dorsal/arxiv
View SchemaFarey Tree and the Frenkel-Kontorova Model
| Authors | Hsien-chung Kao, Shih-Chang Lee, Wen-Jer Tzeng |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9604001 |
| URL | https://arxiv.org/abs/solv-int/9604001 |
| DOI | 10.1103/PhysRevE.55.2628 |
Abstract
We solved the Frenkel-Kontorova model with the potential $V(u)= -\frac{1}{2} |\lambda|(u-{\rm Int}[u]-\frac{1}{2})^2$ exactly. For given $|\lambda|$, there exists a positive integer $q_c$ such that for almost all values of the tensile force $\sigma$, the winding number $\omega$ of the ground state configuration is a rational number in the $q_c$-th level Farey tree. For fixed $\omega=p/q$, there is a critical $\lambda_c$ when a first order phase transition occurs. This phase transition can be understood as the dissociation of a large molecule into two smaller ones in a manner dictated by the Farey tree. A kind of ``commensurate-incommensurate'' transition occurs at critical values of $\sigma$ when two sizes of molecules co-exist. ``Soliton'' in the usual sense does not exist but induces a transformation of one size of molecules into the other.
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"abstract": "We solved the Frenkel-Kontorova model with the potential $V(u)= -\\frac{1}{2}\n|\\lambda|(u-{\\rm Int}[u]-\\frac{1}{2})^2$ exactly. For given $|\\lambda|$, there\nexists a positive integer $q_c$ such that for almost all values of the tensile\nforce $\\sigma$, the winding number $\\omega$ of the ground state configuration\nis a rational number in the $q_c$-th level Farey tree. For fixed $\\omega=p/q$,\nthere is a critical $\\lambda_c$ when a first order phase transition occurs.\nThis phase transition can be understood as the dissociation of a large molecule\ninto two smaller ones in a manner dictated by the Farey tree. A kind of\n``commensurate-incommensurate\u0027\u0027 transition occurs at critical values of\n$\\sigma$ when two sizes of molecules co-exist. ``Soliton\u0027\u0027 in the usual sense\ndoes not exist but induces a transformation of one size of molecules into the\nother.",
"arxiv_id": "solv-int/9604001",
"authors": [
"Hsien-chung Kao",
"Shih-Chang Lee",
"Wen-Jer Tzeng"
],
"categories": [
"solv-int",
"cond-mat",
"nlin.SI"
],
"doi": "10.1103/PhysRevE.55.2628",
"title": "Farey Tree and the Frenkel-Kontorova Model",
"url": "https://arxiv.org/abs/solv-int/9604001"
},
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