Lecture Series

Venue: HIM, Poppelsdorfer Allee 45, Lecture Hall

Lecture Series: July, 10-12

Lecturers: Martin Evans and Stefan Grosskinsky


Martin Evans: Condensation in the zero-range process and related models

The following four topics will be covered in the three lecture hours:

1. Zero range process

  • Definition
  • History and applications
  • Factorised Stationary state
  • Condensation

2. Dynamics of Condensation

  • Emergence of condensate from uniform background
  • Extinction of condensate

3. Misanthrope process

  • Definition
  • Proof of conditions for factorised stationary state
  • Solution of conditions
  • Explosive Condensation

4. Condensation with multiple constraints

  • Constraints on mass and variance
  • Interaction driven condensation
  • Random walks with area constraint

Stefan Grosskinsky: Stochastic and kinetic models of condensation

We review recent and classical results on aggregation/clustering models on a microscopic scale in the framework of stochastic particle systems, and on a mesoscopic scale using kinetic equations for discrete mass distributions. Both approaches are used to describe phenomena such as condensation, (instantaneous) gelation and coarsening, and we show how they are related in a mean-field scaling limit of stochastic particle systems. We also discuss differences between both descriptions such as volume constraints, and introduce size-biased versions of the dynamics to study the coarsening behaviour for condensing systems with bounded activity. In the last lecture we focus models with product aggregation kernels and unbounded activity, and summarize results for condensation dynamics and instantaneous condensation. Examples that will be used as illustrations throughout include the zero-range process, the inclusion process and generalizations, and Becker-Doering models.
Since my background is in probability rather than kinetic theory, I naturally will have a bit more to say about particle systems than kinetic models. Part of the lecture is based on recent work with Watthanan Jatuviriyapornchai (Bangkok) and Paul Chleboun (Warwick).

Rough outline

Lecture 1:

  • Intro of aggregation models in stochastic particle systems (SPS) and kinetic theory (rate equations)
  • Definition of condensation in SPS, previous results and examples
  • Definition of gelation in kinetic models and examples
  • Connection between descriptions and the role of volume in kinetic models

Lecture 2:

  • Kinetic models as general mean-field scaling limits of SPS for kernels bounded by a bilinear function
  • Results on coarsening dynamics for zero-range process
  • Size-biased dynamics and Becker-Doering-type models

Lecture 3:

  • Focus on models with product kernels of the form K(n,m)=n^g (d+m^g ) with g>=1 and d>=0
  • The inclusion process as an 'exactly solvable' SPS
  • Summary of results on instantaneous condensation for SPS and on gelation for kinetic models
  • Mean-field scaling limit for \gamma >1 and d=0 (symmetric product kernel)

Literature Review