Renormalization of Drift and Diffusivity in Random Gradient Flows

We investigate the relationship between the effective diffusivity and effective drift of a particle moving in a random medium. The velocity of the particle combines a white noise diffusion process with a local drift term that depends linearly on the gradient of a gaussian random field with homogeneous statistics. The theoretical analysis is confirmed by numerical simulation. For the purely isotropic case the simulation, which measures the effective drift directly in a constant gradient background field, confirms the result previously obtained theoretically, that the effective diffusivity and effective drift are renormalized by the same factor from their local values. For this isotropic case we provide an intuitive explanation, based on a {\it spatial} average of local drift, for the renormalization of the effective drift parameter relative to its local value. We also investigate situations in which the isotropy is broken by the tensorial relationship of the local drift to the gradient of the random field. We find that the numerical simulation confirms a relatively simple renormalization group calculation for the effective diffusivity and drift tensors.Comment: Latex 16 pages, 5 figures ep

Effective diffusion constant in a two dimensional medium of charged point scatterers

We obtain exact results for the effective diffusion constant of a two dimensional Langevin tracer particle in the force field generated by charged point scatterers with quenched positions. We show that if the point scatterers have a screened Coulomb (Yukawa) potential and are uniformly and independently distributed then the effective diffusion constant obeys the Volgel-Fulcher-Tammann law where it vanishes. Exact results are also obtained for pure Coulomb scatterers frozen in an equilibrium configuration of the same temperature as that of the tracer.Comment: 9 pages IOP LaTex, no figure

Some observations on the renormalization of membrane rigidity by long-range interactions

We consider the renormalization of the bending and Gaussian rigidity of model membranes induced by long-range interactions between the components making up the membrane. In particular we analyze the effect of a finite membrane thickness on the renormalization of the bending and Gaussian rigidity by long-range interactions. Particular attention is paid to the case where the interactions are of a van der Waals type.Comment: 11 pages RexTex, no figure

Aging on Parisi's tree

We present a detailed study of simple `tree' models for off equilibrium dynamics and aging in glassy systems. The simplest tree describes the landscape of a random energy model, whereas multifurcating trees occur in the solution of the Sherrington-Kirkpatrick model. An important ingredient taken from these models is the exponential distribution of deep free-energies, which translate into a power-law distribution of the residence time within metastable `valleys'. These power law distributions have infinite mean in the spin-glass phase and this leads to the aging phenomenon. To each level of the tree are associated an overlap and the exponent of the time distribution. We solve these models for a finite (but arbitrary) number of levels and show that a two level tree accounts very well for many experimental observations (thermoremanent magnetisation, a.c susceptibility, second noise spectrum....). We introduce the idea that the deepest levels of the tree correspond to equilibrium dynamics whereas the upper levels correspond to aging. Temperature cycling experiments suggest that the borderline between the two is temperature dependent. The spin-glass transition corresponds to the temperature at which the uppermost level is put out of equilibrium but is subsequently followed by a sequence of (dynamical) phase transitions corresponding to non equilibrium dynamics within deeper and deeper levels. We tentatively try to relate this `tree' picture to the real space `droplet' model, and speculate on how the final description of spin-glasses might look like.Comment: 30 pages, RevTeX, 9 figures, available on request, report # 077 / SPEC / 199

Weak non-linear surface charging effects in electrolytic films

A simple model of soap films with nonionic surfactants stabilized by added electrolyte is studied. The model exhibits charge regularization due to the incorporation of a physical mechanism responsible for the formation of a surface charge. We use a Gaussian field theory in the film but the full non-linear surface terms which are then treated at a one-loop level by calculating the mean-field Poisson-Boltzmann solution and then the fluctuations about this solution. We carefully analyze the renormalization of the theory and apply it to a triple layer model for a thin film with Stern layer of thickness $h$. For this model we give expressions for the surface charge $\sigma(L)$ and the disjoining pressure $P_d(L)$ and show their dependence on the parameters. The influence of image charges naturally arise in the formalism and we show that predictions depend strongly on $h$ because of their effects. In particular, we show that the surface charge vanishes as the film thickness $L \to 0$. The fluctuation terms about this class of theories exhibit a Casimir-like attraction across the film and although this attraction is well known to be negligible compared with the mean-field component for thick films in the presence of electrolyte, in the model studied here these fluctuations also affect the surface charge regulation leading to a fluctuation component in the disjoining pressure which has the same behavior as the mean-field component even for large film thickness.Comment: 17 pages, 12 figures, latex sourc

Boundary Effects in the One Dimensional Coulomb Gas

We use the functional integral technique of Edwards and Lenard to solve the statistical mechanics of a one dimensional Coulomb gas with boundary interactions leading to surface charging. The theory examined is a one dimensional model for a soap film. Finite size effects and the phenomenon of charge regulation are studied. We also discuss the pressure of disjunction for such a film. Even in the absence of boundary potentials we find that the presence of a surface affects the physics in finite systems. In general we find that in the presence of a boundary potential the long distance disjoining pressure is positive but may become negative at closer interplane separations. This is in accordance with the attractive forces seen at close separations in colloidal and soap film experiments and with three dimensional calculations beyond mean field. Finally our exact results are compared with the predictions of the corresponding Poisson-Boltzmann theory which is often used in the context of colloidal and thin liquid film systems.Comment: 28 pages, LATEX2e, 11 figures, uses styles[12pt] resubmission because of minor corrections to tex

Metastable states of spin glasses on random thin graphs

In this paper we calculate the mean number of metastable states for spin glasses on so called random thin graphs with couplings taken from a symmetric binary distribution $\pm J$. Thin graphs are graphs where the local connectivity of each site is fixed to some value $c$. As in totally connected mean field models we find that the number of metastable states increases exponentially with the system size. Furthermore we find that the average number of metastable states decreases as $c$ in agreement with previous studies showing that finite connectivity corrections of order $1/c$ increase the number of metastable states with respect to the totally connected mean field limit. We also prove that the average number of metastable states in the limit $c\to\infty$ is finite and converges to the average number of metastable states in the Sherrington-Kirkpatrick model. An annealed calculation for the number of metastable states $N_{MS}(E)$ of energy $E$ is also carried out giving a lower bound on the ground state energy of these spin glasses. For small $c$ one may obtain analytic expressions for $$.Comment: 13 pages LateX, 3 figures .ep

Dynamical transition for a particle in a squared Gaussian potential

We study the problem of a Brownian particle diffusing in finite dimensions in a potential given by $\psi= \phi^2/2$ where $\phi$ is Gaussian random field. Exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power-law fashion at the dynamical transition temperature. Our results are confronted with numerical simulations where the Gaussian field is constructed, in a standard way, as a sum over random Fourier modes. We show that when the number of Fourier modes is finite the low temperature diffusion constant becomes non-zero and has an Arrhenius form. Thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition. In addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model.Comment: 18 pages, 4 figures .eps, JPA styl

A perturbative path integral study of active and passive tracer diffusion in fluctuating fields

We study the effective diffusion constant of a Brownian particle linearly coupled to a thermally fluctuating scalar field. We use a path integral method to compute the effective diffusion coefficient perturbatively to lowest order in the coupling constant. This method can be applied to cases where the field is affected by the particle (an active tracer), and cases where the tracer is passive. Our results are applicable to a wide range of physical problems, from a protein diffusing in a membrane to the dispersion of a passive tracer in a random potential. In the case of passive diffusion in a scalar field, we show that the coupling to the field can, in some cases, speed up the diffusion corresponding to a form of stochastic resonance. Our results on passive diffusion are also confirmed via a perturbative calculation of the probability density function of the particle in a Fokker-Planck formulation of the problem. Numerical simulations on simplified systems corroborate our results.Comment: 13 pages RevTex, 4 figure

Path integrals for stiff polymers applied to membrane physics

Path integrals similar to those describing stiff polymers arise in the Helfrich model for membranes. We show how these types of path integrals can be evaluated and apply our results to study the thermodynamics of a minority stripe phase in a bulk membrane. The fluctuation induced contribution to the line tension between the stripe and the bulk phase is computed, as well as the effective interaction between the two phases in the tensionless case where the two phases have differing bending rigidities.Comment: 11 pages RevTex, 4 figure