Diffusion of Particles with Short-Range Interactions

A system of interacting Brownian particles subject to short-range repulsive potentials is considered. A continuum description in the form of a nonlinear diffusion equation is derived systematically in the dilute limit using the method of matched asymptotic expansions. Numerical simulations are performed to compare the results of the model with those of the commonly used mean-field and Kirkwood-superposition approximations, as well as with Monte Carlo simulation of the stochastic particle system, for various interaction potentials. Our approach works best for very repulsive short-range potentials, while the mean-field approximation is suitable for long-range interactions. The Kirkwood-superposition approximation provides an accurate description for both short- and long-range potentials but is considerably more computationally intensive.

[1]  Massimo Fornasier,et al.  Particle, kinetic, and hydrodynamic models of swarming , 2010 .

[2]  Maria Bruna,et al.  Excluded-volume effects in the diffusion of hard spheres. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  I. R. Mcdonald,et al.  Theory of simple liquids , 1998 .

[4]  A. Bertozzi,et al.  Self-propelled particles with soft-core interactions: patterns, stability, and collapse. , 2006, Physical review letters.

[5]  A. Mogilner,et al.  A non-local model for a swarm , 1999 .

[6]  Alex James,et al.  Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias , 2015, Journal of The Royal Society Interface.

[7]  Maria Bruna,et al.  Diffusion of Finite-Size Particles in Confined Geometries , 2012, Bulletin of mathematical biology.

[8]  Chun Liu,et al.  PNP equations with steric effects: a model of ion flow through channels. , 2012, The journal of physical chemistry. B.

[9]  Antti-Pekka Hynninen,et al.  Phase diagrams of hard-core repulsive Yukawa particles. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[11]  Matthew J Simpson,et al.  Correcting mean-field approximations for birth-death-movement processes. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Ramon Grima,et al.  A continuum approximation to an off-lattice individual-cell based model of cell migration and adhesion. , 2014, Journal of theoretical biology.

[13]  Peter A. Markowich,et al.  On Singular Limits of Mean-Field Equations , 2001 .

[14]  D. Heyes,et al.  The influence of potential softness on the transport coefficients of simple fluids. , 2005, The Journal of chemical physics.

[15]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[16]  Deborah C Markham,et al.  Incorporating spatial correlations into multispecies mean-field models. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  B. U. Felderhof,et al.  Diffusion of interacting Brownian particles , 1978 .

[18]  Leonid Berlyand,et al.  Complexity Reduction in Many Particle Systems with Random Initial Data , 2013, SIAM/ASA J. Uncertain. Quantification.

[19]  조준학,et al.  Growth of human bronchial epithelial cells at an air-liquid interface alters the response to particle exposure , 2013, Particle and Fibre Toxicology.

[20]  Hinch.  Perturbation Methods , 1991 .

[21]  A. Singer,et al.  Maximum entropy formulation of the Kirkwood superposition approximation. , 2004, The Journal of chemical physics.

[22]  Angel Mulero,et al.  Theory and Simulation of Hard-Sphere Fluids and Related Systems , 2008 .

[23]  J. Kirkwood.  Statistical Mechanics of Fluid Mixtures , 1935 .

[24]  J. A. Carrillo,et al.  Numerical Simulation of Diffusive and Aggregation Phenomena in Nonlinear Continuity Equations by Evolving Diffeomorphisms , 2009, SIAM J. Sci. Comput..

[25]  Martin Burger,et al.  Nonlinear Cross-Diffusion with Size Exclusion , 2010, SIAM J. Math. Anal..