Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking
arXiv:2509.12571 · doi:10.1088/1361-6633/ae358c
Abstract
Diffusive motion is a fundamental transport mechanism in physical and biological systems, governing dynamics across a wide range of scales -- from molecular transport to animal foraging. In many complex systems, however, diffusion deviates from classical Brownian behaviour, exhibiting striking phenomena such as Brownian yet non-Gaussian diffusion (BYNGD) and anomalous diffusion. BYNGD describes a frequently observed statistical feature characterised by the coexistence of linear mean-square displacement (MSD) and non-Gaussian displacement distributions. Anomalous diffusion, in contrast, involves a nonlinear time dependence of the MSD and often reflects mechanisms such as trapping, viscoelasticity, heterogeneity, or active processes. Both phenomena challenge the conventional framework based on constant diffusivity and Gaussian statistics. This review focuses the theoretical modelling of such behaviour via the Langevin equation with fluctuating diffusivity (LEFD) -- a flexible stochastic framework that captures essential features of diffusion in heterogeneous media. LEFD not only accounts for BYNGD but also naturally encompasses a wide range of anomalous transport phenomena, including subdiffusion, ageing, and weak ergodicity breaking. Ergodicity is discussed in terms of the correspondence between time and ensemble averages, as well as the trajectory-to-trajectory variability of time-averaged observables. The review further highlights the empirical relevance of LEFD and related models in explaining diverse experimental observations and underscores their value to uncovering the physical mechanisms governing transport in complex systems.
52 pages, 16 figures
References in corpus (97)
- Theoretical perspective on the glass transition and amorphous materials
- Superstatistics
- Intermittent search strategies
- In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
- Random Time-Scale Invariant Diffusion and Transport Coefficients
- Geometrical explanation and scaling of dynamical heterogeneities in glass forming systems
- Universal nature of particle displacements close to glass and jamming transitions
- Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
- "Diffusing diffusivity": A model for anomalous and "anomalous yet Brownian" diffusion
- Ergodic Properties of Fractional Brownian-Langevin Motion
- State-dependent diffusion: thermodynamic consistency and its path integral formulation
- Coarse-grained microscopic model of glass formers
- Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins
- Slow dynamics in glassy soft matter
- Mode-Coupling Theory (MCT) Lecture Notes
- Stochastic Ergodicity Breaking: a Random Walk Approach
- Viscoelastic subdiffusion: from anomalous to normal
- Nonergodic Subdiffusion from Brownian Motion in an Inhomogeneous Medium
- Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity
- Elucidating the Origin of Heterogeneous Anomalous Diffusion in the Cytoplasm of Mammalian Cells
- Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
- Diffusion-limited reactions in dynamic heterogeneous media
- Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion
- Multiple Excitation of Confined Graphene Plasmons by Single Free Electrons
- Machine learning method for single trajectory characterization
- Universal relation between instantaneous diffusivity and radius of gyration of proteins in aqueous solution
- Spectral content of a single non-Brownian trajectory
- What does the potential energy landscape tell us about the dynamics of supercooled liquids and glasses?
- Non-ergodicity, fluctuations, and criticality in heterogeneous diffusion processes
- Packets of Diffusing Particles Exhibit Universal Exponential Tails
- From the area under the Bessel excursion to anomalous diffusion of cold atoms
- Noisy continuous time random walks
- Anomalous diffusion in fractal globules
- Infinite Ergodic Theory for Heterogeneous Diffusion Processes
- Development of anomalous diffusion among crowding proteins
- Origin of subdiffusion of water molecules on cell membrane surfaces
- Heterogeneous diffusion in a reversible gel
- Loopy Lévy flights enhance tracer diffusion in active suspensions
- Bayesian deep learning for error estimation in the analysis of anomalous diffusion
- Anomalous Polymer Dynamics Is Non-Markovian: Memory Effects and The Generalized Langevin Equation Formulation
- Fluctuation Analysis of Time-Averaged Mean-Square Displacement for Langevin Equation with Time-Dependent and Fluctuating Diffusivity
- Langevin equation with fluctuating diffusivity: a two-state model
- Fluid mixing by curved trajectories of microswimmers
- Two Coupled Mechanisms Produce Fickian, yet non-Gaussian Diffusion in Heterogeneous Media
- Distribution of particle displacements due to swimming microorganisms
- Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion
- Langevin dynamics in inhomogeneous media: Re-examining the Itô-Stratonovich Dilemma
- First passage statistics for diffusing diffusivity
- The Hitchhiker model for Laplace diffusion processes in the cell environment
- Distributional Response to Biases in Deterministic Superdiffusion
- Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories
- Dynamic phase coexistence in glass-forming liquids
- Ergodic properties of continuous-time random walks: finite-size effects and ensemble dependences
- Systems-level approach to uncovering diffusive states and their transitions from single particle trajectories
- Universal Fluctuations of Single-Particle Diffusivity in Quenched Environment
- Anomalous diffusion in random dynamical systems
- Large deviations for continuous time random walks
- Distributional Ergodicity in Stored-Energy-Driven Lévy Flights
- Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation
- Length scales in Brownian yet non-Gaussian dynamics
- Distributional Behaviors of Time-averaged Observables in Langevin Equation with Fluctuating Diffusivity: Normal Diffusion but Anomalous Fluctuations
- Anomalous diffusion in a quenched-trap model on fractal lattices
- Relaxation Functions of Ornstein-Uhlenbeck Process with Fluctuating Diffusivity
- Non-Gaussian diffusion in static disordered media
- Phase Diagram in Stored-Energy-Driven Lévy Flight
- Non-Gaussian diffusion of mixed origins
- Quantitative evaluation of methods to analyze motion changes in single-particle experiments
- Anisotropic dynamics of a self-assembled colloidal chain in an active bath
- Non-Gaussian diffusion near surfaces
- Ultraslow Convergence to Ergodicity in Transient Subdiffusion
- Non-Gaussian displacements in active transport on a carpet of motile cells
- Convergence to a Gaussian by narrowing of central peak in Brownian yet non-Gaussian diffusion in disordered environments
- Distributional Behavior of Diffusion Coefficients Obtained by Single Trajectories in Annealed Transit Time Model
- Brownian non-Gaussian polymer diffusion and queing theory in the mean-field limit
- Elucidating fluctuating diffusivity in center-of-mass motion of polymer models with time-averaged mean-square-displacement tensor
- Distributional behavior of time averages of non- observables in one-dimensional intermittent maps with infinite invariant measures
- Quenched trap model on the extreme landscape: the rise of sub-diffusion and non-Gaussian diffusion
- Polymers critical point originates Brownian non-Gaussian diffusion
- Non-self-averaging behaviors and ergodicity in quenched trap model with finite system size
- The random walk of intermittently self-propelled particles
- Crossover Time in Relative Fluctuations Characterizes the Longest Relaxation Time of Entangled Polymers
- Being Heterogeneous Is Advantageous: Extreme Brownian Non-Gaussian Searches
- Detection of Transition Times from Single-particle-tracking Trajectories
- Heterogeneous biological membranes regulate protein partitioning via fluctuating diffusivity
- Fractional Brownian motion with fluctuating diffusivities
- Active cage model of glassy dynamics
- Multifractional Brownian motion with telegraphic, stochastically varying exponent
- Brownian non-Gaussian diffusion of self-avoiding walks
- Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory
- Being heterogeneous is disadvantageous: Brownian non-Gaussian searches
- Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling
- Fluctuating Diffusivity Emerges even in Binary Gas Mixtures
- Coarse-Graining of Microscopic Dynamics into Mesoscopic Transient Potential Model
- Glassy phases of the Gaussian Core Model
- Change-point detection in anomalous-diffusion trajectories utilising machine-learning-based uncertainty estimates
- Brownian Yet Non-Gaussian Diffusion of a Light Particle in Heavy Gas: Lorentz Gas Based Analysis
- Universal Poisson statistics of a passive tracer diffusing in dilute active suspensions