Anomalous Diffusion and Stochastic Processes in Complex Systems

Anomalous diffusion refers to transport phenomena in which the mean squared displacement of particles deviates from the linear time dependence characteristic of classical Brownian motion. In complex media—ranging from crowded cellular interiors and heterogeneous polymer networks to geological formations and porous catalysts—the underlying heterogeneity, long‐range correlations and trapping events give rise to subdiffusive or superdiffusive dynamics. Stochastic models such as continuous time random walks, fractional Brownian motion and diffusing diffusivity frameworks have been developed to capture non‐Gaussian displacement statistics, ageing effects and ergodicity breaking. These approaches offer insight into intracellular trafficking, macromolecular crowding, contaminant transport in subsurface environments and the design of advanced functional materials. By linking microscopic structure to macroscopic transport laws, the study of anomalous diffusion informs diverse fields including biophysics, materials science, environmental engineering and soft condensed matter physics.

Research from Nature Portfolio

Recent studies have introduced a universal framework for exploration dynamics in random walks, demonstrating that the time required to discover new sites follows analytic laws that apply across normal, anomalous and disordered diffusion. This work unifies disparate transport regimes into common universality classes and provides compact expressions for visitation statistics on complex substrates. Separately, investigations of membrane proteins on live cells have revealed dynamic heterogeneity and non‐Gaussian displacement distributions arising from active remodelling of the cortical actin network. These findings highlight how local structural rearrangements in the membrane environment drive deviations from Gaussian diffusion and implicate stochastic heterogeneity as a regulatory mechanism in cellular signalling and receptor function.

Research from all publishers

A comprehensive review of heterogeneous anomalous transport in cellular and molecular biology emphasises advances in experimental single‐molecule methods and high‐throughput image analysis, together with generative theoretical models that integrate viscoelasticity and spatially varying exponents. In environmental systems, measurements of molecular diffusion through diverse rock samples have demonstrated persistent long‐time tailing and non‐Fickian tracer advance, successfully described by continuous time random walk formalisms, with implications for contaminant migration and resource recovery. Fundamental theoretical work on Brownian yet non‐Gaussian diffusion has established a minimal diffusing diffusivity model that bridges superstatistical and subordination approaches, explaining crossover regimes and providing a versatile platform for interpreting non‐Gaussian yet normal‐like transport in soft and biological matter.

Anomalous Diffusion and Stochastic Processes in Complex Systems publication trend

The graph below shows the total number of articles in anomalous diffusion and stochastic processes in complex systems across all publications each year (not limited to Nature Index journals).

Technical terms

Anomalous diffusion: Transport where mean squared displacement scales non‐linearly with time (〈x²〉 ∝ t^α, α ≠ 1).

Continuous time random walk (CTRW): A stochastic model characterised by random waiting times and step lengths, used to describe trapping and hopping dynamics.

Fractional Brownian motion (FBM): A Gaussian process with long‐range correlations, yielding subdiffusive or superdiffusive behaviour depending on the correlation exponent.

Diffusing diffusivity: A framework in which the diffusion coefficient itself fluctuates in time or space, producing non‐Gaussian displacement distributions.

Mean squared displacement (MSD): A statistical measure of the average squared distance travelled by a particle as a function of time.

References

1. Heterogeneous anomalous transport in cellular and molecular biology. Reports on Progress in Physics (2023).
2. Universal exploration dynamics of random walks. Nature Communications (2023).
3. Diffusion in Porous Rock Is Anomalous. Environmental Science and Technology (2024).
4. Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities. Physical Review X (2017).
5. Dynamic heterogeneity and non-Gaussian statistics for acetylcholine receptors on live cell membrane. Nature Communications (2016).