arXiv subjects

Claude Godrèche

Publications and source records attributed to Claude Godrèche.

25 records · Page 2Linked to original sources

arXiv2019

Condensation for random variables conditioned by the value of their sum

We revisit the problem of condensation for independent, identically distributed random variables with a power-law tail, conditioned by the value of their sum. For large values of the sum, and for a large number of summands, a condensation transition occurs where the largest summand accommodates the excess difference between the value of the sum and its mean. This simple scenario of condensation underlies a number of studies in statistical physics, such as, e.g., in random allocation and urn models, random maps, zero-range processes and mass transport models. Much of the effort here is devoted to presenting the subject in simple terms, reproducing known results and adding some new ones. In particular we address the question of the quantitative comparison between asymptotic estimates and exact finite-size results. Simply stated, one would like to know how accurate are the asymptotic estimates of the observables of interest, compared to their exact finite-size counterparts, to the extent that they are known. This comparison, illustrated on the particular exemple of a distribution with Lévy index equal to $3/2$, demonstrates the role of the contributions of the dip and large deviation regimes. Except for the last section devoted to a brief review of extremal statistics, the presentation is self-contained and uses simple analytical methods.

cond-mat.stat-mech↗

arXiv2018

Characterising the nonequilibrium stationary states of Ornstein-Uhlenbeck processes

We characterise the nonequilibrium stationary state of a generic multivariate Ornstein-Uhlenbeck process involving $N$ degrees of freedom. The irreversibility of the process is encoded in the antisymmetric part of the Onsager matrix. The linearity of the Langevin equations allows us to derive closed-form expressions in terms of the latter matrix for many quantities of interest, including the entropy production rate and the fluctuation-dissipation ratio matrix. This general setting is then illustrated by two classes of systems. First, we consider the one-dimensional ferromagnetic Gaussian spin model endowed with a stochastic dynamics where spatial asymmetry results in irreversibility. The stationary state on a ring is independent of the asymmetry parameter, whereas it depends continuously on the latter on an open chain. Much attention is also paid to finite-size effects, especially near the critical point. Second, we consider arrays of resistively coupled electrical circuits. The entropy production rate is evaluated in the regime where the local temperatures of the resistors have small fluctuations. For $RL$ networks the entropy production rate grows linearly with the size of the array. For $RC$ networks a quadratic growth law violating extensivity is predicted.

cond-mat.stat-mech↗

arXiv2018

Freezing in stripe states for kinetic Ising models: a comparative study of three dynamics

We present a comparative study of the fate of an Ising ferromagnet on the square lattice with periodic boundary conditions evolving under three different zero-temperature dynamics. The first one is Glauber dynamics, the two other dynamics correspond to two limits of the directed Ising model, defined by rules that break the full symmetry of the former, yet sharing the same Boltzmann-Gibbs distribution at stationarity. In one of these limits the directed Ising model is reversible, in the other one it is irreversible. For the kinetic Ising-Glauber model, several recent studies have demonstrated the role of critical percolation to predict the probabilities for the system to reach the ground state or to fall in a metastable state. We investigate to what extent the predictions coming from critical percolation still apply to the two other dynamics.

cond-mat.stat-mech↗

arXiv2017

Generic phase coexistence in the totally asymmetric kinetic Ising model

The physical analysis of generic phase coexistence in the North-East-Center Toom model was originally given by Bennett and Grinstein. The gist of their argument relies on the dynamics of interfaces and droplets. We revisit the same question for a specific totally asymmetric kinetic Ising model on the square lattice. This nonequilibrium model possesses the remarkable property that its stationary-state measure in the absence of a magnetic field coincides with that of the usual ferromagnetic Ising model. We use both analytical arguments and numerical simulations in order to make progress in the quantitative understanding of the phenomenon of generic phase coexistence. At zero temperature a mapping onto the TASEP allows an exact determination of the time-dependent shape of the ballistic interface sweeping a large square minority droplet of up or down spins. At finite temperature, measuring the mean lifetime of such a droplet allows an accurate measurement of its shrinking velocity $v$, which depends on temperature $T$ and magnetic field $h$. In the absence of a magnetic field, $v$ vanishes with an exponent $Δ_v\approx2.5\pm0.2$ as the critical temperature $T_c$ is approached. At fixed temperature in the ordered phase, $v$ vanishes at the phase-boundary fields $\pm h_{\rm b}(T)$ which mark the limits of the coexistence region. The latter fields vanish with an exponent $Δ_h\approx3.2\pm0.3$ as $T_c$ is approached.

cond-mat.stat-mech↗

arXiv2017

Two-time correlation and occupation time for the Brownian bridge and tied-down renewal processes

Tied-down renewal processes are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time $t$. We give an analytical derivation of the two-time correlation function for such processes in the Laplace space of all temporal variables. This yields the exact asymptotic expression of the correlation in the Porod regime of short separations between the two times and in the persistence regime of large separations. We also investigate other quantities, such as the backward and forward recurrence times, as well as the occupation time of the process. The latter has a broad distribution which is determined exactly. Physical implications of these results for the Poland Scheraga and related models are given. These results also give exact answers to questions posed in the past in the context of stochastically evolving surfaces.

cond-mat.stat-mech↗

arXiv2017

Longest interval between zeros of the tied-down random walk, the Brownian bridge and related renewal processes

The probability distribution of the longest interval between two zeros of a simple random walk starting and ending at the origin, and of its continuum limit, the Brownian bridge, was analysed in the past by Rosén and Wendel, then extended by the latter to stable processes. We recover and extend these results using simple concepts of renewal theory, which allows to revisit past or recent works of the physics literature.

cond-mat.stat-mech↗

arXiv2007

Tagged Particle Correlations in the Asymmetric Simple Exclusion Process: Finite Size Effects

We study finite size effects in the variance of the displacement of a tagged particle in the stationary state of the Asymmetric Simple Exclusion Process (ASEP) on a ring of size $L$. The process involves hard core particles undergoing stochastic driven dynamics on a lattice. The variance of the displacement of the tagged particle, averaged with respect to an initial stationary ensemble and stochastic evolution, grows linearly with time at both small and very large times. We find that at intermediate times, it shows oscillations with a well defined size-dependent period. These oscillations arise from sliding density fluctuations (SDF) in the stationary state with respect to the drift of the tagged particle, the density fluctuations being transported through the system by kinematic waves. In the general context of driven diffusive systems, both the Edwards-Wilkinson (EW) and the Kardar-Parisi-Zhang (KPZ) fixed points are unstable with respect to the SDF fixed point, a flow towards which is generated on adding a gradient term to the EW and the KPZ time-evolution equation. We also study tagged particle correlations for a fixed initial configuration, drawn from the stationary ensemble, following earlier work by van Beijeren. We find that the time dependence of this correlation is determined by the dissipation of the density fluctuations. We show that an exactly solvable linearized model captures the essential qualitative features seen in the finite size effects of the tagged particle correlations in the ASEP. Moreover, this linearized model also provides an exact coarse-grained description of two other microscopic models.

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