Diffusion in disordered media

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.     

Effect of macromolecular crowding on the rate of diffusion-limited enzymatic reaction

The cytoplasm of a living cell is crowded with several macromolecules of different shapes and sizes. Molecular diffusion in such a medium becomes anomalous due to the presence of macromolecules and diffusivity is expected to decrease with increase in macromolecular crowding. Moreover, many cellular processes are dependent on molecular diffusion in the cell cytosol. The enzymatic reaction rate has been shown to be affected by the presence of such macromolecules. A simple numerical model is proposed here based on percolation and diffusion in disordered systems to study the effect of macromolecular crowding on the enzymatic reaction rates. The model qualitatively explains some of the experimental observations.      

Locally ordered regions in the phase transition in the systems with a finite-range correlated quenched disorder

The locally ordered regions (LOR) in the phase transition in disordered systems are studied. There are two parts in this paper. One part is to report our numerical results on the one-dimensional saddle point equation of the Ginzburg-Landau Hamiltonian with random temperature in the presence of an ordering field. The disordered system is modelled as a lattice, on which each cell has a local reduced temperature. The random part of the local reduced temperature is distributed in the Gaussian form. The one-dimensional saddle point equation is solved numerically. The average, the fluctuation and the correlation length of the solution are calculated. The scaling relations for these quantities with the temperature, the ordering field and the disorder strength are derived. The numerical data are fitted with the scaling relations well. Another part is to discuss qualitatively the phase diagram of the finite-range correlated disordered systems. There are two proposed classes for the phase transition in connection with the LOR. One class is described by the percolative scenario, in which the phase transition is inhomogeneous. In the percolative scenario the percolation of the LOR dominates the phase transition. In another class, the phase transition is homogeneous, and can be described by the renormalization group (RG) with replica symmetry breaking (RSB). In the RG with RSB, there is nothing to do with the percolation of LOR. We shall show that these two theories, which seem contradictory, may describe two parts of the whole phase diagram. Whether the phase transition is homogeneous or inhomogeneous depends on the interaction between the LOR. If the interaction between the LOR is strong enough, the phase transition is percolative and inhomogeneous. If the interaction between the LOR is weak, the phase transition is homogeneous. The interaction between the LOR is discussed with the numerical solution on the saddle point equation.     

Molecular dynamics studies of slow dynamics in random media: Type A-B and reentrant transitions

Molecular dynamics simulations of mobile particles confined in disordered immobile particles are carried out. Slow dynamics in random media are characterized by two types of dynamics: Type B dynamics for large mobile particle density and Type A dynamics for small mobile particle density. The crossover from Type A to B dynamics is studied by the mean square displacement and the density correlation function. Our results are qualitatively consistent with the results of recent numerical and theoretical studies on relevant spatially heterogeneous systems. We also investigate the effect of random matrix generation on the dynamics of mobile particles in order to examine the reentrant transition predicted by the recent mode-coupling theory. Our simulations demonstrate that the diffusion of the mobile particles largely depends on the protocol of the random matrix generation and that the reentrant transition is observed for a particular protocol.     

Optical percolation in ceramics assisted by porous clusters

We investigated optical transparency in ceramics assisted by disordered porous clusters. The structure and statistical properties of three-dimensional (3D) well porous ceramics is studied. Theoretical model based on the percolation theory and numerical simulations are applied to interpret the observed phase transition from an optically opaque state to a transparent state. The porous ceramic samples were fabricated by the technique of slurry casting. The transmission of optical radiation (optical percolation) over the entire porous samples is observed since the critical concentration of porosity was exceeded. We explain this effect by the rising of the spanning cluster inside of the porous structure that produces a network of porous voids. Our experimental results are in good agreement with the numerical simulations.     

Propagation and trapping of excitations on percolation clusters

A study is presented of migration of optical or magnetic excitations on percolation clusters which terminates upon reaching a trapping site. The theory is based on the extension of results from the theory of random walks to systems without translational invariance, together with the use of scaling concepts. For the case of an excitation which resides on one type of atom in a randomly mixed crystal near the percolation threshold, new power laws for the time and concentration dependences of the mean number of sites visited at timet of the kinetics of arrival at traps are obtained. Some of these results are also tested for the first time by numerical simulations.     

Corrections to scaling for diffusion exponents on three-dimensional percolation systems at criticality

Recent results of Monte Carlo simulations of the ant-in-the-labyrinth method in three-dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz, motivated by inconsistent results that have appeared in the literature. The results are observed to be sensitive to the form of the scaling correction terms. Using a single correction term, we estimate the valuek=0.197±0.004 for the anomalous diffusion exponent at criticality. When two correction terms are included,k=0.200±0.002 is obtained. These new estimates are consistent with known theoretical bounds, with recent series expansion results, and with numerical calculations of the conductance of random resistor networks above criticality.     

Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal

This report introduces a novel method to characterize the diffusion-time dependence of the diffusion-weighted magnetic resonance (MR) signal in biological tissues. The approach utilizes the theory of diffusion in disordered media where two parameters, the random walk dimension and the spectral dimension, describe the evolution of the average propagators obtained from q-space MR experiments. These parameters were estimated, using several schemes, on diffusion MR spectroscopy data obtained from human red blood cell ghosts and nervous tissue autopsy samples. The experiments demonstrated that water diffusion in human tissue is anomalous, where the mean-square displacements vary slower than linearly with diffusion time. These observations are consistent with a fractal microstructure for human tissues. Differences observed between healthy human nervous tissue and glioblastoma samples suggest that the proposed methodology may provide a novel, clinically useful form of diffusion MR contrast.     

Conductivity exponent in three-dimensional percolation by diffusion based on molecular trajectory algorithm and blind-ant rules

Diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method, respectively. The classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods. Our Monte Carlo simulations by the simple lattice random walk method give the conductivity exponent μ/ν=2.32±0.02 for diffusion on the incipient infinite clusters and μ/ν=2.21±0.03 for diffusion on a percolating lattice above the threshold. However, while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold, the result is found to be μ/ν=2.26±0.02. In addition, it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk.     

Anomalous Diffusion in a Model with a Variable Percolation Radius

Anomalous diffusion models for random 1-D cluster and comb structures of length L = 100 with finite fingers and different boundary conditions are considered. The effect of electric field on anomalous diffusion is discussed. The cases with different percolation radii are compared. The comb-structure model with periodic boundary conditions is shown to be useful in studying various types of anomalous diffusion. A new diffusion type, where the average rate is higher than the typical rate, is predicted. Physical causes for this diffusion are revealed.     

Nodal domain statistics for quantum maps, percolation, and stochastic Loewner evolution

We develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting statistical properties of the nodal domains is demonstrated for perturbed cat maps and supports the use of percolation theory to describe the wave functions of general Hamiltonian systems. We also demonstrate that the nodal domains of the perturbed cat maps obey the Cardy crossing formula and find evidence that the boundaries of the nodal domains are described by stochastic Loewner evolution with diffusion constant close to the expected value of 6, suggesting that quantum chaotic wave functions may exhibit conformal invariance in the semiclassical limit.     

Diffusion in disordered lattices and related heisenberg ferromagnets

We study the diffusion of classical hard-core particles in disordered lattices within the formalism of a quantum spin representation. This analogy enables an exact treatment of noninstantaneous correlation functions at finite particle densities in terms of single spin excitations in disordered ferromagnetic backgrounds. Applications to diluted chains and percolation clusters are discussed. It is found that density fluctuations in the former exhibit a stretched exponential decay while an anomalous power law asymptotic decay is conjectured for the latter.     

Fractional Feynman-Kac Equation with Space-Dependent Anomalous Exponent

To describe the distribution of functionals for inhomogeneous subdiffusion in space- and time-dependent force field, we derive forward and backward time-fractional Feynman-Kac equations with space-dependent anomalous exponent based on the space-jump random walk model. In our examples, we get the statistic of occupation fraction and first passage time for anomalous infiltration of free particles in disordered systems by applying the backward version.     

Constraint theory,vector percolation and glass formation

We connect the Phillips constraint theory of glass formation to numerical simulations of vector percolation with nearest neighbor central forces. Feng and Sen have shown that the addition of bond-bending forces shifts the vector percolation threshold to the scalar value in two dimensions. Using constraint theory we show that in the mean-field approximation for d > 2 the correct non-central vector threshold lies between the scalar and central vector thresholds. For a fully equilibrated statically disordered network mean-field theory may be almost exact.     

Quasi Bernoulli fluctuations in random and disordered systems

It is shown that quasi Bernoulli fluctuations, which appear at a morphological phase transition, can be considered as a statistical basis for multifractal processes with constant multifractal specific heat in a wide class of random and disordered systems. This class contains at least following processes: percolation, diffusion-limited aggregation and corrosion, Lorenz like attractors, and mesoscopic systems with Anderson transition. Received: 14 April 1998 / Revised and Accepted: 20 April 1998     

An introduction to percolation theory

Percolation theory, the theory of the properties of classical particles interacting with a random medium, is of wide applicability and provides a simple picture exhibiting critical behaviour, the features of which are well understood and amenable to detailed calculation. In this review the concepts of percolation theory and the general features associated with the critical region about the onset of percolation are developed in detail. In particular, several dimensional invariants are examined which make it possible to unify much of the available information, and to extend the insights of percolation theory to processes which have not yet received numerical study. The compilation of the results of percolation theory, both exact and numerical, is believed to be complete through 1970. A selective bibliography is given. In a concluding chapter several recent applications of percolation theory to classical and to quantum mechanical problems are discussed.     

Incorporation of Effects of Diffusion into Advection-Mediated Dispersion in Porous Media

The distribution of solute arrival times, W(t;x), at position x in disordered porous media does not generally follow Gaussian statistics. A previous publication determined W(t;x) in the absence of diffusion from a synthesis of critical path, percolation scaling, and cluster statistics of percolation. In that publication, W(t;x) as obtained from theory, was compared with simulations in the particular case of advective solute transport through a two-dimensional model porous medium at the percolation threshold for various lengths x. The simulations also did not include the effects of diffusion. Our prediction was apparently verified. In the current work we present numerical results related to moments of W(x;t), the spatial solute distribution at arbitrary time, and extend the theory to consider effects of molecular diffusion in an asymptotic sense for large Peclet numbers, P_e. However, results for the scaling of the dispersion coefficient in the range 1<P_e<100 agree with those of other authors, while results for the dispersivity as a function of spatial scale also appear to explain experiment.     

Acoustic attenuation in glasses and its relation with the boson peak

A theory for the vibrational dynamics in disordered solids [W. Schirmacher, Europhys. Lett. 73, 892 (2006), based on the random spatial variation of the shear modulus, has been applied to determine the wave vector (k) dependence of the Brillouin peak position (Omega(k)) and width (Gamma(k)), as well as the density of vibrational states [g(omega)], in disordered systems. As a result, we give a firm theoretical ground to the ubiquitous k2 dependence of Gamma(k) observed in glasses. Moreover, we derive a quantitative relation between the excess of the density of states (the boson peak) and Gamma(k), two quantities that were not considered related before. The successful comparison of this relation with the outcome of experiments and numerical simulations gives further support to the theory.     

运用CTRW-Metropolis模型数值研究亚稳势中粒子逃逸问题

基于连续时间随机行走(CTRW)理论,实现反常扩散条件下对跳跃步长和等待时间分布函数的抽样,改进Metropolis抽样判定方法以适用于存在非线性势的情况.数值研究布朗粒子在亚稳势下的逃逸速率.结果显示,稳定逃逸速率γ_st随反常指数α非单调变化,在超扩散条件下存在极大值和位垒相消现象.     

Flux front penetration in disordered superconductors

We investigate flux front penetration in a disordered type-II superconductor by molecular dynamics simulations of interacting vortices and find scaling laws for the front position and the density profile. The scaling can be understood by performing a coarse graining of the system and writing a disordered nonlinear diffusion equation. Integrating numerically the equation, we observe a crossover from flat to fractal front penetration as the system parameters are varied. The value of the fractal dimension indicates that the invasion process is described by gradient percolation.