C08  Stochastic spatial coagulation particle processes
Head(s): Dr. Robert Patterson (WIAS)
Project member(s): Dr. Luisa Andreis, Dr. Michiel Renger
Participating institution(s): WIAS
Project Summary
We use large deviations principles for analysing large systems of stochastic interacting particles and then performing further scaling limits. Our particle systems are models for (bio)chemical reactions, aggregate coagulation and the evolution of convective towers in the atmosphere. In each case some discrete entities, which we term particles, experience binary interactions when they are ‘close’ in an appropriate sense. These entities have internal structure of some kind, such as mass or chemical composition or the height of a convective tower, which play an important role in determining the interaction rates. We consider three scenarios where a parameter N proportional to the number of ‘particles’ becomes large: Firstly, where the particles are contained in a single, small and wellmixed volume, with size independent of N. Secondly, where the particles are contained in many small volumes, whose sizes are bounded and possibly vanishing, and thirdly, where the volume containing the particles is of size proportional to N.
Each of our motivating applications has, in addition to N and the scales it generates, a second small parameter ε, which generates an additional cascade of scales: In the first two applications it is possible to describe systems by the concentration C^{(N)} (t, x, y) of the particles of type y at time t at the site x ∈ IR^{d} defined as the quotient of their number with N (here interpreted as a kind of volume). Application 1 is a family of chemical reactions with subfamilies accelerated by different negative powers of ε to model biochemical processes on very different time scales. Application 2 deals with accelerated bulk diffusion and slow diffusion in a vanishing membrane separating two compartments. We expect to interpret an existing reduced model and associated numerical method for the nucleus and cytoplasm of a cell as a limiting case of a more general approach.
In the third application concentrations are not an appropriate mathematical description, because we have to work in the thermodynamic limit N → ∞ of N coagulating Brownian motions in a large container with volume V (N) ≍ N. Here in the simple setting of spheres that merge on reaction/coagulation we seek to understand the statistics of the coagulation events now that we model the positions of the particles explicitly. We use a novel mathematical approach with marked point processes. A simplified form of the point process model will allow us to model the evolution of convective towers, which are triggered due to surface layer turbulence in the atmosphere and play an important role in tropical storm formation as studied in C06.
In all three applications large scale effects arise from small scale randomness in the presence of additional scales. In this project we seek to use these applications to guide the development of tools from the theory of large deviations principles (LDPs) and Gconvergence to rigorously understand the micro–macro transition in conjunction with additional scaling limits.

Project surrounding publications
 Patterson, R. I. A (2016) Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries. Journal of Evolution Equations, 16 . pp. 261291.
 Yapp, E.K.Y. and Patterson, R. I. A and Akroyd, J. and Mosbach, S. and Adkins, E.M. and Miller, J.H. and Kraft, M. (2016) Numerical simulation and parametric sensitivity study of optical band gap in a laminar coflow ethylene diffusion flame. Combustion and Flame, 167 . pp. 320334.