dorsal/arxiv
View SchemaThree basic issues concerning interface dynamics in nonequilibrium pattern formation
| Authors | Wim van Saarloos |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9801002 |
| URL | https://arxiv.org/abs/patt-sol/9801002 |
Abstract
These are lecture notes of a course given at the 9th International Summer School on Fundamental Problems in Statistical Mechanics, held in Altenberg, Germany, in August 1997. In these notes, we discuss at an elementary level three themes concerning interface dynamics that play a role in pattern forming systems: (i) We briefly review three examples of systems in which the normal growth velocity is proportional to the gradient of a bulk field which itself obeys a Laplace or diffusion type of equation (solidification, viscous fingers and streamers), and then discuss why the Mullins-Sekerka instability is common to all such gradient systems. (ii) Secondly, we discuss how underlying an effective interface description of systems with smooth fronts or transition zones, is the assumption that the relaxation time of the appropriate order parameter field(s) in the front region is much smaller than the time scale of the evolution of interfacial patterns. Using standard arguments we illustrate that this is generally so for fronts that separate two (meta)stable phases: in such cases, the relaxation is typically exponential, and the relaxation time in the usual models goes to zero in the limit in which the front width vanishes. (iii) We finally summarize recent results that show that so-called ``pulled'' or ``linear marginal stability'' fronts which propagate into unstable states have a very slow universal power law relaxation. This slow relaxation makes the usual ``moving boundary'' or ``effective interface'' approximation for problems with thin fronts, like streamers, impossible.
{
"annotation_id": "b60ed87e-fc28-4f0c-8ecb-f979b2034f68",
"date_created": "2026-03-02T18:00:29.291000Z",
"date_modified": "2026-03-02T18:00:29.291000Z",
"file_hash": "6afb045bfb99a77739d74df0a3f2ded4db2a11b78eaeddc330fb9637766d8128",
"private": false,
"record": {
"abstract": "These are lecture notes of a course given at the 9th International Summer\nSchool on Fundamental Problems in Statistical Mechanics, held in Altenberg,\nGermany, in August 1997. In these notes, we discuss at an elementary level\nthree themes concerning interface dynamics that play a role in pattern forming\nsystems: (i) We briefly review three examples of systems in which the normal\ngrowth velocity is proportional to the gradient of a bulk field which itself\nobeys a Laplace or diffusion type of equation (solidification, viscous fingers\nand streamers), and then discuss why the Mullins-Sekerka instability is common\nto all such gradient systems. (ii) Secondly, we discuss how underlying an\neffective interface description of systems with smooth fronts or transition\nzones, is the assumption that the relaxation time of the appropriate order\nparameter field(s) in the front region is much smaller than the time scale of\nthe evolution of interfacial patterns. Using standard arguments we illustrate\nthat this is generally so for fronts that separate two (meta)stable phases: in\nsuch cases, the relaxation is typically exponential, and the relaxation time in\nthe usual models goes to zero in the limit in which the front width vanishes.\n(iii) We finally summarize recent results that show that so-called ``pulled\u0027\u0027\nor ``linear marginal stability\u0027\u0027 fronts which propagate into unstable states\nhave a very slow universal power law relaxation. This slow relaxation makes the\nusual ``moving boundary\u0027\u0027 or ``effective interface\u0027\u0027 approximation for problems\nwith thin fronts, like streamers, impossible.",
"arxiv_id": "patt-sol/9801002",
"authors": [
"Wim van Saarloos"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Three basic issues concerning interface dynamics in nonequilibrium pattern formation",
"url": "https://arxiv.org/abs/patt-sol/9801002"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "ed5ab667-ed07-4c4f-9498-81f64335e949",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}