Pattern formation with mass conservation From passive to active models

Doctoral examination procedure finished at: Doctoral examination procedure at University of Münster

Start date of doctoral examination procedure: 01/10/2017

End date of doctoral examination procedure: 21/04/2023

Name of the doctoral candidate: Frohoff-Hülsmann, Tobias

Doctoral subject: Physik

Doctoral degree: Dr. rer. nat.

Awarded by: Department 11 - Physics

List of all reviewers: Thiele, Uwe; Gurevich, Svetlana; Krenner, Hubert; Bär, Markus

If particles can neither be consumed nor produced, the system is particle-, i.e., mass-conserving. In this thesis we theoretically elaborate on how mass-conservation influences pattern formation in spatially extended systems. For this, we employ different continuum mean-field models and analytically study their linear and weakly nonlinear behavior. This is amended by investigations of stationary patterns, localized states, coexisting phases and time-periodic behavior in the fully nonlinear regime by means of both numerical path continuation and time simulation. We study passive and active models, i.e., models that have a gradient dynamics structure and models where this structure is broken, respectively. Specifically, the latter is employed by introducing nonreciprocal interactions that break Newton's third law and resemble a predator-prey interaction scheme. This represents an energy flow into the system, thus describing active, e.g., living matter. In addition to the specific results, the present work also provides conceptual insights regarding the role of mass conservation in pattern-forming systems. In the first part, we review and amend the widely used classification of linear instabilities that describe the onset of pattern formation. As a consequence, eight basic linear behaviors can be identified where in four cases conservation laws are involved. Further, we review the corresponding amplitude equations that describe the weakly nonlinear regime and in seven of the eight cases briefly discuss their phenomenology. We identify the conserved-Hopf instability, a large-scale oscillatory linear instability with two conservation laws, as the eighth case for which an amplitude equation has not been determined, yet. Through a detailed weakly-nonlinear analysis we determine the general nonreciprocal Cahn-Hilliard model as the missing amplitude equation. It represents a two-species model where each species is conserved, i.e., described by a continuity equation. In the uncoupled limit two classical Cahn-Hilliard equations are reproduced, in the general case they are asymmetrically, i.e., nonreciprocally coupled. In the second part, we consider two basic processes in the presence of mass conservation, namely phase separation and crystallization, that are modeled by the Cahn-Hilliard and the phase-field crystal equation, respectively. We study the equilibrium states in the vicinity of the corresponding phase transition, unfold their bifurcation structure in finite-sized systems and shed light on the formation of the Maxwell construction in the thermodynamic limit. In the third part, we phenomenologically reintroduce the general nonreciprocal Cahn-Hilliard model as a reasonable active matter model with a passive, i.e., ``dead'' limit. We study its linear regime where, in the active case, rich behavior is encountered that involves conserved-Turing and conserved-Hopf instabilities. We find a 1:1 correspondence between the occurrence of primary instabilities in conserved and non-conserved dynamics. Then, we proceed with the analyses on the fully nonlinear behavior for a specific linear coupling. We investigate how the coarsening, i.e., the phase separation process of the original Cahn-Hilliard equation is suppressed in the two-species model and identify three mechanisms. Furthermore, we investigate the phase behavior in the reciprocal case and show that linear coupling allows us to define active pendants to chemical potentials and pressure also in the nonreciprocal case. Further, we elaborate on the fully nonlinear behavior close to the conserved-Hopf instability that, e.g., leads to large-scale oscillations and phase-separated (modulated) traveling states. We also investigate localized states related to the conserved-Turing instability. We encounter homoclinic slanted snaking and discover the occurrence of spatially localized time-periodic states. The analysis of the conserved-Hopf-Turing resonance that leads to intricate nonstationary behavior completes our study on the linear nonreciprocal Cahn-Hilliard model. In the fourth and final part of this thesis we elaborate on the reliability of simplified models. We discover that unphysical asymmetric stationary states are common in various popular minimal active matter models that include the active phase-field crystal model, the linear nonreciprocal Cahn-Hilliard model, and the FitzHugh-Nagumo model. We determine the underlying shared structure and systematically show how the unphysical behavior can be prohibited.