Anisotropic random walks and asymptotically one-dimensional diffusion onabc-gaskets

Asymptotically one-dimensional diffusion processes are studied on the class of fractals calledabc-gaskets. The class is a set of certain variants of the Sierpiński gasket containing infinitely many fractals without any nondegenerate fixed point of renoramalization maps. While the “standard” method of constructing diffusions on the Sierpiński gasket and on nested fractals relies on the existence of a nondegenerate fixed point and hence it is not applicable to allabc-gaskets, the asymptotically one-dimensional diffusion is constructed on anyabc-gasket by means of an unstable degenerate fixed point. To this end, the generating functions for numbers of steps of anisotropic random walks on theabc-gaskets are analyzed, along the line of the authors' previous studies. In addition, a general stategy of handling random walk sequences with more than one parameter for the construction of asymptotically one-dimensional diffusion is proposed.     

Convection-enhanced diffusion for random flows

We analyze the effective diffusivity of a passive scalar in a two-dimensional, steady, incompressible random flow that has mean zero and a stationary stream function. We show that in the limit of small diffusivity or large Peclet number, with convection dominating, there is substantial enhancement of the effective diffusivity. Our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to be correct but have not been proved yet. We show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of Isichenko and Kalda that the effective diffusivity behaves likeɛ ^3/13 when the molecular diffusivityɛ is small, assuming some percolation-theoretic facts. We also analyze the effective diffusivity for a special class of convective flows, random cellular flows, where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows.     

Multiple scattering in random media

The paper is an application of a general microscopic approach to the theory of the average scattering matrix for a particle interacting with random scatterers. We present a detailed treatment for the case of uncorrelated positions of the scatterers. First, the general two-body additive approximation is used to truncate the hierarchy of correlation functions for fluctuations. It is shown that the self-energy is accurate through the fourth power of the individual scattering amplitude. Second, the hierarchy is terminated at the next stage. The self-energy is correct to the sixth power of the scattering amplitude.Work supported in part by the National Science Foundation under Contract No. NSF DMR 79-23213.     

Anomalous diffusion in a random velocity field

We construct diffusions in random velocity fields which present anomalous superdiffusive behavior. The mean square displacement can be made to have any power lawt for 1<2. Higher moments and characteristic functions are also investigated.     

Distribution functions for fluids in random media

A random medium is considered, composed of identifiable interactive sites or obstacles equilibrated at a high temperature and then quenched rapidly to form a rigid structure, statistically homogeneous on all but molecular length scales. The equilibrium statistical mechanics of a fluid contained inside this quenched medium is discussed. Various particle-particle and particle-obstacle correlation functions, which differ from the corresponding functions for a fully equilibrated binary mixture, are defined through an averaging process over the static ensemble of obstacle configurations and application of topological reduction techniques. The Ornstein-Zernike equations also differ from their equilibrium counterparts.     

Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

A hierarchical model for random walks in random media

We show that the random walk generated by a hierarchical Laplacian in ^d has standard diffusive behavior. Moreover, we show that this behavior is stable under a class of random perturbations that resemble an off-diagonal disordered lattice Laplacian. The density of states and its asymptotic behavior around zero energy are computed: singularities appear in one and two dimensions.     

Microwave propagation in dense random media

Summary Using simple, approximate arguments, we obtain a formula that relates the average spacing between peaks in the transmitted intensityvs. wave frequency distribution of a single configuration of a random distribution of scatterers to the diffusion constant, sample thickness, and effective absorption length. The value of the diffusion constant obtained this way is found to be within 20% of the value obtained via intensity-intensity autocorrelation function techniques. The author of this paper has agreed to not receive the proofs for correction.     

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.     

Diffusion of a Test Chain in a Quenched Background of Semidilute Polymers

Based on a recently established formalism [U. Ebert, J. Stat. Phys. 82:183 (1996)], we analyze the diffusive motion of a long polymer in a quenched random medium. The medium is modeled by a frozen semidilute polymer system. In the framework of standard renormalization group (RG) theory we present a systematic perturbative approach to handle such a many-chain system. In contrast to the work cited above, here we deal with long-range correlated disorder and find an attractive RG fixed point. Unlike in polymer statics, the semidilute limit here yields new nontrivial power laws for dynamic quantities. The exponents are intermediate between the Rouse and reptation results. An explicit one-loop calculation for the center-of-mass motion is given.     

Dielectric polarization in random media

The theory of dielectric polarization in random media is systematically formulated in terms of response kernels. The primary response kernel K(12) governs the mean dielectric response at the pointr ₁ to the external electric field at the pointr ₂ in an infinite system. The inverse of K(12) is denoted by L(12); it is simpler and more fundamental than K(12) itself. Rigorous expressions are obtained for the effective dielectric constant ^* in terms of L(12) and K(12). The latter expression involves the Onsager-Kirkwood function ( ^*– ₀)(2 ^*+ ₀) /₀^* (where ₀ is an arbitrary reference value), and appears to be new to the random medium context. A wide variety of series representations for ^* are generated by means of general perturbation expansions for K(12) and L(12). A discussion is given of certain pitfalls in the theory, most of which are related to the fact that the response kernels are long ranged. It is shown how the dielectric behavior of nonpolar molecular fluids may be treated as a special case of the general theory. The present results for ^* apply equally well to other effective phenomenological coefficients of the same generic type, such as thermal and electrical conductivity, magnetic susceptibility, and diffusion coefficients.Work performed under the auspices of the United States Department of Energy. A preliminary report on this work was given at the Eighth West Coast Statistical Mechanics Conference, University of California, Berkeley, 22 June 1982.     

Multiple scattering in random media I. Restricted two-body additive approximation

We present a microscopic theory of the problem of finding the properties of a particle interacting with potentials located at random sites. The sites are governed by a general probability distribution. The starting point is the multiple scattering equations for the amplitude k ₁|T |k ₂ in terms of the individual scattering amplitudes k ₁|T |k ₂. We work with quantitiesA defined by k ₁|T |k ₂=k ₁|T |k ₂exp[i(k ₁–k ₂)R ]. The theory is based on a splitting of the fundamental equation forA into equations for the mean A and the fluctuationsAA . Neglect of the fluctuations yields the quasicrystalline approximation. We rearrange the equation forAA to isolate the collective part of the fluctuations. We then make the simplest microscopic truncation which is thatAA is a restricted two-body additive function of the site positions. With the contribution of the collective fluctuations, this yields results forA that are accurate to ordert ⁴.Work supported in part by the National Science Foundation under Contract No. NSF DMRWork supported in part by the National Science Foundation under Contract No. NSF DMR     

Long time tails in stationary random media. I. Theory

Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t^–(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays liket ^–d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ^d/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.     

Localization estimates for a random discrete wave equation at high frequency

It is shown that at high frequencies matrix elements of the Green's function of a random discrete wave equation decay exponentially at long distances. This is the input to the proof of dense point spectrum with localized eigenfunctions in this frequency range. The proof uses techniques of Fröhlich and Spencer. A sequence of renormalization transformations shows that large regions where wave propagation is easily maintained become increasingly sparse as resonance is approached.     

McKean-Vlasov limit for interacting random processes in random media

We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.     

Two-state random walk model of diffusion. 2. Oscillatory diffusion

Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed with localized oscillations during its halts (e.g., at lattice sites). Earlier approaches proceedvia approximation schemes for the solution of the Fokker-Planck equation for diffusion in a periodic potential. In contrast, we visualize a two-state random walk in velocity space with the particle alternating between a state of flight and one of localized oscillation. Using simple, physically plausible inputs for the primary quantities characterising the random walk, we employ the powerful continuous-time random walk formalism to derive convenient and tractable closed-form expressions for all the objects of interest: the velocity autocorrelation, generalized diffusion constant, dynamic mobility, mean square displacement, dynamic structure factor (in the Gaussian approximation), etc. The interplay of the three characteristic times in the problem (the mean residence and flight times, and the period of the ‘local mode’) is elucidated. The emergence of a number of striking features of oscillatory diffusion (e.g., the local mode peak in the dynamic mobility and structure factor, and the transition between the oscillatory and diffusive regimes) is demonstrated.     

Persistent random walks in a one-dimensional random environment

Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (grant No. 476/82).     

Diffusion and reaction in random media and models of evolution processes

A diffusion equation including source terms, representing randomly distributed sources and sinks is considered. For quasilinear growth rates the eigenvalue problem is equivalent to that of the quantum mechanical motion of electrons in random fields. Correspondingly there exist localized and extended density distributions dependent on the statistics of the random field and on the dimension of the space. Besides applications in physics (nonequilibrium processes in pumped disordered solid materials) a new evolution model is discussed which considers evolution as hill climbing in a random landscape.We dedicate this work to the memory of Ilya M. Lifshitz.     

Trapping aspects in enhanced diffusion

We study the superlinear diffusion x ²(t) t , > 1, in layered media containing random velocity fields. The superlinear behavior holds in the case of random velocities along thex direction accompanied by diffusional motion in the space transverse to it. The transverse space can be either Euclidean, fractal, or ultrametric. For a one-dimensional transverse space we derive exact expressions for the higher moments of the displacement. Furthermore, we investigate the propagatorP(x, t) along thex direction and establish its scaling behavior. Our analysis highlights the resemblance between the stretched-Gaussian behavior of the propagator and the stretched-exponential form of the survivial probability in the trapping problem; both show late crossover behavior.This work is dedicated to Prof. George H. Weiss.