Periodic Reporting for period 1 - ENDRISP (Energy Driven Pattern Formation: Continuous Symmetry Breaking and Positive Temperature)
Reporting period: 2022-10-01 to 2024-10-31
Unveiling the mathematical mechanisms behind pattern formation is a central and long-standing open problem due to both its mathematical complexity and its wide range of applications. From the physical point of view, pattern formation at micro- and mesoscopic scale is universally believed to stem from the competition between short-range attractive forces (SA), favouring pure phases, and long-range repulsive forces (LR), favouring oscillations.
When considering the physical systems at zero temperature the models used are variational models, i.e. the physical states are represented by minimizers of a free-energy functional. At positive temperature the physical states are described by Gibbs measures, which are probability measures on the state space. At zero temperature, the problem of detecting pattern formation can be broadly resumed in the following energy-driven pattern formation conjecture: the emergence of periodic and regular structures is due solely to the minimization of the total energy in presence of symmetric competing SALR interactions. In the positive temperature setting the conjecture consists in proving that the corresponding Gibbs measures “concentrate” on a certain class of periodic and regular structures, and in quantifying this concentration.
A crucial aspect, both at zero and positive temperature, is that in dimension d ≥ 2 the physical states retain less symmetries than the interactions involved in the models. This phenomenon, known as symmetry breaking, coupled with nonlocality makes the problem mathematically extremely challenging and still largely not accessible with the current analytical tools.
The goal of ENDRISP is to gain deep insight into the mathematical mechanisms behind the phenomenon of energy-driven pattern formation, attacking long-standing conjectures while developing robust and versatile analytical techniques of groundbreaking nature, eventually fruitful in other interesting geometric and analytical settings.
ENDRISP will focus on the following foremost questions in the field:
* Explaining continuous symmetry breaking mechanisms for isotropic functionals;
* Show pattern formation and symmetry breaking in the positive temperature setting.
Among the wide variety of patterns occurring under different density and mutual strength conditions between the competing forces, the focus of ENDRISP will be on the spontaneous formation of periodic one-dimensional structures in general dimension (usually in the form of striped/lamellar structures), as they are generally the first type of patterns to emerge from uniform phases when symmetry breaking occurs. Notice that they appear in regimes in which the SA and the LR term are of the same order, thus making the balance between the competing terms delicate to detect.
The problem naturally falls in the interesection of several branches in mathematics. Thus we need to use/develop new techniques in Geometric Measure Theory, Calculus of Variations, Partial Differential Equations, Probablity and Statistical Mechanics.
When considering the physical systems at zero temperature the models used are variational models, i.e. the physical states are represented by minimizers of a free-energy functional. At positive temperature the physical states are described by Gibbs measures, which are probability measures on the state space. At zero temperature, the problem of detecting pattern formation can be broadly resumed in the following energy-driven pattern formation conjecture: the emergence of periodic and regular structures is due solely to the minimization of the total energy in presence of symmetric competing SALR interactions. In the positive temperature setting the conjecture consists in proving that the corresponding Gibbs measures “concentrate” on a certain class of periodic and regular structures, and in quantifying this concentration.
A crucial aspect, both at zero and positive temperature, is that in dimension d ≥ 2 the physical states retain less symmetries than the interactions involved in the models. This phenomenon, known as symmetry breaking, coupled with nonlocality makes the problem mathematically extremely challenging and still largely not accessible with the current analytical tools.
The goal of ENDRISP is to gain deep insight into the mathematical mechanisms behind the phenomenon of energy-driven pattern formation, attacking long-standing conjectures while developing robust and versatile analytical techniques of groundbreaking nature, eventually fruitful in other interesting geometric and analytical settings.
ENDRISP will focus on the following foremost questions in the field:
* Explaining continuous symmetry breaking mechanisms for isotropic functionals;
* Show pattern formation and symmetry breaking in the positive temperature setting.
Among the wide variety of patterns occurring under different density and mutual strength conditions between the competing forces, the focus of ENDRISP will be on the spontaneous formation of periodic one-dimensional structures in general dimension (usually in the form of striped/lamellar structures), as they are generally the first type of patterns to emerge from uniform phases when symmetry breaking occurs. Notice that they appear in regimes in which the SA and the LR term are of the same order, thus making the balance between the competing terms delicate to detect.
The problem naturally falls in the interesection of several branches in mathematics. Thus we need to use/develop new techniques in Geometric Measure Theory, Calculus of Variations, Partial Differential Equations, Probablity and Statistical Mechanics.
In [DR24], the PI together with S. Daneri introduced a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition.
We consider a general class of nonlocal variational problems in dimension $d\geq 1$, in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (\eg stripes or lamellae).
The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory.
Among others, we identify a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries.
The power of decay of the considered kernels at infinity is $p\geq d+3$, and it is related to pattern formation in synthetic antiferromagnets.
[DR24]: Sara Daneri and Eris Runa. “A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactions”. In: preprint arXiv:2406.13773 (2024).
We consider a general class of nonlocal variational problems in dimension $d\geq 1$, in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (\eg stripes or lamellae).
The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory.
Among others, we identify a nonlocal curvature-type quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries.
The power of decay of the considered kernels at infinity is $p\geq d+3$, and it is related to pattern formation in synthetic antiferromagnets.
[DR24]: Sara Daneri and Eris Runa. “A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactions”. In: preprint arXiv:2406.13773 (2024).
Showing Pattern formation and symmetry breaking on an isotropic model is considered to be a very challenging task from the mathematical community.
In particular to archieve the task, we introduce several new geometric quantities like the nonlocal curvature term and introduce several new mathematical techniques which we believe to be of independent interest in the community and will open to the study of patterns in other models.
In particular to archieve the task, we introduce several new geometric quantities like the nonlocal curvature term and introduce several new mathematical techniques which we believe to be of independent interest in the community and will open to the study of patterns in other models.