Parrondo games as disordered systems

Parrondo's paradox refers to the counter-intuitive situation where a winning strategy results from a suitable combination of losing ones. Simple stochastic games exhibiting this paradox have been introduced around the turn of the millennium. The common setting of these Parrondo games is that two rules, $A$ and $B$, are played at discrete time steps, following either a periodic pattern or an aperiodic one, be it deterministic or random. These games can be mapped onto 1D random walks. In capital-dependent games, the probabilities of moving right or left depend on the walker's position modulo some integer $K$. In history-dependent games, each step is correlated with the $Q$ previous ones. In both cases the gain identifies with the velocity of the walker's ballistic motion, which depends non-linearly on model parameters, allowing for the possibility of Parrondo's paradox. Calculating the gain involves products of non-commuting Markov matrices, which are somehow analogous to the transfer matrices used in the physics of 1D disordered systems. Elaborating upon this analogy, we study a paradigmatic Parrondo game of each class in the neutral situation where each rule, when played alone, is fair. The main emphasis of this systematic approach is on the dependence of the gain on the remaining parameters and, above all, on the game, i.e., the rule pattern, be it periodic or aperiodic, deterministic or random. One of the most original sides of this work is the identification of weak-contrast regimes for capital-dependent and history-dependent Parrondo games, and a detailed quantitative investigation of the gain in the latter scaling regimes.Comment: 17 pages, 10 figures, 2 table

An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is shown to measure the degree of equilibration of the system launched from a typical state, from the standpoint of the chosen basis. This approach is substantiated by an in-depth study of the example of a tight-binding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.Comment: 19 pages, 7 figures, 1 tabl

Single-spin-flip dynamics of the Ising chain

We consider the most general single-spin-flip dynamics for the ferromagnetic Ising chain with nearest-neighbour influence and spin reversal symmetry. This dynamics is a two-parameter extension of Glauber dynamics corresponding respectively to non-linearity and irreversibility. The associated stationary state measure is given by the usual Boltzmann-Gibbs distribution for the ferromagnetic Hamiltonian of the chain. We study the properties of this dynamics both at infinite and at finite temperature, all over its parameter space, with particular emphasis on special lines and points.Comment: 31 pages, 18 figure

Universality in survivor distributions: Characterising the winners of competitive dynamics

We investigate the survivor distributions of a spatially extended model of competitive dynamics in different geometries. The model consists of a deterministic dynamical system of individual agents at specified nodes, which might or might not survive the predatory dynamics: all stochasticity is brought in by the initial state. Every such initial state leads to a unique and extended pattern of survivors and non-survivors, which is known as an attractor of the dynamics. We show that the number of such attractors grows exponentially with system size, so that their exact characterisation is limited to only very small systems. Given this, we construct an analytical approach based on inhomogeneous mean-field theory to calculate survival probabilities for arbitrary networks. This powerful (albeit approximate) approach shows how universality arises in survivor distributions via a key concept -- the {\it dynamical fugacity}. Remarkably, in the large-mass limit, the survival probability of a node becomes independent of network geometry, and assumes a simple form which depends only on its mass and degree.Comment: 12 pages, 6 figures, 2 table

Nonequilibrium dynamics of the zeta urn model

We consider a mean-field dynamical urn model, defined by rules which give the rate at which a ball is drawn from an urn and put in another one, chosen amongst an assembly. At equilibrium, this model possesses a fluid and a condensed phase, separated by a critical line. We present an analytical study of the nonequilibrium properties of the fluctuating number of balls in a given urn, considering successively the temporal evolution of its distribution, of its two-time correlation and response functions, and of the associated \fd ratio, both along the critical line and in the condensed phase. For well separated times the \fd ratio admits non-trivial limit values, both at criticality and in the condensed phase, which are universal quantities depending continuously on temperature.Comment: 30 pages, 1 figur

A column of grains in the jamming limit: glassy dynamics in the compaction process

We investigate a stochastic model describing a column of grains in the jamming limit, in the presence of a low vibrational intensity. The key control parameter of the model, $\epsilon$, is a representation of granular shape, related to the reduced void space. Regularity and irregularity in grain shapes, respectively corresponding to rational and irrational values of $\epsilon$, are shown to be centrally important in determining the statics and dynamics of the compaction process.Comment: 29 pages, 14 figures, 1 table. Various minor changes and updates. To appear in EPJ