Deterministic Diffusion Research Articles

Generalized autocorrelation function in the family of deterministic and stochastic anomalous diffusion processes

We investigate the observables of the one-dimensional model for anomalous transport in semiconductor devices where diffusion arises from scattering at dislocations at fixed random positions, known as the Lévy-Lorentz gas. To gain insight into the microscopic properties of such a stochastically complex system, deterministic dynamics known as the slicer map and fly-and-die dynamics are used. We analytically derive the generalized position autocorrelation function of these dynamics and study the special case, the 3-point position correlation function. For this, we derive single-parameter-dependent scaling and compare it with the numerically estimated 3-point position autocorrelation of the Lévy-Lorentz gas, for which the analytical expression is still an open question. Here we obtained a remarkable agreement between them, irrespective of any functional relationship with time. Moreover, we demonstrate that the position moments and the position autocorrelations of these systems scale in the same fashion, provided the times are large enough and far enough apart. Other observables, such as velocity moments and correlations, are reported to distinguish the systems. Published by the American Physical Society 2024

Mechanism of multistability in chaotic maps.

This research aims to investigate the mechanisms of multistability in chaotic maps. The study commences by examining the fundamental principles governing the development of homogeneous multistability using a basic one-dimensional chain-climbing map. Findings suggest that the phase space can be segmented into distinct uniform mediums where particles exhibit consistent movement. As critical parameter values are reached, channels emerge between these mediums, resulting in deterministic chaotic diffusion. Additionally, the study delves into the topic of introducing heterogeneous factors on the formation of heterogeneous multistability in the one-dimensional map. A thorough examination of phenomena such as multistate intermittency highlights the intimate connection between specific phase transition occurrences and channel formation. Finally, by analyzing two instances-a memristive chaotic map and a hyperchaotic map-the underlying factors contributing to the emergence of multistability are scrutinized. This study offers an alternative perspective for verifying the fundamental principles of homogenous and heterogeneous multistability in complex high-dimensional chaotic maps.

Understanding Anxiety in Cervical Dystonia: An Imaging Study.

Anxiety may precede motor symptoms in cervical dystonia (CD) and is associated with an earlier onset of dystonia. Our understanding of anxiety in CD is inadequate. To investigate brain networks associated with anxiety in CD. Twenty-six subjects with idiopathic CD underwent MRI Brain without contrast. Correlational tractography was derived using Diffusion MRI connectometry. Quantitative Anisotropy (QA) was used in deterministic diffusion fiber tracking. Correlational tractography was then used to correlate QA with State-Trait Anxiety Inventory (STAI) state (STAI-S) and trait (STAI-T) subscales. Connectometry analysis showed direct correlation between state anxiety and QA in tracts from amygdala to thalamus/ pulvinar bilaterally, and trait anxiety and QA in tracts from amygdala to motor cortex, sensorimotor cortex and parietal association area bilaterally (FDR ≤0.05). Our efforts to map anxiety to brain networks in CD highlight the role of the amygdala in the pathophysiology of anxiety in CD.

Graph theory-based analysis reveals neural anatomical network alterations in chronic post-traumatic stress disorder

Abstract Multimodal imaging using network connectivity techniques shows promise for investigating neuropathology influencing Post-Traumatic Stress Disorder (PTSD) symptom maintenance and course. We recruited World Trade Center (WTC) responders who continued to suffer from chronic PTSD into a diffusion tensor neuroimaging protocol (n=100), along with nine unexposed controls without PTSD from other sources. Using a graph theory approach to probe network alterations in brain diffusion images, we calculated weighted characteristics path length as a surrogate marker for the effective neuroanatomical distance between anatomical nodes. The sample (N=109; 47 with chronic PTSD) was in their mid-fifties, and the majority were male. Responders were matched in terms of cognitive performance, occupation, and demographics. The anatomical connectivity graph was constructed for each participant using deterministic diffusion tractography. We identified a significant difference in weighted Characteristic Path Length (wCPL) between trauma-exposed WTC responders (Cohen’s d = 0.42, P<0.001) that was highest in people with PTSD, and not explained by WTC exposure severity or duration. We also found that wCPL was associated with PTSD symptom severity in responders with PTSD. In the largest study to date to examine the relationship between chronic PTSD and anatomy, we examined the anatomical topography of neural connections and found that wCPL differed between the PTSD+ and PTSD- diagnostic categories.

A Score-Based Deterministic Diffusion Algorithm with Smooth Scores for General Distributions

Score matching based diffusion has shown to achieve the state of art results in generation modeling. In the original score matching based diffusion algorithm, the forward equation is a differential equation for which the probability density equation evolves according to a linear partial differential equation, the Fokker-Planck equation. A drawback of this approach is that one needs the data distribution to have a Lipschitz logarithmic gradient. This excludes a large class of data distributions that have a compact support. We present a deterministic diffusion process for which the vector fields are always Lipschitz and hence the score does not explode for probability measures with compact support. This deterministic diffusion process can be seen as a regularization of the porous media equation equation, which enables one to guarantee long term convergence of the forward process to the noise distribution. Though the porous media equation is itself not always guaranteed to have a Lipschitz vector field, it can be used to understand the closeness of the output of the algorithm to the data distribution as a function of the the time horizon and score matching error. This analysis enables us to show that the algorithm has better dependence on the score matching error than approaches based on stochastic diffusions. Using numerical experiments we verify our theoretical results on example one and two dimensional data distributions which are compactly supported. Additionally, we validate the approach on a modified MNIST data set for which the distribution is concentrated on a compact set. In each of the experiments, the approach using deterministic diffusion performs better that the diffusion algorithm with stochastic forward process, when considering the FID scores of the generated samples.

Towards Explainability of the Latent Space by Disentangled Representation Learning

Deep neural networks are widely used in computer vision for image classification, segmentation and generation. They are also often criticised as “black boxes” because their decision-making process is often not interpretable by humans. However, learning explainable representations that explicitly disentangle the underlying mechanisms that structure observational data is still a challenge. To further explore the latent space and achieve generic processing, we propose a pipeline for discovering the explainable directions in the latent space of generative models. Since the latent space contains semantically meaningful directions and can be explained, we propose a pipeline to fully resolve the representation of the latent space. It consists of a Dirichlet encoder, conditional deterministic diffusion, a group-swap and a latent traversal module. We believe that this study provides an insight into the advancement of research explaining the disentanglement of neural networks in the community.

Disrupted Small-World Networks in Children with Drug-Naïve Attention-Deficit/Hyperactivity Disorder: A DTI-Based Network Analysis

Attention-deficit/hyperactivity disorder (ADHD) is one of the most common neurodevelopmental disorders, while the potential neurological mechanisms are poorly understood. To explore the alterations in the white matter (WM) structural connectome in children with drug-naïve ADHD, forty-nine ADHD and 51 age- and gender-matched typically developing (TD) children aged 6–14 years were enrolled. WM structural connectivity based on deterministic diffusion tensor imaging (DTI) was constructed in 90 cortical and subcortical regions, and topological parameters of the resulting graphs were calculated. Network metrics were compared between two groups. The concentration index and the total cancellation test scores of digit cancellation test were used to evaluate clinical symptom severity in ADHD. Then, a partial correlation analysis was performed to explore the relationship between significant topologic metrics and clinical symptom severity. Compared to TD group, ADHD showed an increase in the characteristic path length (Lp), normalized clustering coefficient (γ), small worldness (σ), and a decrease in the global efficiency (Eglob) (all p < 0.05). Furthermore, ADHD showed reduced nodal centralities mainly in the regions of default mode network (DMN), central executive network (CEN), basal ganglia, and bilateral thalamus (all p < 0.05). After performing Benjamini-Hochberg’s procedure, only the left orbital part of superior frontal gyrus and the left caudate were statistically significant (p < 0.05, FDR-corrected). In addition, the concentration index of ADHD was negatively correlated with the nodal betweenness of the left orbital part of the middle frontal gyrus (r = −0.302, p = 0.042). Our findings revealed an ADHD-related shift of WM network topology toward “regularization” pattern, characterized by decreased global network integration, which is also reflected by changed nodal centralities involving DMN, CEN, basal ganglia, and bilateral thalamus. ADHD could be understood by examining the dysfunction of large-scale spatially distributed neural networks.

Bioprinted Multicomponent Hydrogel Co-culture Tumor-Immune Model for Assessing and Simulating Tumor-Infiltrated Lymphocyte Migration and Functional Activation.

The immune response against a tumor is characterized by the interplay among components of the immune system and neoplastic cells. Here, we bioprinted a model with two distinct regions containing gastric cancer patient-derived organoids (PDOs) and tumor-infiltrated lymphocytes (TILs). The initial cellular distribution allows for the longitudinal study of TIL migratory patterns concurrently with multiplexed cytokine analysis. The chemical properties of the bioink were designed to present physical barriers that immune T-cells must breech during infiltration and migration toward a tumor with the use of an alginate, gelatin, and basal membrane mix. TIL activity, degranulation, and regulation of proteolytic activity reveal insights into the time-dependent biochemical dynamics. Regulation of the sFas and sFas-ligand present on PDOs and TILs, respectively, and the perforin and granzyme longitudinal secretion confirms TIL activation when encountering PDO formations. TIL migratory profiles were used to create a deterministic reaction-advection diffusion model. The simulation provides insights that decouple passive from active cell migration mechanisms. The mechanisms used by TILs and other adoptive cell therapeutics as they infiltrate the tumor barrier are poorly understood. This study presents a pre-screening strategy for immune cells where motility and activation across ECM environments are crucial indicators of cellular fitness.

Prediction of the Topography of the Corticospinal Tract on T1-Weighted MR Images Using Deep-Learning-Based Segmentation.

Tractography is an invaluable tool in the planning of tumor surgery in the vicinity of functionally eloquent areas of the brain as well as in the research of normal development or of various diseases. The aim of our study was to compare the performance of a deep-learning-based image segmentation for the prediction of the topography of white matter tracts on T1-weighted MR images to the performance of a manual segmentation. T1-weighted MR images of 190 healthy subjects from 6 different datasets were utilized in this study. Using deterministic diffusion tensor imaging, we first reconstructed the corticospinal tract on both sides. After training a segmentation model on 90 subjects of the PIOP2 dataset using the nnU-Net in a cloud-based environment with graphical processing unit (Google Colab), we evaluated its performance using 100 subjects from 6 different datasets. Our algorithm created a segmentation model that predicted the topography of the corticospinal pathway on T1-weighted images in healthy subjects. The average dice score was 0.5479 (0.3513-0.7184) on the validation dataset. Deep-learning-based segmentation could be applicable in the future to predict the location of white matter pathways in T1-weighted scans.

Innovation Diffusion Processes: Concepts, Models, and Predictions

Innovation diffusion processes have attracted considerable research attention for their interdisciplinary character, which combines theories and concepts from disciplines such as mathematics, physics, statistics, social sciences, marketing, economics, and technological forecasting. The formal representation of innovation diffusion processes historically used epidemic models borrowed from biology, departing from the logistic equation, under the hypothesis that an innovation spreads in a social system through communication between people like an epidemic through contagion. This review integrates basic innovation diffusion models built upon the Bass model, primarily from the marketing literature, with a number of ideas from the epidemiological literature in order to offer a different perspective on innovation diffusion by focusing on critical diffusions, which are key for the progress of human communities. The article analyzes three key issues: barriers to diffusion, centrality of word-of-mouth, and the management of policy interventions to assist beneficial diffusions and to prevent harmful ones. We focus on deterministic innovation diffusion models described by ordinary differential equations.

Application of model-order reduction of non-linear time-dependent neutronics via POD-Galerkin projection and matrix discrete empirical interpolation

Presented here is an application of model-order reduction to non-linear neutronic transients. The effort described extends previous work (Elzohery and Roberts, 2021b) aimed to develop and assess a Reduced-order model (ROM) framework that can be used for different purposes, such as design optimization and propagating input parameter uncertainties like nuclear data. Here, the target applications is non-linear. The ROM was built with the use of the POD-Galerkin projection method and the matrix version of the Discrete Empirical Interpolation Method (MDEIM) for improved efficiency. Where applications of the ROM to the diffusion equation have been introduced before, this work focuses on treating the non-linearity induced by coupling with different physics. Previous work has used DEIM to approximate non-linear functions resulting from multiphysics coupling such as the temperature and the velocity field (German et al., 2019, 2022). Here, the MDEIM is used to approximate the projection of the problem operator that otherwise must be computed at each non-linear iteration within each time step, which increases the ROM cost. By applying the method to the LRA benchmark, speed ups from 6 to 40 were achieved depending on the full-order model (FOM) spatial fidelity. The maximum relative error in the assemblies power was in the order of 1×10−5%. Moreover, the ROM was parameterized by POD-greedy sampling, and a case study was performed where the kinetic data, i.e., the precursor decay constants and delayed-neutron fraction were assumed to be the model parameters of interest. For the parameterized ROM, the maximum relative error in the assembly power was 1×10−3%, The computational gain from the MDEIM-ROM was compared with that of a direct projection ROM and showed that employing the MDEIM is computationally advantageous, and the advantage grows with dimension of the problem. The method was implemented in the open-source deterministic transport code Detran (Roberts, 2014) where only access to the system operator was needed to construct the ROM, and; hence, the method is trivially invasive and could easily be implemented in many modern deterministic diffusion codes.

Multisite Harmonization of Structural DTI Networks in Children: An A-CAP Study.

The analysis of large, multisite neuroimaging datasets provides a promising means for robust characterization of brain networks that can reduce false positives and improve reproducibility. However, the use of different MRI scanners introduces variability to the data. Managing those sources of variability is increasingly important for the generation of accurate group-level inferences. ComBat is one of the most promising tools for multisite (multiscanner) harmonization of structural neuroimaging data, but no study has examined its application to graph theory metrics derived from the structural brain connectome. The present work evaluates the use of ComBat for multisite harmonization in the context of structural network analysis of diffusion-weighted scans from the Advancing Concussion Assessment in Pediatrics (A-CAP) study. Scans were acquired on six different scanners from 484 children aged 8.00–16.99 years [Mean = 12.37 ± 2.34 years; 289 (59.7%) Male] ~10 days following mild traumatic brain injury (n = 313) or orthopedic injury (n = 171). Whole brain deterministic diffusion tensor tractography was conducted and used to construct a 90 x 90 weighted (average fractional anisotropy) adjacency matrix for each scan. ComBat harmonization was applied separately at one of two different stages during data processing, either on the (i) weighted adjacency matrices (matrix harmonization) or (ii) global network metrics derived using unharmonized weighted adjacency matrices (parameter harmonization). Global network metrics based on unharmonized adjacency matrices and each harmonization approach were derived. Robust scanner effects were found for unharmonized metrics. Some scanner effects remained significant for matrix harmonized metrics, but effect sizes were less robust. Parameter harmonized metrics did not differ by scanner. Intraclass correlations (ICC) indicated good to excellent within-scanner consistency between metrics calculated before and after both harmonization approaches. Age correlated with unharmonized network metrics, but was more strongly correlated with network metrics based on both harmonization approaches. Parameter harmonization successfully controlled for scanner variability while preserving network topology and connectivity weights, indicating that harmonization of global network parameters based on unharmonized adjacency matrices may provide optimal results. The current work supports the use of ComBat for removing multiscanner effects on global network topology.

An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems

In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Loève decomposition, then a probabilistic collocation method (PCM) with sparse grids is employed to sample the stochastic process. On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. In terms of analysis, we first show the analytic regularity of solutions with respect to parameters in the diffusion coefficients and then present the rigorous convergence analysis for the proposed numerical scheme, i.e., the scheme is asymptotic compatible spatially and achieves an algebraic or sub-exponential convergence rate in the random coefficients space as the number of collocation points grows. These results are further confirmed by a number of benchmark problems. Finally, to validate the applicability of this approach we consider a randomly heterogeneous nonlocal problem with a given spatial correlation structure, demonstrating that the proposed PCM approach achieves substantial speed-up compared to conventional Monte Carlo simulations.

Magnetic resonance imaging and diffusion tensor imaging reconstruction of connectomes in a macropod, the quokka (Setonix brachyurus).

The diversity of the diprotodontids provides an excellent opportunity to study how a basic marsupial cortical plan has been modified for the needs of the mammals living in different habitats. Very little is known about the connections of the cerebral cortex with the deep brain structures (basal ganglia and thalamus) in this evolutionarily significant group of mammals. In this study, we performed mapping of brain regions and connections in a diprotodontid marsupial from data obtained from an excised brain scanned in high-field (9.4T) microstructural magnetic resonance imaging (MRI) instrument. The analysis was based on two MRI methodologies. First, high-resolution structural scans were used to map MRI visible brain regions from T1w and T2w images. Second, extensive diffusion tensor imaging (DTI) data were obtained to elucidate connectivity between brain areas using deterministic diffusion tracking of neuronal brain fibers. From the data, we were able to identify corticostriate connections between the frontal association and dorsomedial isocortex and the head of the caudate, and between the lateral somatosensory cortex and the putamen. We were also able to follow the olfactory and limbic connections by tracing fibers in the fornix, cingulum, intrabulbar part of the anterior commissure, and lateral olfactory tract. There was segregation of fibers in the anterior commissure such that olfactory connections passed through the rostroventral part and successively more dorsal cortical areas connected through more dorsal parts of the commissure. Our findings confirm a common pattern of cortical connectivity in therian mammals, even where brain expansion has occurred independently in diverse groups.

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan–Majda model and Taylor–Aris Dispersion

We study the long time behavior of an advection–diffusion equation with a random shear flow which depends on a stationary Ornstein–Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary N-point correlator using the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). In turn, appealing to the conclusion of the Hausdorff moment problem (Shohat and Tamarkin, 1943), we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability density function at long times. The strategy we presented is not only restricted to the parallel-plate channel domain. The same methods can derive effective equations for a straight channel with uniform arbitrary cross-section. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar’s long time limiting probability density function (PDF) for different types of initial conditions (e.g. deterministic and random).

Neutronic modeling and calculation of the Nuclear Heating Reactor NHR-5 by the deterministic codes DRAGON5 & DONJON5

The main objective of this study is to assess the neutronic modeling and calculations of the nuclear heating reactor NHR-5. To this end, the neutronic parameters of NHR-5 reactor underwent a comparative and validation protocols/methods through the analysis of the finite multiplication factor keff and some neutronic parameters including D,νΣf,Σa as well as the power and flux distributions, using the evaluated nuclear data libraries ENDF/B-VII.1, JEFF3.1 and JENDL4.0 based on 172 energy groups.In this study, the collision probability approach is used to simulate and calculate the neutronic parameters in NHR5 reactor using the deterministic core diffusion code DONJON5 and the lattice transport code DRAGON5. The group constants of the reactor components were produced by DRAGON5. The DONJON5 code was then used to calculate the effective multiplication factor, excess reactivity, and power and flux distributions by introducing these group constants. The results are compared to those of the experiment to validate the calculation scheme. The results of the simulations reveal that the computed neutronic parameters consistently produce reasonable and consistent results of flux and power distributions, excess reactivity and all other parameters when compared to the experiment values. Therefore, the reactor models of NHR5 developed by DRAGON and DONJON codes were good in predicting the effective multiplication factor as well as the studied neutronic parameters. In this study, we being able to show the potential validation of the reactor physics lattice transport code DRAGON5 and the core diffusion code DONJON5, as well as the nuclear data libraries ENDF/B-VII.1, JEFF3.1 and JENDL4.0.

Stochastic discontinuous Galerkin methods with low–rank solvers for convection diffusion equations

We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of deterministic convection diffusion equations, is used to handle the stochastic domain in this study, whereas discontinuous Galerkin method is used to discretize spatial domain due to its local mass conservativity. A priori error estimates of the stationary problem and stability estimate of the unsteady model problem are derived in the energy norm. To address the curse of dimensionality of stochastic Galerkin method, we take advantage of the low–rank Krylov subspace methods, which reduce both the storage requirements and the computational complexity by exploiting a Kronecker–product structure of system matrices. The efficiency of the proposed methodology is illustrated by numerical experiments on the benchmark problems.

Topological Alterations in White Matter Structural Networks in Blepharospasm

AbstractBackgroundAccumulating evidence indicates regional structural changes in the white matter (WM) of brains in patients with blepharospasm (BSP); however, whether large‐scale WM structural networks undergo widespread reorganization in these patients remains unclear.ObjectiveWe investigated topology changes and global and local features of large‐scale WM structural networks in BSP patients compared with hemifacial spasm (HFS) patients or healthy controls (HCs).MethodsThis cross‐sectional study applied graph theoretical analysis to assess deterministic diffusion tensor tractography findings in 41 BSP patients, 41 HFS patients, and 41 HCs. WM structural connectivity in 246 cortical and subcortical regions was assessed, and topological parameters of the resulting graphs were calculated. Networks were compared among BSP, HFS, and HCs groups.ResultsCompared to HCs, both BSP and HFS patients showed alterations in network integration and segregation characterized by increased global efficiency and modularity and reduced shortest path length. Moreover, increased nodal efficiency in multiple cortical and subcortical regions was found in BSP and HFS patients compared with HCs. However, these differences were not found between BSP and HFS patients. Whereas all participants showed highly similar hub distribution patterns, BSP patients had additional hub regions not present in either HFS patients or HCs, which were located in the primary head and face motor cortex and basal ganglia.ConclusionsOur findings suggest that the large‐scale WM structural network undergoes an extensive reorganization in BSP, probably due to both dystonia‐specific abnormalities and facial hyperkinetic movements. © 2021 International Parkinson and Movement Disorder Society

Unsteady dynamics of a classical particle-wave entity

A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations, and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.

Accounting for Variability in ULF Wave Radial Diffusion Models

AbstractMany modern outer radiation belt models simulate the long‐time behavior of high‐energy electrons by solving a three‐dimensional Fokker‐Planck equation for the drift‐ and bounce‐averaged electron phase space density that includes radial, pitch‐angle, and energy diffusion. Radial diffusion is an important process, often characterized by a deterministic diffusion coefficient. One widely used parameterization is based on the median of statistical ultralow frequency (ULF) wave power for a particular geomagnetic index Kp. We perform idealized numerical ensemble experiments on radial diffusion, introducing temporal and spatial variability to the diffusion coefficient through stochastic parameterization, constrained by statistical properties of its underlying observations. Our results demonstrate the sensitivity of radial diffusion over a long time period to the full distribution of the radial diffusion coefficient, highlighting that information is lost when only using median ULF wave power. When temporal variability is included, ensembles exhibit greater diffusion with more rapidly varying diffusion coefficients, larger variance of the diffusion coefficients and for distributions with heavier tails. When we introduce spatial variability, the variance in the set of all ensemble solutions increases with larger spatial scales of variability. Our results demonstrate that the variability of diffusion affects the temporal evolution of phase space density in the outer radiation belt. We discuss the need to identify important temporal and length scales to constrain variability in diffusion models. We suggest that the application of stochastic parameterization techniques in the diffusion equation may allow the inclusion of natural variability and uncertainty in modeling of wave‐particle interactions in the inner magnetosphere.