Phase transitions

We study the long time behaviour and the number and structure of stationary solutions for the McKean-Vlasov equation, a nonlinear nonlocal Fokker-Planck type equation that describes the mean field limit of a system of weakly interacting diffusions. We consider two cases: the McKean-Vlasov equation in a multiscale confining potential with quadratic, Curie-Weiss, interaction (the so-called Dasai-Zwanzig model), and the McKean-Vlasov dynamics on the torus with periodic boundary conditions and with a localized interaction.

Biological macromolecules are often charged and perform their functions in ionic environments. For this reason the mechanical and electrostatic behavior of macromolecules are connected. In this talk we will shed light on the interplay of mechanics and electrostatics of rod-like macromolecules and cells. In the first part of the talk we will study the deformations of a cell constrained between two plates and subject to a change in potential difference across its membrane.

Plant pollen are cells which exhibit a plethora of surface patterns, including stripes, hexagons, foam-like ridges, and arrangements of spikes. The spherical shape of the cells ensures that these various patterns have defects. We study the development of these patterns and conjecture that the diversity of the observed patterns, and their reproducibility within a single species, may be explained by a first-order phase transition from an unpatterned to a patterned state.

The edge-triangle exponential random graph model has been a topic of continued research interest. We review recent developments in the study of this classic model and concentrate on the phenomenon of phase transitions. We first describe the asymptotic feature of the model along general straight lines. We show that as we continuously vary the slopes of these lines, a typical graph exhibits quantized behavior, jumping from one complete multipartite structure to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets.

In the talk we describe two applications important for global climate
and energy studies: methane hydrates and coalbed methane. Methane
hydrates also known as ice that burns are present in large amounts
along continental slopes and in permafrost regions and, therefore, are
a possible source of energy and at the same time a potential
environmental hazard. Their evolution critically depends on how the
hydrate formation and dissociation affects the porescale properties.
This so far has been only modeled with ad-hoc phenomenological

Large scale conformational rearrangements represent both a
challenge and an opportunity for rational drug design.
Exploring the conformational space of a target protein with sufficient detail is computationally very demanding and most current docking programs are very limited in this respect.
If it were possible, however, it could open the avenue to the design of
more selective drug candidates. Here we show how atomistic molecular
dynamics together with methods developed to accelerate rare events can

We establish the existence of the phase transition in site percolation on pseudo-random d-regular graphs. Let G=(V,E) be an (n,d,lambda)-graph, that is, a d-regular graph on n vertices in which all eigenvalues of the adjacency matrix, but the first one, are at most lambda in their absolute values. Form a random subset R of V by putting every vertex v in V into R independently with probability p.

The study of phase transitions in random graphs was initiated by Erdos and Renyi in 1960. They proved among other things that a uniform random graph undergoes a drastic change in the size and structure of the largest component, caused by altering a critical edge density. Since the seminal work of Erdos and Renyi, various random graph models have been introduced and studied over the last five decades. In this talk I will discuss some recent results in phase transitions in random hypergraphs. (Joint work with Oliver Cooley and Christoph Koch.)

We discuss a nonlocal Fokker-Planck equation that describes energy minimisation in a double well-potential and is driven by a time-dependent constraint. Via formal asymptotic analysis we identify different small parameter regimes that correspond to hysteretic and non-hysteretic phase transitions respectively. For the fast reaction regime that is related to Kramers-type phase transitions we also indicate how can rigorously derive a rate-independent evolution equation in a small parameter limit.

This is joint work with Michael Herrmann and Juan Velazquez.

Random matrices are measured in many areas of engineering, social science, and natural science. When the rows of the matrix are random samples of a vector of dependent variables the sample correlations between the columns of the matrix specify a correlation graph that can be used to explore the dependency structure.