Nonextensive statistics in viscous fingering

Measurements in turbulent flows have revealed that the velocity field in nonequilibrium systems exhibits $q$-exponential or power law distributions in agreement with theoretical arguments based on nonextensive statistical mechanics. Here we consider Hele-Shaw flow as simulated by the Lattice Boltzmann method and find similar behavior from the analysis of velocity field measurements. For the transverse velocity, we obtain a spatial $q$-Gaussian profile and a power law velocity distribution over all measured decades. To explain these results, we suggest theoretical arguments based on Darcy's law combined with the non-linear advection-diffusion equation for the concentration field. Power law and $q$-exponential distributions are the signature of nonequilibrium systems with long-range interactions and/or long-time correlations, and therefore provide insight to the mechanism of the onset of fingering processes.Comment: 8 pages including 3 figures; to appear in PHYSICA

Propagation-Dispersion Equation

A {\em propagation-dispersion equation} is derived for the first passage distribution function of a particle moving on a substrate with time delays. The equation is obtained as the continuous limit of the {\em first visit equation}, an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as {\em time-delayers}. The propagation-dispersion equation should be contrasted with the advection-diffusion equation (or the classical Fokker-Planck equation) as it describes a dispersion process in {\em time} (instead of diffusion in space) with a drift expressed by a propagation speed with non-zero bounded values. The {\em temporal dispersion} coefficient is shown to exhibit a form analogous to Taylor's dispersivity. Physical systems where the propagation-dispersion equation applies are discussed.Comment: 12 pages+ 5 figures, revised and extended versio

Viscous fingering in miscible, immiscible and reactive fluids

With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of Hele-Shaw flow under conditions corresponding to various experimental systems. We discuss the onset of the instability (dispersion relation), the static properties (characterization of the interface) and the dynamic properties (growth of the mixing zone) of simulated Hele-Shaw systems. We examine the role of reactive processes (between the two fluids) and we show that they have a sharpening effect on the interface similar to the effect of surface tension.Comment: 6 pages with 2 figure, to be published in J.Mod.Phys

Propagation and organization in lattice random media

We show that a signal can propagate in a particular direction through a model random medium regardless of the precise state of the medium. As a prototype, we consider a point particle moving on a one-dimensional lattice whose sites are occupied by scatterers with the following properties: (i) the state of each site is defined by its spin (up or down); (ii) the particle arriving at a site is scattered forward (backward) if the spin is up (down); (iii) the state of the site is modified by the passage of the particle, i.e. the spin of the site where a scattering has taken place, flips ($\uparrow \Leftrightarrow \downarrow$). We consider one dimensional and triangular lattices, for which we give a microscopic description of the dynamics, prove the propagation of a particle through the scatterers, and compute analytically its statistical properties. In particular we prove that, in one dimension, the average propagation velocity is $= 1/(3-2q)$, with $q$ the probability that a site has a spin $\uparrow$, and, in the triangular lattice, the average propagation velocity is independent of the scatterers distribution: $= 1/8$. In both cases, the origin of the propagation is a blocking mechanism, restricting the motion of the particle in the direction opposite to the ultimate propagation direction, and there is a specific re-organization of the spins after the passage of the particle. A detailed mathematical analysis of this phenomenon is, to the best of our knowledge, presented here for the first time.Comment: 30 pages, 15 separate figures (in PostScript); submitted to J. Stat. Phy

Dependence of the liquid-vapor surface tension on the range of interaction: a test of the law of corresponding states

The planar surface tension of coexisting liquid and vapor phases of a fluid of Lennard-Jones atoms is studied as a function of the range of the potential using both Monte Carlo simulations and Density Functional Theory. The interaction range is varied from $r_c^* = 2.5$ to $r_c^* = 6$ and the surface tension is determined for temperatures ranging from $T^* = 0.7$ up to the critical temperature in each case. The results are shown to be consistent with previous studies. The simulation data are well-described by Guggenheim's law of corresponding states but the agreement of the theoretical results depends on the quality of the bulk equation of state.Comment: 13 pages, 5 figure

Phase behavior of a confined nano-droplet in the grand-canonical ensemble: the reverse liquid-vapor transition

The equilibrium density distribution and thermodynamic properties of a Lennard-Jones fluid confined to nano-sized spherical cavities at constant chemical potential was determined using Monte Carlo simulations. The results describe both a single cavity with semipermeable walls as well as a collection of closed cavities formed at constant chemical potential. The results are compared to calculations using classical Density Functional Theory (DFT). It is found that the DFT calculations give a quantitatively accurate description of the pressure and structure of the fluid. Both theory and simulation show the presence of a ``reverse'' liquid-vapor transition whereby the equilibrium state is a liquid at large volumes but becomes a vapor at small volumes.Comment: 13 pages, 8 figures, to appear in J. Phys. : Cond. Mat

Long-range correlations in non-equilibrium systems: Lattice gas automaton approach

In systems removed from equilibrium, intrinsic microscopic fluctuations become correlated over distances comparable to the characteristic macroscopic length over which the external constraint is exerted. In order to investigate this phenomenon, we construct a microscopic model with simple stochastic dynamics using lattice gas automaton rules that satisfy local detailed balance. Because of the simplicity of the automaton dynamics, analytical theory can be developed to describe the space and time evolution of the density fluctuations. The exact equations for the pair correlations are solved explicitly in the hydrodynamic limit. In this limit, we rigorously derive the results obtained phenomenologically by fluctuating hydrodynamics. In particular, the spatial algebraic decay of the equal-time fluctuation correlations predicted by this theory is found to be in excellent agreement with the results of our lattice gas automaton simulations for two different types of boundary conditions. Long-range correlations of the type described here appear generically in dynamical systems that exhibit large scale anisotropy and lack detailed balance.Comment: 23 pages, RevTeX; to appear in Phys. Rev.

Effects of protein size on the highconcentration/low-concentration phase transition

SCOPUS: ar.kinfo:eu-repo/semantics/publishe

Hydrodynamique statistique des gaz sur réseau

Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe