First-Passage Exponent in Two-Dimensional Soap Froth

Abstract

We report an experimental measurement of the temporal dependence of the area A us in a twodimensional soap froth which has not been swept by the passage of soap films up to time t, as the froth coarsens from an initial time t 0 within the scaling regime. We find A us scales with the mean cell area ͗A͘ as A us~͗ A͘ 2u 0 , with a first-passage exponent u 0 1. The study of domain coarsening following a quench from a high-temperature disordered phase to a low temperature ordered one is relevant in many areas of physics ranging from the evolution of crystalline domains in metallurgy, to the initial stages of evolution of the Universe in cosmology [1]. Typically, domain patterns evolve into a scaling regime in which statistical pattern properties are time invariant and the average domain size increases as a power law t a . While these features have been well established experimentally and theoretically in classic statistical mechanical problems such as spinodal decomposition and Lifshitz-Slyozov processes, attention has been shifted more recently to properties related to the zerotemperature relaxational dynamics of certain theoretical models, which being unrelated to the usual response functions, are characterized by new nontrivial universal exponents [2-4]. These properties are related to the survival probability of domains, namely, how long until they are overrun by domain walls. The exponents have been calculated in various ways in the case of one-dimensional q-state Potts models [5][6][7], and estimated numerically from computer simulations of the two-dimensional versions of these models [8]. An approximate method for computing these exponents for arbitrary dimensions (for the Ohta-JasnowKawasaki model [9]) was given in [10,11].To our knowledge, only one experimental work has addressed the measurement of an exponent of this type in a physical system [12]. The system consists of patterns of droplets condensing on a surface called breath figures, whose evolution is driven by droplet condensation and coalescence. Breath figures cannot be adequately modeled by domain coarsening of Potts models, and therefore a comparison between existing theoretical predictions based on these models and the experimental results was precluded.In this paper, we present the results of an experimental measurement of exponents related to the zero-temperature Potts models. Our experiment was performed on a two-dimensional soap froth, a system whose evolution is well mimicked by q-state Potts model for large q

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

“…We have checked numerically this exact asymptotic behavior (115) for the three classes of random polynomials (3,6,7). In all the three cases the pdf of the maximum λ max was obtained by averaging over 10 5 different realizations of the random Gaussian variables a i 's.…”

“…In conclusion, we have investigated different statistical properties of the real roots of three different types of real random polynomials (3,6,7). In these three classes, the mean density of real roots exhibit a rich variety of behaviors, as shown in Figs 1-3.…”

“…Section III is devoted to real random polynomials, where our main results are presented. In section III-A, we present a detailed study of the density of real roots for these three classes of polynomials, which turns out to behave quite differently in all the the three cases under investigations (3,6,7) . In section III-B, we will turn to the analysis of gap probabilities, which we will first analyse from the point of view of two-point correlations.…”

“…Among them, the persistence probability p 0 (t) received a particular attention, especially in the context of many-body non-equilibrium statistical physics, both analytically [1] as well as experimentally [2,3,4,5]. The persistence p 0 (t) for a time dependent stochastic process with zero mean is defined as the probability that it has not changed sign up to time t. In various physical situations, p 0 (t) has a power law tail p 0 (t) ∼ t −θ where θ turns out to be a non-trivial exponent whenever the stochastic process under study has a non Markovian dynamics.…”

Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensionalXYmodel

“…Persistence phenomena have been widely studied in recent years [1][2][3][4][5][6][7][8][9][10]. Theoretical and computational studies include spin systems in one [1] and higher [2] dimensions, diffusion fields [3], fluctuating interfaces [4], phaseordering dynamics [5], and reaction-diffusion systems [6].…”