Entropic aging and extreme value statistics

Entropic aging consists in a progressive slowing down of the low-temperature dynamics of a glassy system due to the rarefaction of downwards directions on the energy landscape, as lower and lower energy levels are reached. A prototypical model exhibiting this scenario is the Barrat-M\'ezard model. We argue that in the zero-temperature limit, this model precisely corresponds to a dynamical realization of extreme value statistics, providing an interesting connection between the two fields. This mapping directly yields the long-time asymptotic shape of the dynamical energy distribution, which is then one of the standard extreme value distributions (Gumbel, Weibull or Fr\'echet), thus restricting the class of asymptotic energy distributions with respect to the original preasymptotic results. We also briefly discuss similarities and differences between the Barrat-M\'ezard model and undriven dissipative systems like granular gases.Comment: 8 pages, to appear in J. Phys.

Participation ratio for constraint-driven condensation with superextensive mass

Broadly distributed random variables with a power-law distribution $f(m) \sim m^{-(1+\alpha)}$ are known to generate condensation effects, in the sense that, when the exponent $\alpha$ lies in a certain interval, the largest variable in a sum of $N$ (independent and identically distributed) terms is for large $N$ of the same order as the sum itself. In particular, when the distribution has infinite mean ($0<\alpha<1$) one finds unconstrained condensation, whereas for $\alpha>1$ constrained condensation takes places fixing the total mass to a large enough value $M=\sum_{i=1}^N m_i > M_c$. In both cases, a standard indicator of the condensation phenomenon is the participation ratio $Y_k=\langle \sum_i m_i^k / (\sum_i m_i)^k\rangle$ ($k>1$), which takes a finite value for $N \to \infty$ when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value $M \sim N^{1+\delta}$ ($\delta >0$), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as $M\sim N^{1/\alpha}$ for $\alpha<1$) and the extensive constrained mass. In particular we show that for exponents $\alpha \delta_c=1/\alpha-1$ is separated from a homogeneous phase at $\delta < \delta_c$ by a transition line, $\delta=\delta_c$, where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass.Comment: 11 pages, 2 figures, to appear in Entrop

The glass transition in a nutshell: a source of inspiration to describe the subcritical transition to turbulence

The starting point of the present work is the observation of possible analogies, both at the phenomenological and at the methodological level, between the subcritical transition to turbulence and the glass transition. Having recalled the phenomenology of the subcritical transition to turbulence, we review the theories of the glass transition at a very basic level, focusing on the history of their development as well as on the concepts they have elaborated. Doing so, we aim at attracting the attention on the above mentioned analogies, which we believe could inspire new developments in the theory of the subcritical transition to turbulence. We then briefly describe a model inspired by one of the simplest and most inspiring model of the glass transition, the so-called Random Energy Model, as a first step in that direction.Comment: 9 pages, 1 figure; to appear in a topical issue of Eur. Phys. J. E dedicated to Paul Mannevill

Dynamical fluctuations in a simple housing market model

We consider a simple stochastic model of a urban rental housing market, in which the interaction of tenants and landlords induces rent fluctuations. We simulate the model numerically and measure the equilibrium rent distribution, which is found to be close to a lognormal law. We also study the influence of the density of agents (or equivalently, the vacancy rate) on the rent distribution. A simplified version of the model, amenable to analytical treatment, is studied and leads to a lognormal distribution of rents. The predicted equilibrium value agrees quantitatively with numerical simulations, while a qualitative agreement is obtained for the standard deviation. The connection with non-equilibrium statistical physics models like ratchets is also emphasized.Comment: 12 pages, 5 figures, to appear in J. Stat. Mec

Aging of the frictional properties induced by temperature variations

The dry frictional contact between two solid surfaces is well-known to obey Coulomb friction laws. In particular, the static friction force resisting the relative lateral (tangential) motion of solid surfaces, initially at rest, is known to be proportional to the normal force and independent of the area of the macroscopic surfaces in contact. Experimentally, the static friction force has been observed to slightly depend on time. Such an aging phenomenon has been accounted for either by the creep of the material or by the condensation of water bridges at the microscopic contacts points. Studying a toy-model, we show that the small uncontrolled temperature changes of the system can also lead to a significant increase of the static friction force.Comment: 8 pages, 5 figures, final version, to appear in Phys. Rev.

Symmetry-breaking phase transition in a dynamical decision model

We consider a simple decision model in which a set of agents randomly choose one of two competing shops selling the same perishable products (typically food). The satisfaction of agents with respect to a given store is related to the freshness of the previously bought products. Agents select with a higher probability the store they are most satisfied with. Studying the model from a statistical physics perspective, both through numerical simulations and mean-field analytical methods, we find a rich behaviour with continuous and discontinuous phase transitions between a symmetric phase where both stores maintain the same level of activity, and a phase with broken symmetry where one of the two shops attracts more customers than the other.Comment: 13 pages, 6 figures, submitted to JSTA

Statistics of sums of correlated variables described by a matrix product ansatz

We determine the asymptotic distribution of the sum of correlated variables described by a matrix product ansatz with finite matrices, considering variables with finite variances. In cases when the correlation length is finite, the law of large numbers is obeyed, and the rescaled sum converges to a Gaussian distribution. In constrast, when correlation extends over system size, we observe either a breaking of the law of large numbers, with the onset of giant fluctuations, or a generalization of the central limit theorem with a family of nonstandard limit distributions. The corresponding distributions are found as mixtures of delta functions for the generalized law of large numbers, and as mixtures of Gaussian distributions for the generalized central limit theorem. Connections with statistical physics models are emphasized.Comment: 6 pages, 1 figur

Matrix product representation and synthesis for random vectors: Insight from statistical physics

Inspired from modern out-of-equilibrium statistical physics models, a matrix product based framework permits the formal definition of random vectors (and random time series) whose desired joint distributions are a priori prescribed. Its key feature consists of preserving the writing of the joint distribution as the simple product structure it has under independence, while inputing controlled dependencies amongst components: This is obtained by replacing the product of distributions by a product of matrices of distributions. The statistical properties stemming from this construction are studied theoretically: The landscape of the attainable dependence structure is thoroughly depicted and a stationarity condition for time series is notably obtained. The remapping of this framework onto that of Hidden Markov Models enables us to devise an efficient and accurate practical synthesis procedure. A design procedure is also described permitting the tuning of model parameters to attain targeted properties. Pedagogical well-chosen examples of times series and multivariate vectors aim at illustrating the power and versatility of the proposed approach and at showing how targeted statistical properties can be actually prescribed.Comment: 10 pages, 4 figures, submitted to IEEE Transactions on Signal Processin

Dependence of the fluctuation-dissipation temperature on the choice of observable

On general grounds, a nonequilibrium temperature can be consistently defined from generalized fluctuation-dissipation relations only if it is independent of the observable considered. We argue that the dependence on the choice of observable generically occurs when the phase-space probability distribution is non-uniform on constant energy shells. We relate quantitatively this observable dependence to a fundamental characteristics of nonequilibrium systems, namely the Shannon entropy difference with respect to the equilibrium state with the same energy. This relation is illustrated on a mean-field model in contact with two heat baths at different temperatures.Comment: 4 pages, 2 figures, final versio