The present paper develops the approach to the famous problem presented by L. Onsager (1944) and its further investigation proposed in a recent work by Z.-D. Zhang (2007). The above works give quaternion-based two- and three-dimensional (quantum) models of order-disorder transition and simple orthorhombic Ising lattices (1925). The general methods applied by Zhang refer to opening knots by a rotation in a higher dimensional space, introduction of weight factor (his Conjecture 1 and 2) and important commutators. The main objective of the present paper is to reformulate the algebraic part of the theory in terms of the quaternionic sequence of Jordan algebras and to look at some of the geometrical aspects of simple orthorhombic Ising-Onsager-Zhang lattices. The present authors discuss the relationship with Bethe-type fractals, Kikuchi-type fractals, and fractals of the algebraic structure and, moreover, the duality for fractal sets and lattice models on fractal sets. A simple description in terms of fractals corresponding to algebraic structure involving the quaternionic sequence (H(q)(4)) of P. Jordan's algebras appears to be possible. Physically we obtain models of (H(q)(4)) for q = 5 . 2(2) = 20 for the melting, q = 9 . 2(6) = 576 for binary alloys, and q = 13 . 2(10) = 13 312 for ternary alloys.

An Approach to Models of Order-Disorder and Ising Lattices / J., Lawrynowicz; Marchiafava, Stefano; A., Niemczynowicz. - In: ADVANCES IN APPLIED CLIFFORD ALGEBRAS. - ISSN 0188-7009. - STAMPA. - 20:3-4(2010), pp. 733-743. ((Intervento presentato al convegno 8th International Conference on Clifford Algebras and their Applications in Mathematical Physics tenutosi a Campinas, BRAZIL nel MAR 26-30, 2008 [10.1007/s00006-010-0219-7].

An Approach to Models of Order-Disorder and Ising Lattices

MARCHIAFAVA, Stefano;
2010

Abstract

The present paper develops the approach to the famous problem presented by L. Onsager (1944) and its further investigation proposed in a recent work by Z.-D. Zhang (2007). The above works give quaternion-based two- and three-dimensional (quantum) models of order-disorder transition and simple orthorhombic Ising lattices (1925). The general methods applied by Zhang refer to opening knots by a rotation in a higher dimensional space, introduction of weight factor (his Conjecture 1 and 2) and important commutators. The main objective of the present paper is to reformulate the algebraic part of the theory in terms of the quaternionic sequence of Jordan algebras and to look at some of the geometrical aspects of simple orthorhombic Ising-Onsager-Zhang lattices. The present authors discuss the relationship with Bethe-type fractals, Kikuchi-type fractals, and fractals of the algebraic structure and, moreover, the duality for fractal sets and lattice models on fractal sets. A simple description in terms of fractals corresponding to algebraic structure involving the quaternionic sequence (H(q)(4)) of P. Jordan's algebras appears to be possible. Physically we obtain models of (H(q)(4)) for q = 5 . 2(2) = 20 for the melting, q = 9 . 2(6) = 576 for binary alloys, and q = 13 . 2(10) = 13 312 for ternary alloys.
clifford algebra; crystal; disordered system; ising lattice; jordan algebra; quaternion
01 Pubblicazione su rivista::01a Articolo in rivista
An Approach to Models of Order-Disorder and Ising Lattices / J., Lawrynowicz; Marchiafava, Stefano; A., Niemczynowicz. - In: ADVANCES IN APPLIED CLIFFORD ALGEBRAS. - ISSN 0188-7009. - STAMPA. - 20:3-4(2010), pp. 733-743. ((Intervento presentato al convegno 8th International Conference on Clifford Algebras and their Applications in Mathematical Physics tenutosi a Campinas, BRAZIL nel MAR 26-30, 2008 [10.1007/s00006-010-0219-7].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/356774
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