A coalescence model for freely decaying two-dimensional turbulence

We propose a ballistic coalescence model (punctuated-Hamiltonian approach) mimicking the fusion of vortices in freely decaying two-dimensional turbulence. A temporal scaling behaviour is reached where the vortex density evolves like $t^{-\xi}$. A mean-field analytical argument yielding the approximation $\xi=4/5$ is shown to slightly overestimate the decay exponent $\xi$ whereas Molecular Dynamics simulations give $\xi =0.71\pm 0.01$, in agreement with recent laboratory experiments and simulations of Navier-Stokes equation.Comment: 6 pages, 1 figure, to appear in Europhysics Letter

Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all the solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension 2, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d?3$

Numerical renormalization group of vortex aggregation in 2D decaying turbulence: the role of three-body interactions

In this paper, we introduce a numerical renormalization group procedure which permits long-time simulations of vortex dynamics and coalescence in a 2D turbulent decaying fluid. The number of vortices decreases as $N\sim t^{-\xi}$, with $\xi\approx 1$ instead of the value $\xi=4/3$ predicted by a na\"{\i}ve kinetic theory. For short time, we find an effective exponent $\xi\approx 0.7$ consistent with previous simulations and experiments. We show that the mean square displacement of surviving vortices grows as $\sim t^{1+\xi/2}$. Introducing effective dynamics for two-body and three-body collisions, we justify that only the latter become relevant at small vortex area coverage. A kinetic theory consistent with this mechanism leads to $\xi=1$. We find that the theoretical relations between kinetic parameters are all in good agreement with experiments.Comment: 23 RevTex pages including 7 EPS figures. Submitted to Phys. Rev. E (Some typos corrected; see also cond-mat/9911032

Topological correlations in soap froths

Correlation in two-dimensional soap froth is analysed with an effective potential for the first time. Cells with equal number of sides repel (with linear correlation) while cells with different number of sides attract (with NON-bilinear) for nearest neighbours, which cannot be explained by the maximum entropy argument. Also, the analysis indicates that froth is correlated up to the third shell neighbours at least, contradicting the conventional ideas that froth is not strongly correlated.Comment: 10 Pages LaTeX, 6 Postscript figure

Analytical results for random walk persistence

In this paper, we present the detailed calculation of the persistence exponent $\theta$ for a nearly-Markovian Gaussian process $X(t)$, a problem initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. New resummed perturbative and non-perturbative expressions for $\theta$ are obtained, which suggest a connection with the result of the alternative independent interval approximation (IIA). The perturbation theory is extended to the calculation of $\theta$ for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and non-perturbative expressions for the persistence exponent $\theta(X_0)$, describing the probability that the process remains bigger than $X_0\sqrt{}$.Comment: 23 pages; accepted for publication to Phys. Rev. E (Dec. 98

Nontrivial Exponent for Simple Diffusion

The diffusion equation \partial_t\phi = \nabla^2\phi is considered, with initial condition \phi( _x_ ,0) a gaussian random variable with zero mean. Using a simple approximate theory we show that the probability p_n(t_1,t_2) that \phi( _x_ ,t) [for a given space point _x_ ] changes sign n times between t_1 and t_2 has the asymptotic form p_n(t_1,t_2) \sim [\ln(t_2/t_1)]^n(t_1/t_2)^{-\theta}. The exponent \theta has predicted values 0.1203, 0.1862, 0.2358 in dimensions d=1,2,3, in remarkably good agreement with simulation results.Comment: Minor typos corrected, affecting table of exponents. 4 pages, REVTEX, 1 eps figure. Uses epsf.sty and multicol.st

A precise approximation for directed percolation in d=1+1

We introduce an approximation specific to a continuous model for directed percolation, which is strictly equivalent to 1+1 dimensional directed bond percolation. We find that the critical exponent associated to the order parameter (percolation probability) is beta=(1-1/\sqrt{5})/2=0.276393202..., in remarkable agreement with the best current numerical estimate beta=0.276486(8).Comment: 4 pages, 3 EPS figures; Submitted to Physical Review Letters v2: minor typos + 1 major typo in Eq. (30) correcte

Effect of conduction electron interactions on Anderson impurities

The effect of conduction electron interactions for an Anderson impurity is investigated in one dimension using a scaling approach. The flow diagrams are obtained by solving the renormalization group equations numerically. It is found that the Anderson impurity case is different from its counterpart -- the Kondo impurity case even in the local moment region. The Kondo temperature for an Anderson impurity shows nonmonotonous behavior, increasing for weak interactions but decreasing for strong interactions. The implication of the study to other related impurity models is also discussed.Comment: 10 pages, revtex, 4 figures (the postscript file is included), to appear in Phys. Rev. B (Rapid Commun.

Spectrum and diffusion for a class of tight-binding models on hypercubes

We propose a class of exactly solvable anisotropic tight-binding models on an infinite-dimensional hypercube. The energy spectrum is analytically computed and is shown to be fractal and/or absolutely continuous according to the value hopping parameters. In both cases, the spectral and diffusion exponents are derived. The main result is that, even if the spectrum is absolutely continuous, the diffusion exponent for the wave packet may be anything between 0 and 1 depending upon the class of models.Comment: 5 pages Late