We consider the statistics of volume fluctuations in a one-dimensional classical gas of non-interacting particles confined by a piston, and subjected to an arbitrary external potential. We show that despite the absence of interactions between particles, volume fluctuations of the gas are non-Gaussian, and are described by generalized extreme value distributions. The continuous shape parameter of these distributions is related to the ratio between the force acting on the piston, and the force acting on the particles. Gaussian fluctuations are recovered in the strong compression limit, when the effect of the external potential becomes negligible. Consequences for the thermodynamics are also discussed.
We analyze the equlibrium statistics of a long linear homo-polymer chain confined in between two flat geometrical constraints under good solvent condition. The chain is ocupying two dimensional space and geometrical constraints are two impenetrable lines for the two dimensional space. A fully directed self avoiding walk lattice model is used to derive analytical expression of the partition function for the given value of separation in between the impenetrable lines. The exact values of the critical exponents ($ν_{||}, ν_{\perp}, ν$ and $ γ_1$) were obtained for different value of separations i…
The thermodynamic properties of systems with long-range interactions is still an ongoing challenge, both from the point of view of theory as well as computer simulation. In this work we study a model system, a Coulomb gas confined inside a sphere, by using the Wang-Landau algorithm. We have computed the configurational density of states (CDOS), the thermodynamic entropy and the caloric curve, and compared with microcanonical Metropolis simulations, while showing how concepts such as the configurational inverse temperature can be used to understand some aspects of thermodynamic behavior. A dyna…
In this paper, the effects of boundary and connectivity on ideal gases in two-dimensional confined space and three-dimensional tubes are discussed in detail based on the analytical result. The implication of such effects on the mesoscopic system is also revealed.
We study the behaviour of an ideal non-relativistic Bose gas in a three-dimensional space where one of the dimensions is compactified to form a circle. In this case there is no phase transition like that for the case of an infinite volume, nevertheless Bose-Einstein condensation signified by a sudden buildup of particles in the ground state can occur. We use the grand canonical ensemble to study this problem. In particular, the specific heat is evaluated numerically, as well as analytically in certain limits. We show analytically how the familiar result for the specific heat is recovered as we…
We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise, where Langevin representation is absent. In view of the Lévy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $ρ_*(x) \sim \exp [-Φ(x)]$. Here, we infer pdf $ρ(x,t)$ based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for $ρ(x,t)$ and its target pdf $ρ_*(x)$. T…
The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for non-thermal systems such as cold atoms in optical lattices, where the heat bath is replaced by the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center …
An intriguing connection between extreme value statistics and traveling fronts has been found recently in a number of diverse problems. In this brief review we outline a few such problems and consider their various applications.
It is shown that the Truncated Euler Equations, i.e. a finite set of ordinary differential equations for the amplitude of the large-scale modes, can correctly describe the complex transitional dynamics that occur within the turbulent regime of a confined 2D Navier-Stokes flow with bottom friction and a spatially periodic forcing. In particular, the random reversals of the large scale circulation on the turbulent background involve bifurcations of the probability distribution function of the large-scale circulation velocity that are described by the related microcanonical distribution which dis…
We study the effect of long range algebraic correlations on extreme value statistics and demonstrate that correlations can produce a limit distribution which is indistinguishable from the ubiquitous Bramwell-Holdsworth-Pinton distribution. We also consider the square-width fluctuations of the avalanche signal. We find, as recently predicted by T. Antal, M. Droz G. Gyorgyi and Z. Racz for logarithmic correlated 1/f signals, that these fluctuations follow the Fisher-Tippett-Gumbel distribution from uncorrelated extreme value statistics.
We establish a connection between the level density of a gas of non-interacting bosons and the theory of extreme value statistics. Depending on the exponent that characterizes the growth of the underlying single-particle spectrum, we show that at a given excitation energy the limiting distribution function for the number of excited particles follows the three universal distribution laws of extreme value statistics, namely Gumbel, Weibull and Fréchet. Implications of this result, as well as general properties of the level density at different energies, are discussed.
Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a $d$-dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile dist…
A length dependence of the effective mobility in the form of a power law, B ~ L^(1-1/alpha) is observed in dispersive transport in amorphous substances, with 0 < α< 1. We deduce this behavior as a simple consequence of the statistical theory of extreme events. We derive various quantities related to the largest value in samples of n trials, for the exponential and power-law probability densities of the individual events.
Characterizing the occupation statistics of a radiation flow through confined geometries is key to such technological issues as nuclear reactor design and medical diagnosis. This amounts to assessing the distribution of the travelled length $\ell$ and the number of collisions $n$ performed by the underlying stochastic transport process, for which remarkably simple Cauchy-like formulas were established in the case of branching Pearson random walks with exponentially distributed jumps. In this Letter, we show that such formulas strikingly carry over to the much broader class of branching process…
We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of non-interacting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-T-behavior in contrast to an earlier theory in Feynman's textbook on Statistical Mechanics, in which the special role of the ground state was ignored. The results are compared with an exact quantum mechanical treatment. Furthermore, we derive the fini…
This paper, about a fluid-like system of spatially confined particles, reveals the analytic structure for both, the canonical and grand canonical partition functions. The studied system is inhomogeneously distributed in a region whose boundary is made by planar faces without any particular symmetry. This type of geometrical body in the $d$-dimensional space is a polytope. The presented result in the case of $d=3$ gives the conditions under which the partition function is a polynomial in the volume, surface area, and edges length of the confinement vessel. Equivalent results for the cases $d=1,…
Derrida's model can be used for construction and transmission of error correcting codes, in an optimal way. we use the extreme statistics method to derive the optimal signal to noise ratio.
We consider the $N$ particle classical Riesz gas confined in a one-dimensional external harmonic potential with power law interaction of the form $1/r^k$ where $r$ is the separation between particles. As special limits it contains several systems such as Dyson's log-gas ($k\to 0^+$), Calogero-Moser model ($k=2$), 1d one component plasma ($k=-1$) and the hard-rod gas ($k\to \infty$). Despite its growing importance, only large-$N$ field theory and average density profile are known for general $k$. In this Letter, we study the fluctuations in the system by looking at the statistics of the gap bet…
Graphene nanochannels are relevant for their possible applications, as in water purification, and for the challenge of understanding how they change the properties of confined liquids. Here, we use all-atom molecular dynamics simulations to investigate water confined in an open graphene slit-pore as a function of its width $w$, down to sub-nm scale. We find that the water translational and rotational dynamics exhibits an oscillatory dependence on $w$, due to water layering. The oscillations in dynamics correlate with those in hydration pressure, which can be negative (hydrophobic attraction), …
Recently, the conditional maximum-entropy method (abbreviated as C-MaxEnt) has been proposed for selecting priors in Bayesian statistics in a very simple way. Here, it is examined for extreme-value statistics. For the Weibull type as an explicit example, it is shown how C-MaxEnt can give rise to a prior satisfying Jeffreys' rule.
A simple 1D lattice gas model is presented, which very well describes the equilibrium and kinetic behaviors of water confined in a thin carbon nanotube found in an atomistic molecular dynamics(MD) simulation {[} Nature {\bf 414}, 188 (2001) {]}. The model parameters are corresponding to various physical interactions and can be calculated or estimated in statistic mechanics. The roles of every interaction in the water filling, emptying and transporting processes are clearly understood. Our results indicate that the physical picture of the single-file kinetics is very simple.
We analyse extreme event statistics of experimentally realized Markov chains with various drifts. Our Markov chains are individual trajectories of a single atom diffusing in a one dimensional periodic potential. Based on more than 500 individual atomic traces we verify the applicability of the Sparre Andersen theorem to our system despite the presence of a drift. We present detailed analysis of four different rare event statistics for our system: the distributions of extreme values, of record values, of extreme value occurrence in the chain, and of the number of records in the chain. We observ…
We show that the dynamics of simple disordered models, like the directed Trap Model and the Random Energy Model, takes place at a coexistence point between active and inactive dynamical phases. We relate the presence of a dynamic phase transition in these models to the extreme value statistics of the associated random energy landscape.
We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process.
The fracture stress of materials typically depends on the sample size and is traditionally explained in terms of extreme value statistics. A recent work reported results on the carrying capacity of long polyamide and polyester wires and interpret the results in terms of a probabilistic argument known as the St. Petersburg paradox. Here, we show that the same results can be better explained in terms of extreme value statistics. We also discuss the relevance of rate dependent effects.
In this paper we calculate some exact solutions of the grand partition functions for quantum gases in confined space, such as ideal gases in two- and three-dimensional boxes, in tubes, in annular containers, on the lateral surface of cylinders, and photon gases in three-dimensional boxes. Based on these exact solutions, which, of course, contain the complete information about the system, we discuss the geometry effect which is neglected in the calculation with the thermodynamic limit $V\to \infty $, and analyze the validity of the quantum statistical method which can be used to calculate the g…
We propose a minimal model, based on active Brownian particles, for the dynamics of cells confined in a two-state micropattern, composed of two rectangular boxes connected by a bridge, and investigate the transition statistics. A transition between boxes occurs when the active particle crosses the center of the bridge, and the time between subsequent transitions is the dwell time. By assuming that the rotational diffusion time $τ$ is a function of the position, the main features of the transition statistics observed experimentally are recovered. $τ$ controls the transition from a ballistic reg…
Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of uncorrelated variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this note, we will first re…
We consider the conditional galaxy density around each galaxy, and study its fluctuations in the newest samples of the Sloan Digital Sky Survey Data Release 7. Over a large range of scales, both the average conditional density and its variance show a nontrivial scaling behavior, which resembles to criticality. The density depends, for 10 < r < 80 Mpc/h, only weakly (logarithmically) on the system size. Correspondingly, we find that the density fluctuations follow the Gumbel distribution of extreme value statistics. This distribution is clearly distinguishable from a Gaussian distribution, whic…
We consider a Brownian particle with diffusion coefficient $D$ in a $d$-dimensional ball of radius $R$ with reflecting boundaries. We study the maximum $M_x(t)$ of the trajectory of the particle along the $x$-direction at time $t$. In the long time limit, the maximum converges to the radius of the ball $M_x(t) \to R$ for $t\to \infty$. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations $Δ(t) = [R-M_x(t)]/R$ in the limit of large $t$ in all dimensions. We find that the distribution of $Δ(t)$ exhibits a rich variety of b…
As a rough model for the collective motions of cells and organisms we develop here the statistical mechanics of swarms of self-propelled particles. Our approach is closely related to the recently developed theory of active Brownian motion and the theory of canonical-dissipative systems. Free motion and motion of a swarms confined in an external field is studied. Briefly the case of particles confined on a ring and interacting by repulsive forces is studied. In more detail we investigate self-confinement by Morse-type attracting forces. We begin with pairs N = 2; the attractors and distribution…
An exact analytical description of extreme intensity statistics in complex random states is derived. These states have the statistical properties of the Gaussian and Circular Unitary Ensemble eigenstates of random matrix theory. Although the components are correlated by the normalization constraint, it is still possible to derive compact formulae for all values of the dimensionality N. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix…
We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.
We analyze the statistics of work generated by a gradient flow to stretch a nonlinear polymer. We obtain the Large Deviation Function (LDF) of the work in the full range of appropriate parameters by combining analytical and numerical tools. The LDF shows two distinct asymptotes: "near tails" are linear in work and dominated by coiled polymer configurations, while "far tails" are quadratic in work and correspond to preferentially fully stretched polymers. We find the extreme value statistics of work for several singular elastic potentials, as well as the mean and the dispersion of work near the…
We show that generalised extreme value statistics -the statistics of the k-th largest value among a large set of random variables- can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Frechet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumb…
Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of `uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this review, we will firs…
We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory. When $N\to\infty$, fluctuations around the maximum of the variables are described by the Fisher-Tippett-Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiting forms. Among these is the Gumbel distribution, for which the convergence rate with $N$ is of a logarithmic nature. Here, we present a theory that allows one to use the Gumbel limit to accurately approximate the exact extreme value distrib…
We consider the extreme value statistics of correlated random variables that arise from a Langevin equation. Recently, it was shown that the extreme values of the Ornstein-Uhlenbeck process follow a different distribution than those originating from its equilibrium measure, composed of independent and identically distributed Gaussian random variables. Here, we first focus on the discretely sampled Ornstein-Uhlenbeck process, which interpolates between these two limits. We show that in the limit of large times, its extreme values converge to those of the equilibrium distribution, instead of tho…
