# Ordering and patterns

## Overview

Pattern formation is related to a system which is kept out-of-equilibrium and a heat, mass, momentum macroscopic current is imposed via boundary conditions. Because of that the homogeneous state of the system (typical of its equilibrium condition) is made unstable. Instead, Phase ordering appears when the initial and final state of a relaxing system are separated by a critical point.

Research ranges over different lines: from extended interactions in phase separtion processes to the relevance of conservation laws in strongly out-of-equilibrium systems, from noise tto feedback effects in pattern forming systems.

## People involved

## Recent publications

Gabriele Gotti, **Stefano Iubini**, **Paolo ****Politi**:

Phys. Rev. E 103, 052133 (2021) [PDF]

We study condensation phenomena which are accompanied by a process of localization in the real space: a finite fraction of the energy of the whole system is concentrated on a few sites. The correct order parameter of this transition is the participation ratio Y_2, basically the second moment of the energy distribution divided by the system size N. Localization corresponds to a nonvanishing Y_2 for diverging N. The figure shows that in some models Y_2(N) is not monotonous and displays a minimum, which is due to a maximum in the number K of sites hosting the condensate. This behavior has striking consequences in terms of real condensates and in terms of negative temperatures.

F. Di Patti, L. Lavacchi, R. Arbel-Goren, L. Schein-Lubomirsky, **D. Fanelli**, J. Stavans:

PLoS Biology 16(5): e2004877 (2018)

A mathematical model is proposed that explains the formation mechanism of the cellular pattern in *Anabaena*, a bacterium that carries out photosynthesis. Anabaena is a one dimensional filament, made of adjacent cells. In normal environmental conditions (in the presence of nitrogen) the bacterium is mostly formed by vegetative cells. In the absence of nitrogen, however, is able to differentiate some vegetative cells in others specialized in nitrogen fixation. This differentiation occurs following a certain regularity: every 10â€“15 vegetative cells, a heterocyst one is formed. This ingenious biological process guarantees the necessary nitrogen supply to all the cells that make up the *Anabaena *filament. The process of pattern formation is triggered by endogenous noise: microscopic disturbances yields therefore regular macroscopic motifs.

### Spatio-temporal phenomena in complex systems with time delays

Serhiy Yanchuk and **Giovanni Giacomelli**

*J. Phys. A: Math. Theor.* 50 103001 (2017) [PDF]

Propagation delays in the loop arms of feedback cannot always be neglected in the study of dynamical systems. Complex phenomena have shown to occur in such a case, when the delay is comparable with the internal timescale of the solitary system. A regime of particular interest is represented by the so-called long-delayed systems, where the delay time is much longer that any other timescale. The complicated temporal series resulting from the dynamics can be organized by means of a specific method, named spatio-temporal reconstruction. It allows to map the data on an equivalent pattern which can be thus analyzed in the framework of the spatially extended systems. Several phenomena in different systems have been studied in such an approach, ranging from phase and defect turbulence, chaos, bistable and excitable front dynamics to coarsening and nucleation to directed percolation.