1DSpin-glass_tpel.DVI


Mathematical Problems of Computer Science 35, 86{98, 2011.

Statistical P r oper ties of I deal E nsemble of Disor der ed

1D Ster ic Spin-Chains

A s h o t S . Ge vo r kya n , H a ko b G. A b a jya n a n d H a yk S . S u kia s ya n

Institute for Informatics and Automation Problems of NAS of RA

e-mail g ashot@sci.am, habajyan@ipia.sci.am, haikarin@netsys.am

Abstract

The statistical properties of ensemble of disordered 1D steric spin-chains (SSC) of
various length are investigated. Using 1D spin-glass type classical Hamiltonian, the
recurrent trigonometrical equations for stationary points and corresponding conditions
for the construction of stable 1D SSCs are found. The ideal ensemble of spin-chains
is analyzed and the latent interconnections between random angles and interaction
constants for each set of three nearest-neighboring spins are found. It is analytically
proved and by numerical calculation is shown that the interaction constant satis¯es
Le¶vy's alpha-stable distribution law. Energy distribution in ensemble is calculated de-
pending on di®erent conditions of possible polarization of spin-chains. It is speci¯cally
shown that the dimensional e®ects in the form of set of local maximums in the energy
distribution arise when the number of spin-chains M << N 2x (where Nx is the number
of spins in a chain) while in the case when M / N 2x energy distribution has one global
maximum and the ensemble of spin-chains satis¯es Birkho®'s ergodic theorem. E®ec-
tive algorithm for parallel simulation of problem which includes calculation of di®erent
statistic parameters of 1D SSCs ensemble is elaborated.

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