Master Thesis – Mishael Derla

Analytical Models of Random Organization
finished 2025-12
supervised by Michael Schmiedeberg

Abstract

I present my literature work on and own investigations into the continuous dynamical phase transition displayed by random organization and other closely related models. These systems are constituted by spatially extended grains initially placed at randomly, uniformly and independently distributed locations; they are then subjected to iterative protocols trying to eliminate overlaps. In random organization, in each iteration, every grain overlapping with other grains is shifted by a small, random step in space. This protocol displays two regimes of behaviour, depending on its packing density Φ, separated by a dynamical phase transition at some critical packing density Φc: the absorbing phase Φ Φc, where the system fails to find overlap-free configurations and grains move around indefinitely. My literature work is on continuous transitions far from equilibrium in general and two universality in particular: (non-conserved) directed percolation (DP), which I mostly use to illustrate the concepts and methods in the study of critical dynamical systems, and the Manna or conserved directed percolation (CDP) class, to which the absorbing state transition in random organization probably belongs. I conclude that analytical studies into the universal critical behaviour of CDP are dominated by field-theories of the associated order parameter, the density 𝜌𝐴(𝒓, 𝑡) of active grains. This is because large-scale field-theories boast methods geared towards understanding universal behaviour. They are not, however, immediately suitable for finding precise (highly model-dependent) critical densities. The latter task has been left to simulation studies. My own investigations explore and try to assess possible pathways to analytical theories for predicting critical densities. To the end of calculating Φc on paper, I suggest a reaction system, whose analysis is ongoing, that respects spatial correlations beyond mean-field by formulating it in the operator formalism introduced by Masao Doi in two papers from 1976 that are seminal for theoretical non-equilibrium physics.