One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension. (English) Zbl 1502.82004
Summary: In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains a Modica-Mortola type local term and a nonlocal generalized antiferromagnetic term in competition. The competition between the two terms results in a frustrated system which is believed to lead to the emergence of a wide variety of patterns. The sharp interface limit of our model is considered in [M. Goldman and E. Runa, Calc. Var. Partial Differ. Equ. 58, No. 3, Paper No. 103, 26 p. (2019; Zbl 1415.49033)] and in [S. Daneri and E. Runa, Arch. Ration. Mech. Anal. 231, No. 1, 519–589 (2019; Zbl 1410.82005)]. In the discrete setting it has been previously studied in
[A. Giuliani et al., “Checkerboards, stripes, and corner energies in spin models with competing interactions” Phys. Rev. B 84, No. 6, Article ID 064205, 10 p. (2011; doi:10.1103/PhysRevB.84.064205);
Commun. Math. Phys. 331, No. 1, 333–350 (2014; Zbl 1302.82018);
Commun. Math. Phys. 347, No. 3, 983–1007 (2016; Zbl 1351.82019)]. The model contains two parameters: \( \tau\) and \(\varepsilon\). The parameter \(\tau\) represents the relative strength of the local term with respect to the nonlocal one, while the parameter \(\varepsilon\) describes the transition scale in the Modica-Mortola type term. If \(\tau < 0\) one has that the only minimizers of the functional are constant functions with values in \(\{0, 1\}\). In any dimension \(d \geq 1\) for small but positive \(\tau\) and \(\varepsilon\), it is conjectured that the minimizers are non-constant one-dimensional periodic functions. In this paper we are able to prove such a characterization of the minimizers, thus showing also the symmetry breaking in any dimension \(d > 1\).
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B24 | Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 35R09 | Integro-partial differential equations |
| 35B36 | Pattern formations in context of PDEs |
| 35Q82 | PDEs in connection with statistical mechanics |
References:
| [1] | Bomont, J.; Bretonnet, J.; Costa, D.; Hansen, J., Communication: thermodynamic signatures of cluster formation in fluids with competing interactions, J. Chem. Phys., 137, Article 011101 pp. (2012) |
| [2] | Carrillo, J. A.; Choi, Y. P.; Hauray, M., The derivation of swarming models: mean-field limit and Wasserstein distances, (Collective Dynamics from Bacteria to Crowds. Collective Dynamics from Bacteria to Crowds, CISM International Centre for Mechanical Sciences, vol. 553 (2014), Springer: Springer Vienna) |
| [3] | Carrillo, J. A.; Craig, K.; Patacchini, F. S., A blob method for diffusion, Calc. Var. Partial Differ. Equ., 58, 53 (2019) · Zbl 1442.35324 |
| [4] | Carrillo, J. A.; Di Francesco, M.; Figalli, A.; Laurent, T.; Slepčev, D., Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156, 2, 229-271 (2011) · Zbl 1215.35045 |
| [5] | Chacko, B.; Chalmers, C.; Archer, A. J., Two-dimensional colloidal fluids exhibiting pattern formation, J. Chem. Phys., 143, Article 244904 pp. (2015) |
| [6] | Chen, X.; Oshita, Y., An application of the modular function in nonlocal variational problems, Arch. Ration. Mech. Anal., 186, 1, 109-132 (2007) · Zbl 1147.74024 |
| [7] | Craig, K., Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions, Proc. Lond. Math. Soc., 114, 60-102 (2017) · Zbl 1375.35552 |
| [8] | Craig, K.; Topaloglu, I., Aggregation-diffusion to constrained interaction: minimizers and gradient flows in the slow diffusion limit, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 37, 2 (2019) |
| [9] | Daneri, S.; Radici, E.; Runa, E., Deterministic particle approximation of aggregation-diffusion equations on unbounded domains, J. Differ. Equ., 312, 474-517 (2022) · Zbl 1481.35247 |
| [10] | Daneri, S.; Runa, E., Exact periodic stripes for minimizers of a local/non-local interaction functional in general dimension, Arch. Ration. Mech. Anal., 231, 1, 519-589 (2019) · Zbl 1410.82005 |
| [11] | Daneri, S.; Runa, E., Pattern formation for a local/nonlocal interaction functional arising in colloidal systems, SIAM J. Math. Anal., 52, 3, 2531-2560 (2020) · Zbl 1456.82210 |
| [12] | Daneri, S.; Runa, E., On the symmetry breaking and structure of the minimizers for a family of local/nonlocal interaction functionals, Rend. Semin. Mat. Univ. Politec. Torino, 77, 2, 33-48 (2019) · Zbl 1440.49049 |
| [13] | Daneri, S.; Runa, E., Exact periodic stripes for a local/nonlocal minimization problem with volume constraint (2021), Preprint |
| [14] | Daneri, S.; Runa, E., Periodic striped configurations in the large volume limit (2021), Preprint |
| [15] | Giuliani, A.; Lebowitz, J. L.; Lieb, E. H., Periodic minimizers in 1D local mean field theory, Commun. Math. Phys., 286, 163-177 (2009) · Zbl 1173.82008 |
| [16] | Giuliani, A.; Lebowitz, J. L.; Lieb, E. H., Checkerboards, stripes, and corner energies in spin models with competing interactions, Phys. Rev. B, 84, Article 064205 pp. (2011) |
| [17] | Giuliani, A.; Lieb, E. H.; Seiringer, R., Formation of stripes and slabs near the ferromagnetic transition, Commun. Math. Phys., 331, 1, 333-350 (2014) · Zbl 1302.82018 |
| [18] | Giuliani, A.; Müller, S., Striped periodic minimizers of a two-dimensional model for martensitic phase transitions, Commun. Math. Phys., 309, 2, 313-339 (2012) · Zbl 1448.74082 |
| [19] | Giuliani, A.; Seiringer, R., Periodic striped ground states in Ising models with competing interactions, Commun. Math. Phys., 1-25 (2016) |
| [20] | Godfrin, P.; Castañeda-Priego, R.; Liu, Y.; Wagner, N., Intermediate range order and structure in colloidal dispersions with competing interactions, J. Chem. Phys., 139, Article 154904 pp. (2013) |
| [21] | Goldman, D.; Muratov, C. B.; Serfaty, S., The Γ-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density, Arch. Ration. Mech. Anal., 210, 581-613 (2013) · Zbl 1296.82018 |
| [22] | Goldman, D.; Muratov, C. B.; Serfaty, S., The Γ-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement via the renormalized energy, Arch. Ration. Mech. Anal., 212, 2, 445-501 (2014) · Zbl 1305.35134 |
| [23] | Goldman, M.; Runa, E., On the optimality of stripes in a variational model with non-local interactions, Calc. Var., 58, 3, 103 (2019) · Zbl 1415.49033 |
| [24] | Imperio, A.; Reatto, L., Microphase separation in two-dimensional systems with competing interactions, J. Chem. Phys., 124, Article 164712 pp. (2006) |
| [25] | Kerschbaum, A., Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than \(d + 2 (2021)\), Preprint |
| [26] | Kuperfman, R.; Solomon, J. P., A Riemannian approach to reduced plate, shell, and rod theories, J. Funct. Anal., 266, 5, 2989-3039 (2014) · Zbl 1305.74022 |
| [27] | Lewicka, M.; Mora, M. G.; Pakzad, M. R., Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 9, 2, 253-295 (2010) · Zbl 1425.74298 |
| [28] | Modica, L.; Mortola, S., Un esempio di Γ-convergenza, Boll. Unione Mat. Ital., B, 14, 285-299 (1977) · Zbl 0356.49008 |
| [29] | Morini, M.; Sternberg, P., Cascade of minimizers for a nonlocal isoperimetric problem in thin domains, SIAM J. Math. Anal., 46, 3, 2033-2051 (2014) · Zbl 1301.49121 |
| [30] | Ohta, T.; Kawasaki, K., Equilibrium morphologies of block copolymer melts, Macromolecules, 19 (1986) |
| [31] | Olbermann, H.; Runa, E., Interpenetration of matter in plate theories obtained as Γ-limits, ESAIM Control Optim. Calc. Var., 23, 1, 119-136 (2017) · Zbl 1364.49015 |
| [32] | Peletier, M. A.; Veneroni, M., Stripe patterns in a model for block copolymers, Math. Models Methods Appl. Sci., 20, 6, 843-907 (2010) · Zbl 1203.49018 |
| [33] | Seul, M.; Andelman, D., Domain shapes and patterns: the phenomenology of modulated phases, Science, 267, 5197, 476-483 (1995) |
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