### Project Summary

We consider stochastic models for the transport and coagulation of chemical and colloid particles. Such systems are characterized by a number of different length scales: system size, particle interaction range, particle separation, mean (square) particle displacement as well as a very wide range of particle concentrations. These length scales depend on particle mass, and therefore evolve their own ranges of values as the particle mass spectrum develops due to coagulation. We propose to address questions of how these ranges of scales de.ning the microscopic behavior interact to yield mesoscopic and macroscopic behavior.

Our root model is a large system of Brownian particles carrying masses, which randomly coagulate according to a size and separation dependent rate kernel. Coagulation means the replacement of two particles with a new particle containing their combined mass and located on the line segment between the two positions of the old particles. We are particularly interested in thermodynamic limits under various parameter scaling regimes. A spatially homogeneous simplification of such particle systems leads in an appropriate limit to an ODE first written by Smoluchowski, after whom it is now named.

This can naturally be extended to a PDE as a reaction–diffusion equation modeling the spatial variations in the particle concentration. In this project, we (1) derive a new large-deviation principle for the spatial-temporal evolution of the particles in the thermodynamic limit, (2) derive a description of the particle system in terms of an ergodic marked point process, (3) study the appropriately rescaled behavior of key quantities of the system in the extreme regime of fast diffusion, and in (4) we do the same in the extreme regime of long time stretches.

Our projected result in (1) will give a new interpretation of the solution of the Smoluchowski PDE as the minimizer of a physically meaningful large-deviation rate function. This line of research for general particle models is currently in full swing, see Project C03 and [ADPZ11], e.g.; our focus is on the description of the coagulation mechanism. Our approach in (2) describes all details of the particle model in terms of ergodic limits and provides a new, more detailed, perspective. In (3) and (4), we couple the diffusion parameter and the time parameter, respectively, with the size of the system and aim at the identi.cation of a cascade of arising scales of the most important quantities, like the cardinality and the size of the particles and the number of contacts and of coagulations per time unit, relying on ideas from our previous work [4] for interacting many-body systems. For understanding some of the scales, we will take the Smoluchowski PDE as a fundamental orientation. One of the main motivations behind (3) and (4) is the desire to find an approximation of the (spatially homogeneous) Smoluchowski ODE in terms of such extreme particle behavior.

### Project publications

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)

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).

Liero, M. and Mielke, A. and Peletier, M. A. and Renger, M.
(2015)
*On microscopic origins of generalized gradient structures.*
SFB Preprint
.
(Submitted)

Mielke, A. and Peletier, M. A. and Renger, M.
(2015)
*A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility.*
Journal of Non-Equilibrium Thermodynamics
.
(Submitted)

### 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.