Ultrafast diffusion process characterized by unusually large diffusivities is often occurs on porous media and the mean square displacement grows exponentially in time. This paper clarifies the characteristics of ultrafast diffusion and tackles this perplexing problem using the fractional Brownian motion run with a nonlinear clock model. We employ the Mittag-Leffler function as the nonlinear clock, the increments are dependent and obey Gaussian distribution, and the derived corresponding mean square displacement is more widely than the exponential function. A comparison between the power law model and the proposed model with respect to available experimental data verifies that the proposed model is more effective and accurate. Ultraslow diffusion is also studied with the inverse Mittag-Leffler function as the nonlinear clock. The results show that it can capture the ultraslow diffusion process better than the case of the logarithmic model. As the generalization of fractional Brownian motion, fractional Brownian motion run with a nonlinear clock is an alternative model method for extreme anomalous diffusion in complex systems. (C) 2020 Elsevier Ltd. All rights reserved.

A thermomechanical theory of swelling porous media composed of a solid matrix and adsorbed fluid is derived using a hybridization of the mixture-theoretic approach of Bowen [2] and the Coleman and Noll method of exploitation of the entropy inequality. The microscale consists of macromolecular structures (clay platelets, polymers, shales, biological tissues, gels) in a solvent (adsorbed water), both of which are considered as distinct nonoverlaying continua. These continuum are homogenized in the spirit of hybrid mixture theory (HMT), so that at the macroscale they may be thought of as two overlaying continua. The exploitation of the entropy inequality within the Coleman and Noll method yields constitutive results which capture the physics of swelling. The model is applied to a bentonitic clay used for enginnered barrier of nuclear waste repository. Numerical simulations are presented showing the influence of physicochemical effects on the performance of the clay buffer.

Carbon nanotube brushes (carbon nanotubes attached to a current collector) have been proposed for nanoscale electrodes. While the detailed molecular scale physics is important in its own right, often a more useful scale to examine electrode performances is the micro and meso scale continuum. The detailed structure of microscale material parameters results from statistical mechanics and will not be discussed here, rather we focus on the micro scale homogenization to the meso scale wherein material parameters result from a micro scale “cell” problem for a periodic lattice. We apply a matched asymptotic formulation to obtain the meso scale field equations and material parameters. The finite element method (FEM) and finite volume method (FVM) are employed to solve the micro scale cell problem and the mesoscale field equations, respectively. Numerical results are presented for the electrical potential field and concentration of cations and anions. For a test case, it is observed that a doubling of the electrode area can create a ten-fold increase in the potential which has interesting implications for electrode design.

y Many theoretical and experimental results show that anomalous diffusion/dispersion occurs in porous media. To exactly solve anomalously fast dispersion, we introduce a structural derivative diffusion model to present superfast diffusion via a logarithmic structural function in space. The fundamental solution of the diffusion model is a form of log-normal distribution, and the corresponding analytical mean squared displacement grows like e(t/beta 2), 0 < beta < 1. Compared with the existing models, the proposed model is more effective and accurate in fitting experimental data. (C) 2019 Elsevier Ltd. All rights reserved.

Two approaches to the subject of transport in fractal porous media have been derived using minimal assumptions. One is derived from a thermodynamics and statistical mechanics, and the other utilizes a continuum mechanics approach where the divergence of the velocity is a random function. This chapter explores three additional models of transport in fractal media. The Lagrangian perspective on fractional Brownian motion with a nonlinear clock can be clearly understood from the perspective of fractional Brownian motion combined. The primary purpose of the dispersive component in a transport model is to represent the unresolved portion of the velocity. The chapter presents a number of approaches for modeling transport in highly-heterogeneous, fractal velocity fields. The main drawback of these two approaches is that the transport equations are complex and involve integral kernels that cannot be readily estimated in many applications except through curve fitting.

The thesis put forward here is that the occurrence of Fickian dispersion in geophysical settings is a rare event and consequently should be labeled as anomalous. What people classically call anomalous is really the norm. In a Lagrangian setting, a process with mean square displacement which is proportional to time is generally labeled as Fickian dispersion. With a number of counter examples we show why this definition is fraught with difficulty. In a related discussion, we show an infinite second moment does not necessarily imply the process is super dispersive. By employing a rigorous mathematical definition of Fickian dispersion we illustrate why it is so hard to find a Fickian process. We go on to employ a number of renormalization group approaches to classify non-Fickian dispersive behavior. Scaling laws for the probability density function for a dispersive process, the distribution for the first passage times, the mean first passage time, and the finite-size Lyapunov exponent are presented for fixed points of both deterministic and stochastic renormalization group operators. The fixed points of the renormalization group operators are p-self-similar processes. A generalized renormalization group operator is introduced whose fixed points form a set of generalized self-similar processes. Power-law clocks are introduced to examine multi-scaling behavior. Several examples of these ideas are presented and discussed. (C) 2015 Elsevier B.V. All rights reserved.

Recently, nonlocal generalizations of the classical gradient, divergence, and curl have beenintroduced to examine nonlocal field problems, and develop a nonlocal vector calculus. Here weintroduce, by definition, the concept of a nonlocal field variable and relate it to its classicallocal counterpart. Subsequently, we relate the concept of measurement via a convolution(an upscaling) to the nonlocal field variables and the nonlocal operators. It is shown viaFourier transform that the nonlocal gradient and divergence can be thought of as a very specialconvolution averaging of their classical counterparts. A nonlocal self-diffusion equation isupscaled and written in terms of nonlocal operators.

Brownian motion, the classical diffusive process, maximizes the Boltzmann-Gibbs entropy. The Tsallis q entropy, which is nonadditive, was developed as an alternative to the classical entropy for systems which are nonergodic. A generalization of Brownian motion is provided that maximizes the Tsallis entropy rather than the Boltzmann-Gibbs entropy. This process is driven by a Brownian measure with a random diffusion coefficient. The distribution of this coefficient is derived as a function of q for 1<q<3. Applications to transport in porous media are considered.

The collective molecular reorientations within a nematic liquid crystal fluid bathing a spherical colloid cause the colloid to diffuse anomalously on a short time scale (i.e., as a non-Brownian particle). The deformations and fluctuations of long-range orientational order in the liquid crystal profoundly influence the transient diffusive regimes. Here we show that an anisotropic fractional Brownian process run with a nonlinear multiscaling clock effectively mimics this collective and transient phenomenon. This novel process has memory, Gaussian increments, and a multiscale mean square displacement that can be chosen independently from the fractal dimension of a particle trajectory. The process is capable of modeling multiscale sub-, super-, or classical diffusion. The finite-size Lyapunov exponents for this multiscaling process are defined for future analysis of related mixing processes.

Single particle tracking is a tool that is being increasingly used to study diffusive or dispersive processes in many branches of natural science. Often the ability to collect these trajectories experimentally or produce them numerically outpaces the ability to understand them theoretically. On the other hand many stochastic models have been developed and continue to be developed capable of capturing complex diffusive behavior such as heavy tails, long-range correlations, nonstationarity, and combinations of these things. We describe a computational method for connecting particle trajectory data with stochastic models of diffusion. Several tests are performed to demonstrate the efficacy of the method, and the method is applied to polymer diffusion, RNA diffusion in E. coli, and RAFOS dispersion in the Gulf of Mexico.

Fixed points of the renormalization group operator Rp,rX(t)≡X(rt)/rp are said to be p-self-similar. Here X(t) is an arbitrary stochastic process. The concept of a p-self-similar process is generalized via the renormalization group operator RF,GX(t)=F[X(G(t))], where F and G are bijections on (-∞,∞) and [0,∞), respectively. If X(t) is a fixed point of RF,G, then X(t) is said to be (F,G)-self-similar. We say Y(t) is (F,G)-X(t)-similar if RF,GX(t)=Y(t) in distribution. Exit time distributions and finite-size Lyapunov exponents were obtained for these latter processes. A power law multiscaling process is defined with a multipower-law clock. This process is employed to statistically represent diffusion in a nanopore, a monolayer fluid confined between atomically structured surfaces. The tools presented provide a straightforward method to statistically represent any multiscaling process in time.

Motivated by the need to understand the dynamics of motile particles in porous media, our team has applied renormalization group techniques to both upscale and classify anomalous dispersive/diffusive behavior. Central limit theorems, which lead to a specific type of renormalization group, are employed in several cases to upscale transport of motile particles in porous media that display a specific type of fractal character. The old standby classification for diffusion (which we use interchangeably with dispersion) says a particle is anomalous if its mean square displacement is not linear in time. This physically intuitive concept, is shown to be inadequate, and so is replaced by a scheme that relies on the fixed points of specific renormalization group operators. Various asymptotic limits are examined, and scaling laws for the limits are derived. A random renormalization operator is introduced for processes with multiple asymptotes and unknown self-similarity index, and a Bayesian tool is employed to obtain scaling laws that are weighted averages of power laws. Published by Elsevier Ltd.

Nonstationary random fields such as fractional Brownian motion and fractional Lévy motion have been studied extensively in the hydrology literature. On the other hand, random fields that have nonstationary increments have seen little study. A mathematical argument is presented that demonstrates processes with stationary increments are the exception and processes with nonstationary increments are far more abundant. The abundance of nonstationary increment processes has important implications, e.g., in kriging where a translation‐invariant variogram implicitly assumes stationarity of the increments. An approach to kriging for processes with nonstationary increments is presented and accompanied by some numerical results.

Micropolar field theories provide a systematic approach to modeling traditional materials like elastic bodies or viscous fluids that have microstructure. The downside to accounting for the behavior on the smaller scale is the introduction of many new parameters into the field equations. The difficulty of handling these new parameters can be mitigated by performing a sensitivity analysis to determine which parameters have the greatest impact on the solutions to the field equations. The sensitivity of an incompressible micropolar Stokes fluid to variations in the viscosity coefficients and boundary value of the microinertia is examined. This particular choice of field equations is motivated by an application of micropolar field theories to the deformation of the continental lithosphere, but the same approach can be used for other micropolar equations modeling solids or other types of fluids.

A random renormalization group technique is reviewed, and a related set of Bayesian scaling techniques are presented. The techniques are employed to study the motion of drifters in Lake Michigan on a time scale ranging from 30 min to 5 days and in the Gulf of Mexico on a time scale ranging from 8 h to 85 days. The scaling laws generalize the standard power law scalings. One of the advantages of the Bayesian approach is that scaling laws can be determined even with a paucity of data with the caveat that less data produce greater uncertainty in the scaling laws.

A thermodynamically consistent model for multiphase flow which allows for connected and disconnected phases in a swelling medium is developed using hybrid mixture theory with three spatial scales. The mesoscale of the medium consists of swelling particles and two bulk phases, such as liquid water and vapor. The particles are a combination of a vicinal liquid and a solid, which may swell or shrink as a result of interaction with the other bulk phases; an example is a mixture of montmorillonite platelets and water. The theory defines connected and disconnected bulk phases of liquid and vapor at the mesoscale to create a dual‐porosity type model at the macroscale. The disconnected vapor phase consists of either buoyant bubbles or confined vapor packets. The incorporation of disconnected and connected phases is useful for modeling unsaturated swelling systems. The macroscale solid phase volume fraction is refined from previous hybrid mixture approaches for two‐phase multiscale problems and is fully utilized in the field equations and the constitutive theory. Macroscale equations for each of the six phases are presented with bulk regions separated into connected and disconnected domains. A constitutive theory is derived by exploiting the entropy inequality for the mixture. Generalized Darcy's laws and the final set of field equations for the system are presented and compared with previous hybrid mixture theoretic results. Classical parallel flow models only consider disconnected bulk regions and therefore are only appropriate for drainage; the current system can be useful for both imbibition and drainage, including extremely dry systems.

Fractional Brownian motion (fBm) is a stochastic process that has stationary increments with long‐range correlations and known fractal dimension. We study a multiple‐dimensional extension of fBm with nonstationary increments that allows for trends in the statistical structure while maintaining the Gaussian nature and fractal dimension of fBm. Two methods for simulating this extension are employed and described in detail. One approach combines Cholesky decomposition with a generalization of random midpoint displacement. The other makes repeated use of the Cholesky decomposition. The resulting fields can be employed in various geophysical settings, e.g., as log conductivity fields in hydrology and topographic elevation in geomorphology.

Let X(t) be a fixed point the renormalization group operator (RGO), R p,r X(t)=X(rt)/r p . Scaling laws for the probability density, mean first passage times, finite-size Lyapunov exponents of such fixed points are reviewed in anticipation of more general results. A generalized RGO, \(\mathcal{R}_{P,n}\) where P is a random variable, is introduced. Scaling laws associated with these random RGOs (RRGOs) are demonstrated numerically and applied to subdiffusion in bacterial cytoplasm and a process modeling the transition from subdiffusion to classical diffusion. The scaling laws for the RRGO are not simple power laws, but are a weighted average of power laws. The weighting used in the scaling laws can be determined adaptively via Bayes’ theorem.