M5: Reaction and steric effects in stochastic models of drift diffusion

Project completed March 31, 2012

Background

Life is dominated by systems composed of many particles or individuals with a collective behaviour, for example, the movement of cells (see Figure 1), and animal swarms. Stochastic models describing how these interacting individuals give rise to collective behaviour have become a widely used tool across disciplines.

Simple models of diffusive particles with short-ranged repulsive interactions are relevant in many systems, but despite their conceptual simplicity, particle-based models become computationally intractable for large systems of interacting particles. In such cases, a continuum description based on partial differential equations that can capture overall population density becomes attractive.

For interacting particles, however, a continuum description is confined to a high-dimensional space. The challenge is reducing the description while still maintaining the key attributes. A common approach is to use ad hoc closure approximations, which generally assume that particles are independent at some point. However, this assumption often generates errors in the resulting continuum model.

As an alternative to the ad hoc closure approximation, researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) have developed a systematic method to derive the continuum description based on matched asymptotic expansions in the particle volume fraction. 

Techniques and Challenges

The model focuses on a class of particle systems consisting of diffusive particles with short-ranged repulsive interactions involving hard-spheres – particles that cannot overlap – or soft-spheres. Unlike other continuum methods, this new approach to steric interactions leads to a system with a finite number of particles that are far from equilibrium. 

For a system of identical hard-spheres, the result of the method is a nonlinear diffusion equation for the one-particle density function with excluded-volume effects enhancing the overall collective diffusion rate. Comparisons between the solution to this differential equation and the solution to the full particle system (solved with Monte Carlo methods) show agreement. Moreover, the new method gives insight into how steric interactions at the particle level influence the population dynamics. The systematic nature of this technique means that it is flexible, and this core problem was expanded in several directions.

In order to understand how the interspecies competition emerges at the population level, the work was extended to model the interaction of multiple species with different characteristics. The analysis showed that the continuum description of a system with two types of particles is a nonlinear cross-diffusion model. It also explains two alternative notions of the diffusion coefficient that are often confused, namely collective diffusion and self-diffusion. Moreover, the two-population model proves useful for studying transport through obstacles, which has many applications in porous media, such as in water filtration and treatment of biological tissues.

The method was also extended to take into account the confinement of the domain that the particle occupies. For example, in the case of a narrow channel, analysis yields an effective one-dimensional nonlinear diffusion equation that depends on the channel width. An interesting consequence of this result is that it allows interpolation between qualitatively very different types of diffusion: a free two- or three-dimensional diffusion and a one-dimensional diffusion, known as single-file diffusion. 

Results

The technique was also applied to diffusive soft-spheres, in which case it yields an interaction-dependent nonlinear term in the diffusion equation. Importantly, the results are an improvement from those obtained via common closure approaches, showing that the method of matched asymptotics is preferable for short-range interactions. 

The analysis shows promise in the challenging issue of incorporating long-range interactions – this represents a crucial step forward since many real applications exhibit a combination of short- and long-range interactions. 

The Future

This project has opened many challenging mathematical questions for future work. Intriguing areas to explore include the combination of short- and long-range interactions between particles, as mentioned above, as well as the connection between on- and off-lattice based models of the same stochastic particle system, as well as extensions to Langevin dynamics.

References

[10/65] Bruna M., Chapman S.J.: Excluded-volume effects in the diffusion of hard spheres, Phys. Rev. E, vol. 85, p. 011103, 2012

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Carrillo J.A., McCann R.J., Villani C., Rev. Mat. Iberoam. 19, 971, 2003