Random geometric systems

While in one-dimensional equilibrium systems with short-range particle interactions, transitions between phases of different particle densities cannot occur, this is different in nonequilibrium systems. Already a simple driven lattice gas, the asymmetric simple exclusion process with open boundaries, shows such transitions.
In the first part of the talk, I discuss similar phase transitions in a slightly more complex driven lattice gas with repulsive nearest-neighbor interactions. I will describe how to identify all possible nonequilibrium phases based on extremal current principles and bulk-adapted couplings to particle reservoirs [1], and show that long-range decays of density profiles and long-range anticorrelations between current fluctuations lead to weak pinning and subdiffusive dynamics of domain walls separating extremal current phases [2].
In the second part of the talk, I consider driven Brownian dynamics of hard spheres in periodic potentials. I will demonstrate that nonequilibrium phase transitions can be predicted based on the same principles as in the driven lattice gas, with the particle size entering the problem as a further relevant length scale [3]. I then discuss particle dynamics mediated by solitary waves of particle clusters [4] that can occur at high particle densities. We predicted these waves recently [5] and their occurrence could be confirmed shortly after in experiments [6]. Finally, I will present very recent results on phase-locked many-particle currents under time-periodic driving due to synchronized soliton dynamics.

[1] M. Dierl, M. Einax, P. Maass, Phys. Rev. E 87, 062126 (2013).
[2] S. Schweers, D. F. Locher, G. M. Schütz, P. Maass, Phys. Rev. Lett. 132, 167101 (2024).
[3] D. Lips, A. Ryabov, P. Maass, Phys. Rev. Lett. 121, 160601 (2018).
[4] A. P. Antonov, A. Ryabov, P. Maass, Chaos, Solitons & Fractals 132, 115079 (2024).
[5] A. P. Antonov, A. Ryabov, P. Maass, Phys. Rev. Lett. 129, 080601 (2022).
[6] E. Cereceda-López, A. P. Antonov, A. Ryabov, P. Maass, P. Tierno, Nat. Commun. 14, 6448 (2023).

The standard closed convex hull of a set is defined as the intersection of all images, under the action of a group of rigid motions, of a half-space containing the given set. This talk concerns a generalisation of this classical notion, which is obtained from the above construction by replacing a half-space with some other closed convex subset of the Euclidean space, and a group of rigid motions by a subset of the group of invertible affine transformations. The main focus is on the analysis of generalised convex hulls of random samples.
Joint work with Alexandr Marynych (Kiev) and Zakhar Kabluchko (Müster).