Alladi Ramakrishnan Hall

Data-driven modelling of diffusion in complex, heterogeneous medium using Bayesian inference

Samudrajit Thapa

Max Planck Institute, Dresden

Particle diffusion in heterogeneous systems poses the following question: Can a single available model describe the entire dynamics of a particle in complex biological, soft matter systems? Indeed, often several different physical mechanisms are at work and it is more insightful to rank them based on the likelihood of them explaining the dynamics. The first part of this talk will discuss — within the Bayesian framework—(i) how maximum-likelihood model selection can be done by assigning probabilities to each feasible model and (ii) how to estimate the parameters of each model. In particular, the implementation of this powerful statistical tool using the Nested Sampling algorithm to compare—at the single trajectory level—models of Brownian motion, viscoelastic anomalous diffusion and normal yet non-Gaussian diffusion will be discussed. Finally, the application of this method to experimental data of tracer diffusion in polymer-based hydrogels (mucin) will be presented. Viscoelastic anomalous diffusion is often found to be most probable, followed by Brownian motion, while the model with a diffusing diffusion coefficient is only realised rarely.
The second part of this talk will discuss how the results of Bayesian analysis can lead to meaningful model building. Fractional Brownian motion (FBM), a Gaussian, non-Markovian, self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. Inspired by the results discussed in the first part, generalizations of FBM will be presented to include (a) Hurst index that randomly changes from trajectory to trajectory but remains constant along a given trajectory, and (b) Hurst index that varies stochastically in time along a trajectory. A general mathematical framework for analytical, numerical, and statistical analysis for both (a) and (b) will be discussed. An algorithm to distinguish between the three classes of random motions, namely the canonical FBM and its generalizations (a) and (b) will be presented, and the applicability of this algorithm will be demonstrated by analyzing real-world examples for all the three classes