On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree

Statistical mechanics seeks to explain the macroscopic behavior of matter on the basis of its microscopic structure. This includes the analysis of simplified mathematical models. A phase diagram of a model describes a morphology of phases, stability of phases, transitions from ne phase to another a...

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Main Author:	Ganikhodjaev, Nasir
Format:	Conference or Workshop Item
Language:	English
Published:	2009
Subjects:	QA Mathematics
Online Access:	http://irep.iium.edu.my/13885/ http://irep.iium.edu.my/13885/1/abstract_NUUZ%5B2%5D.pdf

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recordtype	eprints
spelling	iium-138852012-02-07T05:36:53Z http://irep.iium.edu.my/13885/ On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree Ganikhodjaev, Nasir QA Mathematics Statistical mechanics seeks to explain the macroscopic behavior of matter on the basis of its microscopic structure. This includes the analysis of simplified mathematical models. A phase diagram of a model describes a morphology of phases, stability of phases, transitions from ne phase to another and corresponding transitions line. A Potts model just as an Ising model on a Cayley tree with competing interactions [1],[2], has recently been studied extensively because of the appearance of nontrivial magnetic orderings. The Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculation of various quantities possible. For many problems the solution on a tree is much simpler than on a regular lattice and is equivalent to the standard Bethe-Peierls theory. On the Cayley tree one can consider two type of second ( triple) neighbours: prolonged and one(two)level. For given lattice model on a Cayley tree we produce recursive relations obtained following the lines of Inawashiro et al [3]. These recursive relations provide us the numerically exact phase diagram of the model. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous or abrupt, respectively characterizing second- or first- order phase transitions. We present a few typical attractors and show that for some lattice models on a Cayley tree with competing interactions their phase diagram contain modulated phase [1],[2] if and only if the Hamiltonian of corresponding lattice model contains interactions of prolonged neighbours. 2009 Conference or Workshop Item NonPeerReviewed application/pdf en http://irep.iium.edu.my/13885/1/abstract_NUUZ%5B2%5D.pdf Ganikhodjaev, Nasir (2009) On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree. In: Control and Optimization of Dynamical Systems-CODS-2009, 27th September-1st October 2009, Tashkent, Uzbekistan. (Unpublished)
repository_type	Digital Repository
institution_category	Local University
institution	International Islamic University Malaysia
building	IIUM Repository
collection	Online Access
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Ganikhodjaev, Nasir On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree
description	Statistical mechanics seeks to explain the macroscopic behavior of matter on the basis of its microscopic structure. This includes the analysis of simplified mathematical models. A phase diagram of a model describes a morphology of phases, stability of phases, transitions from ne phase to another and corresponding transitions line. A Potts model just as an Ising model on a Cayley tree with competing interactions [1],[2], has recently been studied extensively because of the appearance of nontrivial magnetic orderings. The Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculation of various quantities possible. For many problems the solution on a tree is much simpler than on a regular lattice and is equivalent to the standard Bethe-Peierls theory. On the Cayley tree one can consider two type of second ( triple) neighbours: prolonged and one(two)level. For given lattice model on a Cayley tree we produce recursive relations obtained following the lines of Inawashiro et al [3]. These recursive relations provide us the numerically exact phase diagram of the model. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous or abrupt, respectively characterizing second- or first- order phase transitions. We present a few typical attractors and show that for some lattice models on a Cayley tree with competing interactions their phase diagram contain modulated phase [1],[2] if and only if the Hamiltonian of corresponding lattice model contains interactions of prolonged neighbours.
format	Conference or Workshop Item
author	Ganikhodjaev, Nasir
author_facet	Ganikhodjaev, Nasir
author_sort	Ganikhodjaev, Nasir
title	On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree
title_short	On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree
title_full	On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree
title_fullStr	On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree
title_full_unstemmed	On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree
title_sort	on nonlinear dynamic systems arising in lattice models of statistical mechanics on a cayley tree
publishDate	2009
url	http://irep.iium.edu.my/13885/ http://irep.iium.edu.my/13885/1/abstract_NUUZ%5B2%5D.pdf
first_indexed	2023-09-18T20:23:03Z
last_indexed	2023-09-18T20:23:03Z
_version_	1777408234580082688

On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree

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