# 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 well-mixed 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 sub-families 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 G-convergence to rigorously understand the micro–macro transition in conjunction with additional scaling limits.

### Project publications

- Patterson, R. I. A and Renger, M. (2019)
*Large deviations of jump process fluxes.*Math. Phys. Anal. Geom., 22 (21). pp. 1-32. ISSN 1385-0172; ESSN: 1572-9656 - Renger, M. and Zimmer, J. (2019)
*Orthogonality of fluxes in general nonlinear reaction networks.*SFB1114 Preprint 07/2019 in WIAS No. 2609 . pp. 1-12. (Unpublished) - Heydecker, Daniel and Patterson, Robert I. A. (2019)
*Bilinear Coagulation Equations.*SFB1114 Preprint in arxiv:1902.07686 . (Unpublished) - Andreis, L. and König, W. and Patterson, R. I. A (2019)
*A large-deviations approach to gelation.*SFB 1114 Preprint in arXiv:1901.01876 . pp. 1-22. (Unpublished) - Heida, M. and Patterson, R. I. A and Renger, M. (2018)
*Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space.*J. Evol. Equ. . pp. 1-42. ISSN Online: 1424-3202 Print: 1424-3199 - Renger, M. (2018)
*Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory.*J. Stat. Phys., 172 (5). pp. 1261-1326. ISSN 0022-4715 - Koltai, P. and Renger, M. (2018)
*From Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data.*Journal of Nonlinear Science, 28 (5). pp. 1915-1957. ISSN 1432-1467 (online) - Renger, M. (2018)
*Gradient and Generic systems in the space of fluxes, applied to reacting particle systems.*SFB 1114 Preprint in arXiv:1806.10461 . pp. 1-29. (Unpublished) - Mielke, A. and Patterson, R. I. A and Peletier, M. A. and Renger, M. (2017)
*Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics.*SIAM Journal on Applied Mathematics, 77 (4). pp. 1562-1585. ISSN 1095-712X (online) - Liero, M. and Mielke, A. and Peletier, M. A. and Renger, M. (2017)
*On microscopic origins of generalized gradient structures.*Discrete and Continuous Dynamical Systems - Series S, 10 (1). - Mielke, A. and Peletier, M. A. and Renger, M. (2016)
*A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility.*Journal of Non-Equilibrium Thermodynamics, 41 (2). - Erbar, M. and Maas, J. and Renger, M. (2015)
*From large deviations to Wasserstein gradient flows in multiple dimensions.*Electronic Communications in Probability, 20 (89). -
### 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. 261-291. - 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 co-flow ethylene diffusion flame.*Combustion and Flame, 167 . pp. 320-334.