Crystalline motion of discrete interfaces in the Blume-Emery-Griffiths model
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| Format: | Preprint |
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2025
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| author | Cicalese, Marco Fusco, Giuliana Savaré, Giovanni |
| author_facet | Cicalese, Marco Fusco, Giuliana Savaré, Giovanni |
| contents | We study the discrete-to-continuum evolution of a lattice system consisting of two immiscible phases labelled by -1 and +1 in presence of a surfactant phase labelled by 0. The system's energy is described by the classical Blume-Emery-Griffith model on the lattice epsilon Z^2, and its continuum evolution is obtained as epsilon tends to zero through a minimizing-movements scheme with a time step proportional to epsilon. The dissipation functional we choose contains two contributions: a standard Almgren-Taylor-Wang type term penalizing the distance between successive configurations of the +1 phase, and a term penalizing the variation of the surfactant mass and modeling surfactant evaporation. The latter term depends on a scaling parameter gamma > 0, which determines whether the surfactant mass is conserved at each time step. We focus on the case in which the initial configuration consists of a single crystal of phase 1 completely wetted by the surfactant. For gamma > 2 the surfactant can lose mass and the evolution reduces to the crystalline mean curvature flow of an Ising-type model, while for gamma < 2 the conservation of the surfactant mass leads to a more complex evolution characterized by stronger non-uniqueness and partial pinning. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_18403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Crystalline motion of discrete interfaces in the Blume-Emery-Griffiths model Cicalese, Marco Fusco, Giuliana Savaré, Giovanni Analysis of PDEs 53E10, 49J45, 74A50, 82B24, 82C24 We study the discrete-to-continuum evolution of a lattice system consisting of two immiscible phases labelled by -1 and +1 in presence of a surfactant phase labelled by 0. The system's energy is described by the classical Blume-Emery-Griffith model on the lattice epsilon Z^2, and its continuum evolution is obtained as epsilon tends to zero through a minimizing-movements scheme with a time step proportional to epsilon. The dissipation functional we choose contains two contributions: a standard Almgren-Taylor-Wang type term penalizing the distance between successive configurations of the +1 phase, and a term penalizing the variation of the surfactant mass and modeling surfactant evaporation. The latter term depends on a scaling parameter gamma > 0, which determines whether the surfactant mass is conserved at each time step. We focus on the case in which the initial configuration consists of a single crystal of phase 1 completely wetted by the surfactant. For gamma > 2 the surfactant can lose mass and the evolution reduces to the crystalline mean curvature flow of an Ising-type model, while for gamma < 2 the conservation of the surfactant mass leads to a more complex evolution characterized by stronger non-uniqueness and partial pinning. |
| title | Crystalline motion of discrete interfaces in the Blume-Emery-Griffiths model |
| topic | Analysis of PDEs 53E10, 49J45, 74A50, 82B24, 82C24 |
| url | https://arxiv.org/abs/2510.18403 |