Nonlinear diffusion from Einstein's master equation

We generalize Einstein's master equation for random walk processes by considering that the probability for a particle at position $r$ to make a jump of length $j$ lattice sites, $P_j(r)$ is a functional of the particle distribution function $f(r,t)$. By multiscale expansion, we obtain a generalized advection-diffusion equation. We show that the power law $P_j(r) \propto f(r)^{\alpha - 1}$ (with $\alpha > 1$) follows from the requirement that the generalized equation admits of scaling solutions ($f(r;t) = t^{-\gamma}\phi (r/t^{\gamma})$). The solutions have a $q$-exponential form and are found to be in agreement with the results of Monte-Carlo simulations, so providing a microscopic basis validating the nonlinear diffusion equation. Although its hydrodynamic limit is equivalent to the phenomenological porous media equation, there are extra terms which, in general, cannot be neglected as evidenced by the Monte-Carlo computations.}Comment: 7 pages incl. 3 fig

Nonextensive diffusion as nonlinear response

The porous media equation has been proposed as a phenomenological ``non-extensive'' generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous media equation in that it describes \emph{% generalized classical} diffusion, i.e. with $r/\sqrt Dt$ scaling, but with a generalized Einstein relation, and with power-law probability distributions typical of nonextensive statistical mechanics

Lattice gas with ``interaction potential''

We present an extension of a simple automaton model to incorporate non-local interactions extending over a spatial range in lattice gases. {}From the viewpoint of Statistical Mechanics, the lattice gas with interaction range may serve as a prototype for non-ideal gas behavior. {}From the density fluctuations correlation function, we obtain a quantity which is identified as a potential of mean force. Equilibrium and transport properties are computed theoretically and by numerical simulations to establish the validity of the model at macroscopic scale.Comment: 12 pages LaTeX, figures available on demand ([email protected]

Is the Tsallis entropy stable?

The question of whether the Tsallis entropy is Lesche-stable is revisited. It is argued that when physical averages are computed with the escort probabilities, the correct application of the concept of Lesche-stability requires use of the escort probabilities. As a consequence, as shown here, the Tsallis entropy is unstable but the thermodynamic averages are stable. We further show that Lesche stability as well as thermodynamic stability can be obtained if the homogeneous entropy is used as the basis of the formulation of non-extensive thermodynamics. In this approach, the escort distribution arises naturally as a secondary structure.Comment: 6 page

Statistics of precursors to fingering processes

We present an analysis of the statistical properties of hydrodynamic field fluctuations which reveal the existence of precursors to fingering processes. These precursors are found to exhibit power law distributions, and these power laws are shown to follow from spatial $q$-Gaussian structures which are solutions to the generalized non-linear diffusion equation.Comment: 7 pages incl. 5 figs; tp appear in Europhysics Letter

Heavy Quark Diffusion from the Lattice

We study the diffusion of heavy quarks in the Quark Gluon Plasma using the Langevin equations of motion and estimate the contribution of the transport peak to the Euclidean current-current correlator. We show that the Euclidean correlator is remarkably insensitive to the heavy quark diffusion coefficient and give a simple physical interpretation of this result using the free streaming Boltzmann equation. However if the diffusion coefficient is smaller than $\sim 1/(\pi T)$, as favored by RHIC phenomenology, the transport contribution should be visible in the Euclidean correlator. We outline a procedure to isolate this contribution.Comment: 24 pages, 5 figure

Dynamics of short polymer chains in solution

We present numerical and analytical results describing the effect of hydrodynamic interactions on the dynamics of a short polymer chain in solution. A molecular dynamics algorithm for the polymer is coupled to a direct simulation Monte Carlo algorithm for the solvent. We give an explicit expression for the velocity autocorrelation function of the centre of mass of the polymer which agrees well with numerical results if Brownian dynamics, hydrodynamic correlations and sound wave scattering are included

Computer Simulation Study of the Phase Behavior and Structural Relaxation in a Gel-Former Modeled by Three Body Interactions

We report a computer simulation study of a model gel-former obtained by modifying the three-body interactions of the Stillinger-Weber potential for silicon. This modification reduces the average coordination number and consequently shifts the liquid-gas phase coexistence curve to low densities, thus facilitating the formation of gels without phase separation. At low temperatures and densities, the structure of the system is characterized by the presence of long linear chains interconnected by a small number of three coordinated junctions at random locations. At small wave-vectors the static structure factor shows a non-monotonic dependence on temperature, a behavior which is due to the competition between the percolation transition of the particles and the stiffening of the formed chains. We compare in detail the relaxation dynamics of the system as obtained from molecular dynamics with the one obtained from Monte Carlo dynamics. We find that the bond correlation function displays stretched exponential behavior at moderately low temperatures and densities, but exponential relaxation at low temperatures. The bond lifetime shows an Arrhenius behavior, independent of the microscopic dynamics. For the molecular dynamics at low temperatures, the mean squared displacement and the (coherent and incoherent) intermediate scattering function display at intermediate times a dynamics with ballistic character and we show that this leads to compressed exponential relaxation. For the Monte Carlo dynamics we find always an exponential or stretched exponential relaxation. Thus we conclude that the compressed exponential relaxation observed in experiments is due to the out-of-equilibrium dynamics

Finite Temperature Spectral Densities of Momentum and R-Charge Correlators in N=4\N=4 Yang Mills Theory

We compute spectral densities of momentum and R-charge correlators in thermal $\N=4$ Yang Mills at strong coupling using the AdS/CFT correspondence. For $\omega \sim T$ and smaller, the spectral density differs markedly from perturbation theory; there is no kinetic theory peak. For large $\omega$, the spectral density oscillates around the zero-temperature result with an exponentially decreasing amplitude. Contrast this with QCD where the spectral density of the current-current correlator approaches the zero temperature result like $(T/\omega)^4$. Despite these marked differences with perturbation theory, in Euclidean space-time the correlators differ by only $\sim 10$ from the free result. The implications for Lattice QCD measurements of transport are discussed.Comment: 18 pages, 3 figure

Evidence for compact cooperatively rearranging regions in a supercooled liquid

We examine structural relaxation in a supercooled glass-forming liquid simulated by NVE molecular dynamics. Time correlations of the total kinetic energy fluctuations are used as a comprehensive measure of the system's approach to the ergodic equilibrium. We find that, under cooling, the total structural relaxation becomes delayed as compared with the decay of the component of the intermediate scattering function corresponding to the main peak of the structure factor. This observation can be explained by collective movements of particles preserving many-body structural correlations within compact 3D cooperatively rearranging regions.Comment: 8 pages, 4 figure