Searcharxiv⌕ Search

arXiv subjects

Eric Bertin

Publications and source records attributed to Eric Bertin.

At least 73 records · Page 4Linked to original sources
arXiv2014

Dynamics of the bouncing ball

We describe an experiment dedicated to the study of the trajectories of a ball bouncing randomly on a vibrating plate. The system was originally used, considering a sinusoidal vibration, to illustrate period doubling and the route to chaos. Our experimental device makes it possible to impose, to the plate, arbitrary trajectories and not only sinusoidal or random, as is generally the case. We show that the entire trajectory of the ball can still be reconstructed from the measurement of the collisions times. First, we make use of the experimental system to introduce the notion of dissipative collisions and to propose three different ways to measure the associated restitution coefficient. Then, we report on correlations in the chaotic regime and discuss theoretically the complex pattern which they exhibit in the case of a sinusoidal vibration. At last, we show that the use of an aperiodic motion makes it possible to get rid of part of the correlations and to discuss theoretically the average energy of the ball in the chaotic regime.

cond-mat.stat-mech↗
arXiv2014

Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models

We describe a generic theoretical framework, denoted as the Boltzmann-Ginzburg-Landau approach, to derive continuous equations for the polar and/or nematic order parameters describing the large scale behavior of assemblies of point-like active particles interacting through polar or nematic alignment rules. Our study encompasses three main classes of dry active systems, namely polar particles with 'ferromagnetic' alignment (like the original Vicsek model), nematic particles with nematic alignment ("active nematics"), and polar particles with nematic alignment ("self-propelled rods"). The Boltzmann-Ginzburg-Landau approach combines a low-density description in the form of a Boltzmann equation, with a Ginzburg-Landau-type expansion close to the instability threshold of the disordered state. We provide the generic form of the continuous equations obtained for each class, and comment on the relationships and differences with other approaches.

cond-mat.stat-mech↗
arXiv2014

Large deviations for correlated random variables described by a matrix product ansatz

We study the large deviations of sums of correlated random variables described by a matrix product ansatz, which generalizes the product structure of independent random variables to matrices whose non-commutativity is the source of correlations. We show with specific examples that different large deviation behaviors can be found with this ansatz. In particular, it is possible to construct sums of correlated random variables that violate the Law of Large Numbers, the Central Limit Theorem, as well as sums that have nonconvex rate functions or rate functions with linear parts or plateaux.

cond-mat.stat-mech↗
arXiv2014

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.

cond-mat.stat-mech↗
arXiv2013

Large-scale chaos and fluctuations in active nematics

We show that "dry" active nematics, e.g. collections of shaken elongated granular particles, exhibit large-scale spatiotemporal chaos made of interacting dense, ordered, band-like structures in a parameter region including the linear onset of nematic order. These results are obtained from the study of the relatively simple and well-known (deterministic) hydrodynamic equations describing these systems in a dilute limit, and of a self-propelled particle Vicsek-like model for this class of active matter. In this last case, revisiting the status of the strong fluctuations and long-range correlations now considered as landmarks of orientationally-ordered active phases, we show that the giant number fluctuations observed in the chaotic phase are a trivial consequence of density segregation. However anomalous density fluctuations are present in the homogeneous quasi-ordered nematic phase and characterized by a non-trivial scaling exponent.

cond-mat.soft↗
arXiv2013

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.

physics.flu-dyn↗
arXiv2013

Reinterpreting the development of extensive air showers initiated by nuclei and photons

Ultra-high energy cosmic rays (UHECRs) interacting with the atmosphere generate extensive air showers (EAS) of secondary particles. The depth corresponding to the maximum development of the shower, $\Xmax$, is a well-known observable for determining the nature of the primary cosmic ray which initiated the cascade process. In this paper, we present an empirical model to describe the distribution of $\Xmax$ for EAS initiated by nuclei, in the energy range from $10^{17}$ eV up to $10^{21}$ eV, and by photons, in the energy range from $10^{17}$ eV up to $10^{19.6}$ eV. Our model adopts the generalized Gumbel distribution motivated by the relationship between the generalized Gumbel statistics and the distribution of the sum of non-identically distributed variables in dissipative stochastic systems. We provide an analytical expression for describing the $\Xmax$ distribution for photons and for nuclei, and for their first two statistical moments, namely $\langle \Xmax\rangle$ and $σ^{2}(\Xmax)$. The impact of the hadronic interaction model is investigated in detail, even in the case of the most up-to-date models accounting for LHC observations. We also briefly discuss the differences with a more classical approach and an application to the experimental data based on information theory.

hep-ph↗
arXiv2013

Dissipation induced non-Gaussian energy fluctuations

The influence of dissipation on the fluctuation statistics of the total energy is investigated through both a phenomenological and a stochastic model for dissipative energy-transfer through a cascade of states. In equilibrium the states obey equipartition and the total energy obeys the central limit theorem, giving Gaussian fluctuations. In the presence of dissipation, the fluctuations can be driven non-Gaussian if there is macroscopic energy transfer from large to small scales. We are thus able to equate the non-Gaussian order parameter fluctuations in model equilibrium systems at criticality with energy fluctuations in these dissipative systems. Energy fluctuations in the phenomenological model map directly onto the 1/f^alpha noise problem and numerical simulations of the stochastic model yield results in qualitative agreement with these predictions.

cond-mat.stat-mech↗
arXiv2013

Mesoscopic theory for fluctuating active nematics

The term active nematics designates systems in which apolar elongated particles spend energy to move randomly along their axis and interact by inelastic collisions in the presence of noise. Starting from a simple Vicsek-style model for active nematics, we derive a mesoscopic theory, complete with effective multiplicative noise terms, using a combination of kinetic theory and Itô calculus approaches. The stochastic partial differential equations thus obtained are shown to recover the key terms argued in EPL \textbf{62} (2003) 196 to be at the origin of anomalous number fluctuations and long-range correlations. Their deterministic part is studied analytically, and is shown to give rise to the long-wavelength instability at onset of nematic order (see arXiv:1011.5408). The corresponding nonlinear density-segregated band solution is given in a closed form.

cond-mat.stat-mech↗
arXiv2013

On the existence of a glass transition in a Random Energy Model

We consider a generalized version of the Random Energy Model in which the energy of each configuration is given by the sum of $N$ independent contributions ("local energies") with finite variances but otherwise arbitrary statistics. Using the large deviation formalism, we find that the glass transition generically exists when local energies have a smooth distribution. In contrast, if the distribution of the local energies has a {Dirac mass} at the minimal energy (e.g., if local energies take discrete values), the glass transition ceases to exist if the number of energy levels grows sufficiently fast with system size. This shows that statistical independence of energy levels does not imply the existence of a glass transition.

cond-mat.stat-mech↗
arXiv2012

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.

q-fin.GN↗
arXiv2012

Turbulence lifetimes: what we can learn from the physics of glasses

In this note, we critically discuss the issue of the possible finiteness of the turbulence lifetime in subcritical transition to turbulence in shear flows, which attracted a lot of interest recently. We briefly review recent experimental and numerical results, as well as theoretical proposals, and compare the difficulties arising in assessing this issue in subcritical shear flow with that encountered in the study of the glass transition. In order to go beyond the purely methodological similarities, we further elaborate on this analogy and propose a qualitative mapping between these two apparently unrelated situations, which could possibly foster new directions of research in subcritical shear flows.

cond-mat.soft↗
arXiv2012

Aging in the trap model as a relaxation further away from equilibrium

The aging regime of the trap model, observed for a temperature T below the glass transition temperature T_g, is a prototypical example of non-stationary out-of-equilibrium state. We characterize this state by evaluating its "distance to equilibrium", defined as the Shannon entropy difference ΔS (in absolute value) between the non-equilibrium state and the equilibrium state with the same energy. We consider the time evolution of ΔS and show that, rather unexpectedly, ΔS(t) continuously increases in the aging regime, if the number of traps is infinite, meaning that the "distance to equilibrium" increases instead of decreasing in the relaxation process. For a finite number N of traps, ΔS(t) exhibits a maximum value before eventually converging to zero when equilibrium is reached. The time t* at which the maximum is reached however scales in a non-standard way as t* ~ N^(T_g/2T), while the equilibration time scales as N^(T_g/T). In addition, the curves ΔS(t) for different N are found to rescale as ln t/ln t*, instead of the more familiar scaling t/t*.

cond-mat.stat-mech↗
arXiv2012

Simple and faithful nonlinear field equations for aligning self-propelled rods

We derive a set of minimal yet complete nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for the fronts forming density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.

cond-mat.soft↗
arXiv2012

How far can stochastic and deterministic views be reconciled?

In this short note, we try to provide the reader with a brief pedagogical account of some similarities and differences between stochastic and deterministic processes. A short presentation of some basic notions related to the mathematical description of stochastic processes is also given. Our main aim is to illustrate the somehow surprising fact that the gap between the behaviour of stochastic and deterministic processes might, from a practical perspective, be much smaller than a priori expected.

q-bio.QM↗
arXiv2012

On-off intermittency over an extended range of control parameter

We propose a simple phenomenological model exhibiting on-off intermittency over an extended range of control parameter. We find that the distribution of the 'off' periods has as a power-law tail with an exponent varying continuously between -1 and -2, at odds with standard on-off intermittency which occurs at a specific value of the control parameter, and leads to the exponent -3/2. This non-trivial behavior results from the competition between a strong slowing down of the dynamics at small values of the observable, and a systematic drift toward large values.

cond-mat.stat-mech↗
arXiv2012

Matrix products for the synthesis of stationary time series with a priori prescribed joint distributions

Inspired from non-equilibrium statistical physics models, a general framework enabling the definition and synthesis of stationary time series with a priori prescribed and controlled joint distributions is constructed. Its central feature consists of preserving for the joint distribution the simple product struc- ture it has under independence while enabling to input con- trolled and prescribed dependencies amongst samples. To that end, it is based on products of d-dimensional matrices, whose entries consist of valid distributions. The statistical properties of the thus defined time series are studied in details. Having been able to recast this framework into that of Hidden Markov Models enabled us to obtain an efficient synthesis procedure. Pedagogical well-chosen examples (time series with the same marginal distribution, same covariance function, but different joint distributions) aim at illustrating the power and potential of the approach and at showing how targeted statistical prop- erties can be actually prescribed.

cond-mat.stat-mech↗