen chaotic maps can give rise to nearly ordered behavior

Parrondo’s paradox [J.M.R. Parrondo, G.P. Harmer, D. Abbott, New paradoxical games based on Brownian ratchets, Phys. Rev. Lett. 85 (2000), 5226–5229] (see also [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72]) states that two losing gambling games when combined one after the other (either deterministically or randomly) can result in a winning game: that is, a losing game followed by a losing game = a winning game. Inspired by this paradox, a recent study [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] asked an analogous question in discrete time dynamical system: can two chaotic systems give rise to order, namely can they be combined into another dynamical system which does not behave chaotically? Numerical evidence is provided in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] that two chaotic quadratic maps, when composed with each other, create a new dynamical system which has a stable period orbit. The question of what happens in the case of random composition of maps is posed in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] but left unanswered. In this note we present an example of a dynamical system where, at each iteration, a map is chosen in a probabilistic manner from a collection of chaotic maps. The resulting random map is proved to have an infinite absolutely continuous invariant measure (acim) with spikes at two points. From this we show that the dynamics behaves in a nearly ordered manner. When the foregoing maps are applied one after the other, deterministically as in [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72], the resulting composed map has a periodic orbit which is stable.     

The multi-agent Parrondo’s model based on the network evolution

A multi-agent Parrondo’s model is proposed in the paper. The model includes link A based on the rewiring mechanism (the network evolution) + game B (dependent on the spatial neighbors). Moreover, to produce the paradoxical effect and analyze the “agitating” effect of the network evolution, the dynamic processes of the network evolution + game B are studied. The simulation results and the theoretical analysis both show that the network evolution can make game B which is losing produce the winning paradoxical effect. Furthermore, we obtain the parameter space where the strong or weak Parrondo’s paradox occurs. Each size of the region of the parameter space is larger than the one in the available multi-agent Parrondo’s model of game A + game B. This result shows that the “agitating” effect of rewiring based on the network evolution is better than that of the zero-sum game between individuals.     

Reversals of chance in paradoxical games

We present two collective games with new paradoxical features when they are combined. Besides reproducing the so-called Parrondo effect, where a winning game is obtained from the alternation of two fair games, there also exists a current inversion when varying the mixing probability between the games. We show that this is a new effect insofar one of the games is an unbiased random walk without internal structure. We present a detailed study by means of a discrete-time Markov chain analysis, obtaining analytical expressions for the stationary probabilities for a finite number of players. We also provide qualitative insight into this current inversion effect.     

Switching induced oscillations in the logistic map

In ecological modeling, seasonality can be represented as a switching between different environmental conditions. This switching strategy can be related to the so-called Parrondian games, where the alternation of two losing games yield a winning game. Hence we can consider two dynamics that, by themselves, yield undesirable behaviors, but when alternated yield a desirable oscillatory behavior. In this case, we also consider a noisy switching strategy and find that the desirable oscillatory behavior prevails.     

Seasonality as a Parrondian game

Switching strategies can be related to the so-called Parrondian games, where the alternation of two losing games yields a winning game. We consider two dynamics that by themselves yield undesirable behaviors, but when alternated, yield a desirable oscillatory behavior. In the analysis of the alternate-logistic map, we prove that alternating parameter values yielding extinction with parameter values associated with chaotic dynamics results in periodic trajectories. Ultimately, we consider a four season logistic model with either migration or immigration.     

Can two chaotic systems give rise to order?

The recently discovered Parrondo's paradox claims that two losing games can result, under random or periodic alternation of their dynamics, in a winning game: “losing + losing = winning”. In this paper we follow Parrondo's philosophy of combining different dynamics and we apply it to the case of one-dimensional quadratic maps. We prove that the periodic mixing of two chaotic dynamics originates an ordered dynamics in certain cases. This provides an explicit example (theoretically and numerically tested) of a different Parrondian paradoxical phenomenon: “chaos + chaos = order”.     

Two Party Non-Local Games

In this work we have introduced two party games with respective winning conditions. One cannot win these games deterministically in the classical world if they are not allowed to communicate at any stage of the game. Interestingly we find out that in quantum world, these winning conditions can be achieved if the players share an entangled state. We also introduced a game which is impossible to win if the players are not allowed to communicate in classical world (both probabilistically and deterministically), yet there exists a perfect quantum strategy by following which, one can attain the winning condition of the game.     

过阻尼布朗棘轮的斯托克斯效率研究

根据随机能量理论解析得到阻尼环境中布朗粒子的概率流和斯托克斯效率,并进一步研究布朗粒子的输运性能.详细讨论了空间的不对称性、外偏置力及外势结构等对棘轮定向输运的影响.研究发现,合适的外偏置力能使棘轮的定向输运达到最强.通过调节外势的不对称性可使棘轮中粒子的运动反向,当选择合适的空间不对称性时布朗粒子的反向输运可获得最强.此外,一定条件下合适的外势高度也能增强棘轮输运,且能使粒子克服黏滞阻力的斯托克斯效率达到最大.所得结论能够启发实验上设计合适的外势及外偏置来优化布朗棘轮的定向输运性能,并为生物纳米器件的研制提供一定的理论参考.     

A composite micromotor driven by self-thermophoresis and Brownian rectification

Brownian motors and self-phoretic microswimmers are two typical micromotors, for which thermal fluctuations play different roles. Brownian motors utilize thermal noise to acquire unidirectional motion, while thermal fluctuations randomize the self-propulsion of self-phoretic microswimmers. Here we perform mesoscale simulations to study a composite micromotor composed of a self-thermophoretic Janus particle under a time-modulated external ratchet potential. The composite motor exhibits a unidirectional transport, whose direction can be reversed by tuning the modulation frequency of the external potential. The maximum transport capability is close to the superposition of the drift speed of the pure Brownian motor and the self-propelling speed of the pure self-thermophoretic particle. Moreover, the hydrodynamic effect influences the orientation of the Janus particle in the ratched potential, hence also the performance of the composite motor. Our work thus provides an enlightening attempt to actively exploit inevitable thermal fluctuations in the implementation of the self-phoretic microswimmers.     

Demonstration of a controllable three-dimensional Brownian motor in symmetric potentials

We demonstrate a Brownian motor, based on cold atoms in optical lattices, where isotropic random fluctuations are rectified in order to induce controlled atomic motion in arbitrary directions. In contrast to earlier demonstrations of ratchet effects, our Brownian motor operates in potentials that are spatially and temporally symmetric, but where spatiotemporal symmetry is broken by a phase shift between the potentials and asymmetric transfer rates between them. The Brownian motor is demonstrated in three dimensions and the noise-induced drift is controllable in our system.     

Disordered Markovian Brownian ratchets

A model of a Brownian ratchet coupled to a heat bath and driven by a nonequilibrium Poisson white noise is discussed. The formula describing a generated current in terms of the statistical properties of a possible irregular or random potential is derived within the small nonequilibrium noise approximation and illustrated by a few concrete examples. The perturbation technique for Hilbert space operators is used as a mathematical tool.     

反馈控制棘轮的定向输运效率研究

研究了反馈耦合布朗棘轮中粒子处于负载力、时变外力及噪声作用下的定向输运问题.详细讨论了外力作用时间的不对称性、外势空间的不对称性及外力周期等对反馈耦合棘轮中粒子输运效率的影响.研究发现,外力的时间不对称度能促进反馈棘轮中粒子的定向输运,随时间不对称度的增大,反馈棘轮中粒子能获得较大的效率.然而,外势空间的不对称度能有效抑制耦合棘轮中粒子的扩散,达到增强耦合粒子定向输运的效果.同时还发现,存在最优的噪声强度能使耦合粒子的输运效率达到最大.