Anomalous Diffusion Research Articles

Solvability of some integro-differential equations with the double scale anomalous diffusion in higher dimensions

Solvability of some integro-differential equations with the double scale anomalous diffusion in higher dimensions

The Limiting Dynamics of SGD: Modified Loss, Phase-Space Oscillations, and Anomalous Diffusion.

In this work, we explore the limiting dynamics of deep neural networks trained with stochastic gradient descent (SGD). As observed previously, long after performance has converged, networks continue to move through parameter space by a process of anomalous diffusion in which distance traveled grows as a power law in the number of gradient updates with a nontrivial exponent. We reveal an intricate interaction among the hyperparameters of optimization, the structure in the gradient noise, and the Hessian matrix at the end of training that explains this anomalous diffusion. To build this understanding, we first derive a continuous-time model for SGD with finite learning rates and batch sizes as an underdamped Langevin equation. We study this equationin the setting of linear regression, where we can derive exact, analytic expressions for the phase-space dynamics of the parameters and their instantaneous velocities from initialization to stationarity. Using the Fokker-Planck equation, we show that the key ingredient driving these dynamics is not the original training loss but rather the combination of a modified loss, which implicitly regularizes the velocity, and probability currents that cause oscillations in phase space. We identify qualitative and quantitative predictions of this theory in the dynamics of a ResNet-18 model trained on ImageNet. Through the lens of statistical physics, we uncover a mechanistic origin for the anomalous limiting dynamics of deep neural networks trained with SGD. Understanding the limiting dynamics of SGD, and its dependence on various important hyperparameters like batch size, learning rate, and momentum, can serve as a basis for future work that can turn these insights into algorithmic gains.

Analysis of Velocity Autocorrelation Functions from Molecular Dynamics Simulations of a Small Peptide by the Generalized Langevin Equation with a Power-Law Kernel.

Internal motions play an essential role in the biological functions of proteins and have been the subject of numerous theoretical and spectroscopic studies. Such complex environments are associated with anomalous diffusion where, in contrast to the classical Brownian motion, the relevant correlation functions have power law decays with time. In this work, we investigate the presence of long memory stochastic processes through the analysis of atomic velocity autocorrelation functions. Analytical expressions of the velocity autocorrelation function spectrum obtained through a Mori-Zwanzig projection approach were shown to be compatible with molecular dynamics simulations of a small helical peptide (8-polyalanine).

Results for Nonlinear Diffusion Equations with Stochastic Resetting.

In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.

Landscapes of random diffusivity processes in harmonic potential

A huge number of experimental evidences support that Brownian yet non-Gaussian diffusion prevalently exists in biological and soft matter systems. To interpret the physical mechanism of the Brownian yet non-Gaussian diffusion, the superstatistical approach with random diffusivity and further general random diffusivity processes attract extensive attention. This paper focuses on the anomalous diffusion and nonergodic property of the random diffusivity processes confined in a harmonic potential. Based on the theoretical analyses in the framework of Langevin equation, we find the inherent disagreement between ensemble-averaged mean-squared displacement and time-averaged mean-squared displacement for any form of the random diffusivity D(t), which implies the ergodicity breaking of the confined dynamics. Further, the asymptotic distribution of the time-averaged mean-squared displacement is strictly derived and shows the explicit dependence on the time average of the diffusivity D(t), which means the randomness of the time-averaged mean-squared displacement is completely contributed by the random diffusivity D(t). These results hold true for a general diffusivity D(t), which has been verified through the simulations of three specific kinds of diffusivities D(t).

Numerical solutions of linear time-fractional advection-diffusion equations with modified Mittag-Leffler operator in a bounded domain

Fractional advection-diffusion equations have demonstrated to be a powerful tool in modeling complex anomalous diffusion in applied science. In this paper, we studied novel linear time-fractional advection-diffusion equations associated with an extension of Mittag-Leffler fractional derivative operator. A useful feature of the used extension is to address the limitations of the Mittag-Leffler fractional derivative model. We, mainly, proposed a numerical approach to provide approximate solutions to linear time-fractional advection-diffusion equations with the studied extended fractional derivative operator. The suggested approach is based on discretizing the studied models with respect to spatio-temporal domain using uniform meshes. A new type of solutions for the studied models was generated numerically using the proposed approach. Besides, a comparative study was conducted to verify the accuracy and feasibility of the proposed approach.

Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

Fractional derivative method for anomalous aquitard flow in a leaky aquifer system with depth-decaying aquitard hydraulic conductivity

Aquitards significantly affect groundwater flow in multi-aquifer systems through adjacent aquifer leakage. Despite this, studies focusing on their heterogeneity and the non-conventional diffusion patterns of their flow are still limited. In this study, a factional derivative approach was first extended to explore the time-dependent behavior of flow transport in the aquitard. Two analytical solutions were derived for specific discharges in independent aquitards under different boundary conditions. The findings revealed that aquitard flow exhibits obvious anomalous diffusion behaviors, characterized by slower decay and heavy-tailed specific discharge data. The fractional derivative model provided a more accurate representation of this behavior than traditional models, as evidenced by its superior agreement with experimental data. Moreover, a transient model for pumping tests in a leaky aquifer system was developed, incorporating the memory effect of anomalous flow and vertical heterogeneity in aquitards. Relevant semi-analytical solutions were derived to explore the impacts of memory factor β and decay exponent of aquitard hydraulic conductivity (K) on the leakage aquifer system. Theoretical results demonstrated that stronger memory effect reduces drawdowns in the aquitard and confined aquifer during mid-to-late times. A larger dimensionless decay exponent (Ad) decreases aquitard drawdown and increases aquifer drawdown at late times. Sensitivity analysis showed aquitard drawdown experiences two peaks in sensitivity to β and Ad at early- or mid-times, affected by memory effect and decay exponent of aquitard K, signifying maximal impact at these specific intervals. This study provides a practical model to effectively manage groundwater resources by accurately reflecting aquitard memory and heterogeneity effects.

Kinetic modelling of ultrasound-triggered chemotherapeutic drug release from the surface of gold nanoparticles

Therapeutic ultrasound can be used to trigger the on-demand release of chemotherapeutic drugs from gold nanoparticles (GNPs). In the previous work, our group achieved doxorubicin (DOX) release from the surface of GNPS under low-intensity pulsed ultrasound (LIPUS) exposure. However, the specific release kinetics of ultrasound-triggered DOX release from GNPs is not known. Here, we present a release kinetics study of DOX from GNPs under ultrasound exposure for the first time. A novel dialysis membrane setup was designed to quantify DOX release from LIPUS-activated GNPs at 37.0 °C and 43.4 °C (hyperthermia temperature range). Contributions of thermal and non-thermal mechanisms of LIPUS-triggered DOX release were also quantified. Non-thermal mechanisms accounted for 40 ± 7% and 34 ± 5% of DOX release for 37.0 °C and 43.4 °C trials, respectively. DOX release under LIPUS exposure was found to follow Korsmeyer–Peppas (K–P) kinetics, suggesting a shift from a Fickian (static) to a non-Fickian (dynamic) release profile with the addition of non-thermal interactions. DOX release was attributed to an anomalous diffusion release mechanism from the GNP surface. A finite element model was also developed to quantify the acoustic radiation force, believed to be the driving force of non-thermal DOX release inside the dialysis bag.

A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation

As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion and superdiffusion. Due to the fact that fractional operators act on both temporal and spatial derivative terms in TSFDE, efficient solving for TSFDE is important, where the key is solving the corresponding discrete system efficiently. In this paper, we derive a Galerkin–Legendre spectral all-at-once system from the TSFDE, and then we develop a preconditioner to solve this system. Symmetry property of the coefficient matrix in this all-at-once system is destroyed so that the deduced all-at-once system is more convenient for parallel computing than the traditional timing-step scheme, and the proposed preconditioner can efficiently solve the corresponding all-at-once system from TSFDE with nonsmooth solution. Moreover, some relevant theoretical analyses are provided, and several numerical results are presented to show competitiveness of the proposed method.

Parallel scalability of sparse solvers for sparsified fractional diffusion problems discretized with FEM

Anomalous diffusion is a non-local process that describes a wide spectrum of natural processes and phenomena with many applications in science and technology. It is described mathematically by the fractional Laplace operator. In this work we examine the integral definition of the fractional Laplacian modeled with the Riesz potential and discretized with the finite element method. The thus obtained system of linear algebraic equations is dense due to the non-local nature of the fractional Laplacian and is computationally complex to solve. With LU factorization, for example, solving the problem has an O(N 3) computational complexity, where N is the number of unknowns. However, it can be observed that a large amount of the off-diagonal coefficients have very small absolute values compared to the diagonal coefficients. Those small off-diagonal coefficients can be lumped (set to zero and added to the diagonal coefficient) without significant loss of accuracy. In this work we employ a direct sparse parallel solver to the resultant sparse matrix. We analyze the parallel performance and speed-up, as well as the accuracy, varying the fractional power and the lumping threshold.

Structure-preserving discretization of fractional vector calculus using discrete exterior calculus

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties curlαgradα=0 and divαcurlα=0 hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.

Stability analysis in BWRs with double subdiffusion effects: Reduced order fractional model (DS-F-ROM)

The aim of this work is to explore the effect of the double subdiffusion on the stability in BWRs. A BWR novel reduced order model with double subdiffusion effects: reduced order fractional model (DS-F-ROM) to describe the neutron and heat transfer processes was proposed for this study. The double subdiffusion was developed with a fractional-order two-equation model, and with different fractional-orders and relaxation times. The stability analysis was carried out using the root-locus method and change from the s to the W domain and were confirmed using the time-domain evolution of neutron flux for a unit step change in reactivity. The results obtained using the reduced fractional-order model are presented for different anomalous diffusion coefficient values. Results are compared with normal diffusion and P1 equations, which are obtained straightforwardly with DS-ROM when relaxation time tends to zero, and when the anomalous diffusion coefficient tends to one, respectively.

Stochastic model for the transfer of gaseous particles in polymer-carbon-nanotube nanocomposites with interfacial regions.

In this work, a stochastic model of gaseous transfer in polymer-carbon-nanotube (CNT) nanocomposites is presented. The model takes into account interfacial areas, i.e., polymer depletion regions. The local regime of transport is controlled by the density of the polymer. In a dense polymer, this regime corresponds to the ordinary diffusion, while in free volume regions, it corresponds to the ballistic transport. The introduction of a free volume and/or a depleted polymer layer near to a CNT wall leads to the emergence of anomalous diffusion. We have demonstrated how the anomalous diffusion regime changes in the presence of nanotubes for different distributions of polymer density. The presented approach allows us to describe the threshold effect in the diffusion coefficient as a function of CNTs density in polymer-CNT nanocomposites.

Depolymerized Chitosan-g-[Poly(MMA-co-HEMA-cl-EGDMA)] Based Nanogels for Controlled Local Release of Bupivacaine.

This study is designed to formulate and characterize chitosan-based nanogels that provide the controlled delivery of anesthetic drugs, such as bupivacaine (BPV), for effective postoperative pain management over prolonged periods of time. Drug carriers of chitosan/poly (MMA-co-HEMA-cl-EGDMA) (CsPMH) nanogels were prepared by varying the composition of comonomers such as MMA, HEMA, and redox initiator CAN. The nanogels were then characterized using FTIR, TGA, SEM, and TEM. The CsPMH nanogels showed greater encapsulation efficiencies from 43.20-91.77%. Computational studies were also conducted to evaluate the interaction between the drug and CsPMH nanoparticles. Finally, BPV-loaded nanoparticles were used to examine their in vitro release behavior. At pH 7.4, all the drug carriers displayed the "n" value around 0.7, thus the BPV release follows anomalous diffusion. Drug carrier 7 demonstrated a steady and sustained release of BPV for approximately 24 h and released about 91% of BPV, following the K-P mechanism of drug release. On the other hand, drug carrier 6 exhibited controlled release for approximately 12 h and released only 62% of BPV.

Langevin picture of subdiffusive particles under the joint influence of an expanding medium and an external constant force

It has been found that expanding medium has a nonnegligible influence on particle’s diffusion, the physical mechanism of which is quite different from an external force field. To study the joint influence of the expanding medium and the constant force, we start from the Langevin picture of the subdiffusive particles in the expanding medium, and then introduce the effect of the external constant force in two ways, which are sustainedly or intermittently acting on the particle’s motion. Therefore, we get two models with the only difference on the force terms. Then under the framework of the Langevin equation, we conduct detailed analyses and derive the corresponding Fokker–Planck equations and the moments for the two models. In the case with the sustained constant force, the force behaves as a decoupled force, which is called Galilei invariant. While for the intermittent case, the force changes the diffusion behavior, and the specific effects vary according to the expanding rate of the medium. Our detailed theoretical analyses and simulations present a consistent Langevin framework of investigating anomalous diffusion phenomena in the expanding medium together with the external force.

Diffusion-Controlled Reactions: An Overview.

We review the milestones in the century-long development of the theory of diffusion-controlled reactions. Starting from the seminal work by von Smoluchowski, who recognized the importance of diffusion in chemical reactions, we discuss perfect and imperfect surface reactions, their microscopic origins, and the underlying mathematical framework. Single-molecule reaction schemes, anomalous bulk diffusions, reversible binding/unbinding kinetics, and many other extensions are presented. An alternative encounter-based approach to diffusion-controlled reactions is introduced, with emphasis on its advantages and potential applications. Some open problems and future perspectives are outlined.

Staging liver fibrosis by a continuous-time random-walk diffusion model

PurposeNoninvasive assessment of liver fibrosis holds significant clinical importance. We aimed to evaluate the clinical potential of using a continuous-time random-walk diffusion model (CTRW) for staging liver fibrosis. MethodsThis prospective study included 52 patients suspected of liver disease and scheduled for liver biopsy. All patients underwent multi-b value diffusion-weighted imaging (DWI) using a 1.5 T MR scanner to derive the anomalous diffusion coefficient (D) and temporal (α) and spatial (β) diffusion heterogeneity indexes sourced from the CTRW. The mono-exponential DWI-derived apparent diffusion coefficient (ADC), transient elastography-derived liver stiffness measurement (LSM), aspartate aminotransferase-to-platelet ratio index (APRI), and fibrosis-4 (FIB-4) index were calculated. We assessed and compared the correlations of these parameters with fibrosis stages and their efficacy in staging liver fibrosis. ResultsSignificant correlations with fibrosis stages were found for APRI (r = 0.336), FIB-4 (r = 0.351), LSM (r = 0.523), D (r = −0.458), and ADC (r = −0.473). Significant differences were observed between APRI, LSM, D, and ADC of different fibrosis stages. The diagnostic performance of an index that combined D, α, β, ADC, and LSM was superior to that of ADC or LSM alone for fibrosis stage F ≥ 2 and better than the index that combined D, α, β for fibrosis stage F ≥ 4. ConclusionsAccurate liver fibrosis staging was achieved with a model that combined CTRW-derived parameters (D, α, and β), ADC, and LSM. The model could serve as a reliable tool for noninvasive fibrosis evaluation.

Quantitative analysis of peroxisome tracks using a Hidden Markov Model

Diffusion and mobility are essential for cellular functions, as molecules are usually distributed throughout the cell and have to meet to interact and perform their function. This also involves the cytosolic migration of cellular organelles. However, observing such diffusion and interaction dynamics is challenging due to the high spatial and temporal resolution required and the accurate analysis of the diffusional tracks. The latter is especially important when identifying anomalous diffusion events, such as directed motions, which are often rare. Here, we investigate the migration modes of peroxisome organelles in the cytosol of living cells. Peroxisomes predominantly migrate randomly, but occasionally they bind to the cell's microtubular network and perform directed migration, which is difficult to quantify, and so far, accurate analysis of switching between these migration modes is missing. We set out to solve this limitation by experiments and analysis with high statistical accuracy. Specifically, we collect temporal diffusion tracks of thousands of individual peroxisomes in the HEK 293 cell line using two-dimensional spinning disc fluorescence microscopy at a high acquisition rate of 10 frames/s. We use a Hidden Markov Model with two hidden states to (1) automatically identify directed migration segments of the tracks and (2) quantify the migration properties for comparison between states and between different experimental conditions. Comparing different cellular conditions, we show that the knockout of the peroxisomal membrane protein PEX14 leads to a decrease in the directed movement due to a lowered binding probability to the microtubule. However, it does not eradicate binding, highlighting further microtubule-binding mechanisms of peroxisomes than via PEX14. In contrast, structural changes of the microtubular network explain perceived eradication of directed movement by disassembly of microtubules by Nocodazole-treatment.

The final value problem for anomalous diffusion equations involving weak-valued nonlinearities

We are concerned with the final value problem governed by a class of semilinear anomalous diffusion equations, where the nonlinearity may take values in Hilbert scales with negative orders. The solvability and Hölder regularity results are proved by establishing estimates in Hilbert scales for resolvent operators in connection with nonlinearity function. The obtained results are applicable for some concrete problems modeling subdiffusion phenomena.