Global fluctuations and Gumbel statistics

We explain how the statistics of global observables in correlated systems can be related to extreme value problems and to Gumbel statistics. This relationship then naturally leads to the emergence of the generalized Gumbel distribution G_a(x), with a real index a, in the study of global fluctuations. To illustrate these findings, we introduce an exactly solvable nonequilibrium model describing an energy flux on a lattice, with local dissipation, in which the fluctuations of the global energy are precisely described by the generalized Gumbel distribution.Comment: 4 pages, 3 figures; final version with minor change

Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces

We investigate the statistics of the maximal fluctuation of two-dimensional Gaussian interfaces. Its relation to the entropic repulsion between rigid walls and a confined interface is used to derive the average maximal fluctuation $\sim \sqrt{2/(\pi K)} \ln N$ and the asymptotic behavior of the whole distribution $P(m) \sim N^2 e^{-{\rm (const)} N^2 e^{-\sqrt{2\pi K} m} - \sqrt{2\pi K} m}$ for $m$ finite with $N^2$ and $K$ the interface size and tension, respectively. The standardized form of $P(m)$ does not depend on $N$ or $K$, but shows a good agreement with Gumbel's first asymptote distribution with a particular non-integer parameter. The effects of the correlations among individual fluctuations on the extreme value statistics are discussed in our findings.Comment: 4 pages, 4 figures, final version in PR

Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d=1,2,3d=1,2,3 via a Monte-Carlo procedure in the disorder

In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, K\"orner, Katzgraber and Hartmann \cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo Markov chain in the disorder. In this paper, we combine their Monte-Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution $P_d(E_0)$ for the directed polymer in a random medium of dimension $d=1,2,3$. In $d=1$, we check the validity of the algorithm by a direct comparison with the exact result, namely the Tracy-Widom distribution. In dimensions $d=2$ and $d=3$, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our results are in agreement with Zhang's argument, stating that the negative tail exponent $\eta(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{\eta(d)}$ as $E_0 \to -\infty$ is directly related to the fluctuation exponent $\theta(d)$ (which governs the fluctuations $\Delta E_0(L) \sim L^{\theta(d)}$ of the ground state energy $E_0$ for polymers of length $L$) via the simple formula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on the similarities and differences with spin-glasses.Comment: 13 pages, 16 figure

Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics

In this paper, we view fluctuating fronts made of particles on a one-dimensional lattice as an extreme value problem. The idea is to denote the configuration for a single front realization at time $t$ by the set of co-ordinates $\{k_i(t)\}\equiv[k_1(t),k_2(t),...,k_{N(t)}(t)]$ of the constituent particles, where $N(t)$ is the total number of particles in that realization at time $t$. When $\{k_i(t)\}$ are arranged in the ascending order of magnitudes, the instantaneous front position can be denoted by the location of the rightmost particle, i.e., by the extremal value $k_f(t)=\text{max}[k_1(t),k_2(t),...,k_{N(t)}(t)]$. Due to interparticle interactions, $\{k_i(t)\}$ at two different times for a single front realization are naturally not independent of each other, and thus the probability distribution $P_{k_f}(t)$ [based on an ensemble of such front realizations] describes extreme value statistics for a set of correlated random variables. In view of the fact that exact results for correlated extreme value statistics are rather rare, here we show that for a fermionic front model in a reaction-diffusion system, $P_{k_f}(t)$ is Gaussian. In a bosonic front model however, we observe small deviations from the Gaussian.Comment: 6 pages, 3 figures, miniscule changes on the previous version, to appear in Phys. Rev.

Contest based on a directed polymer in a random medium

We introduce a simple one-parameter game derived from a model describing the properties of a directed polymer in a random medium. At his turn, each of the two players picks a move among two alternatives in order to maximize his final score, and minimize opponent's return. For a game of length $n$, we find that the probability distribution of the final score $S_n$ develops a traveling wave form, ${\rm Prob}(S_n=m)=f(m-v n)$, with the wave profile $f(z)$ unusually decaying as a double exponential for large positive and negative $z$. In addition, as the only parameter in the game is varied, we find a transition where one player is able to get his maximum theoretical score. By extending this model, we suggest that the front velocity $v$ is selected by the nonlinear marginal stability mechanism arising in some traveling wave problems for which the profile decays exponentially, and for which standard traveling wave theory applies

Classical diffusion of N interacting particles in one dimension: General results and asymptotic laws

I consider the coupled one-dimensional diffusion of a cluster of N classical particles with contact repulsion. General expressions are given for the probability distributions, allowing to obtain the transport coefficients. In the limit of large N, and within a gaussian approximation, the diffusion constant is found to behave as N^{-1} for the central particle and as (\ln N)^{-1} for the edge ones. Absolute correlations between the edge particles increase as (\ln N)^{2}. The asymptotic one-body distribution is obtained and discussed in relation of the statistics of extreme events.Comment: 6 pages, 2 eps figure

Distribution of meteoric smoke ? sensitivity to microphysical properties and atmospheric conditions

International audienceMeteoroids entering the Earth's atmosphere experience strong deceleration and ablate, whereupon the resulting material is believed to re-condense to nanometre-size "smoke particles". These particles are thought to be of great importance for many middle atmosphere phenomena, such as noctilucent clouds, polar mesospheric summer echoes, metal layers, and heterogeneous chemistry. The properties and distribution of meteoric smoke depend on poorly known or highly variable factors such as the amount, composition and velocity of incoming meteoric material, the efficiency of coagulation, and the state and circulation of the atmosphere. This work uses a one-dimensional microphysical model to investigate the sensitivities of meteoric smoke properties to these poorly known or highly variable factors. The resulting uncertainty or variability of meteoric smoke quantities such as number density, mass density, and size distribution are determined. It is found that the two most important factors are the efficiency of the coagulation and background vertical wind. The seasonal variation of the vertical wind in the mesosphere implies strong global and temporal variations in the meteoric smoke distribution. This contrasts the simplistic picture of a homogeneous global meteoric smoke layer, which is currently assumed in many studies of middle atmospheric phenomena. In particular, our results suggest a very low number of nanometre-sized smoke particles at the summer mesopause where they are thought to serve as condensation nuclei for noctilucent clouds

Sensitivity of meteoric smoke distribution to microphysical properties and atmospheric conditions

International audienceMeteoroids entering the Earth's atmopsphere experience strong deceleration and ablate, whereupon the resulting material is believed to re-condense to nanometre-size "smoke particles". These particles are thought to be of great importance for many middle atmosphere phenomena, such as noctilucent clouds, polar mesospheric summer echoes, metal layers, and heterogeneous chemistry. The properties and distribution of meteoric smoke depend on poorly known or highly variable factors such as the amount, composition and velocity of incoming meteoric material, the efficiency of coagulation, and the state and circulation of the atmosphere. This work uses a one-dimensional microphysical model to investigate the sensitivities of meteoric smoke properties to these poorly known or highly variable factors. The resulting uncertainty or variability of meteoric smoke quantities such as number density, mass density, and size distribution are determined. It is found that the two most important factors are the efficiency of the coagulation and background vertical wind. The seasonal variation of the vertical wind in the mesosphere implies strong global and temporal variations in the meteoric smoke distribution. This contrasts the simplistic picture of a homogeneous global meteoric smoke layer, which is currently assumed in many studies of middle atmospheric phenomena. In particular, our results suggest a very low number of nanometre-sized smoke particles at the summer mesopause where they are thought to serve as condensation nuclei for noctilucent clouds

Absolute density measurements in the middle atmosphere

In the last ten years a total of 25 sounding rockets employing ionization gauges have been launched at high latitudes ( ~ 70° N) to measure total atmospheric density and its small scale fluctuations in an altitude range between 70 and 110 km. While the determination of small scale fluctuations is unambiguous, the total density analysis has been complicated in the past by aerodynamical disturbances leading to densities inside the sensor which are enhanced compared to atmospheric values. Here, we present the results of both Monte Carlo simulations and wind tunnel measurements to quantify this aerodynamical effect. The comparison of the resulting ‘ram-factor’ profiles with empirically determined density ratios of ionization gauge measurements and falling sphere measurements provides excellent agreement. This demonstrates both the need, but also the possibility, to correct aerodynamical influences on measurements from sounding rockets. We have determined a total of 20 density profiles of the mesosphere-lower-thermosphere (MLT) region. Grouping these profiles according to season, a listing of mean density profiles is included in the paper. A comparison with density profiles taken from the reference atmospheres CIRA86 and MSIS90 results in differences of up to 40%. This reflects that current reference atmospheres are a significant potential error source for the determination of mixing ratios of, for example, trace gas constituents in the MLT region