# Statistical Properties of Ideal Ensemble of Disordered 1D Steric Spin-Chains

## 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 satisfies Lévy's alpha-stable distribution law. Energy distribution in ensemble is calculated depending on different conditions of possible polarization of spin-chains. It is specifically shown that the dimensional effects in the form of set of local maximums in the energy distribution arise when the number of spin-chains M << N2 x (where Nx is the number of spins in a chain) while in the case when M / N2 x energy distribution has one global maximum and the ensemble of spin-chains satisfies Birkhoff's ergodic theorem. Effective algorithm for parallel simulation of problem which includes calculation of different statistic parameters of 1D SSCs ensemble is elaborated.

## References

K. Binder and A.P. Young, “Spin glasses: Experimental facts, theoretical concepts, and open questions", Rev. Mod. Physics, vol. 58, no. 4, pp. 801-976, 1986.

M.Mézard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987.

A.P. Young (ed.), Spin Glasses and Random Fields, World Scientific, Singapore, 1998.

R. Fisch and A.B. Harris, “Spin-glass model in continuous dimensionality", Phys. Rev. Let., 47, p. 620, 1981.

A. Bovier, Statistical Mechanics of Disordered Systems: A Mathematical Perspective, Cambridge Series in Statistical and Probabilistic Mathematics, p. 308, 2006.

Y. Tu, J. Tersoff and G. Grinstein, Structure and Energetic of the Si and SiO2 Interface", Phys. Rev. Lett., 81, p. 4899, 1998.

K.V.R. Chary, G. Govil, “NMR in Biological Systems: From Molecules to Human", Focus on Structural Biology 6, Springer, p. 511, 2008.

E. Baake, M. Baake and H. Wagner, “Ising Quantum Chain is a Equivalent to a Model of Biological Evolution", Phys. Rev. Let., 78 (3), pp. 559-562, 1997.

A.S. Gevorkyan et al., “New Mathematical Conception and Computation Algorithm for Study of Quantum 3D Disordered Spin System Under the Influence of External Field", Trans. On Comput. Sci., VII, LNCS 132-153, Spinger-Verlage, 10.1007/978-3-642-11389-58.

S.F. Edwards and P.W. Anderson, Theory of spin glasses, J. Phys. F9, p. 965, 1975.

J. von Neuman, “Physical applications of the ergodic hypothesis", Proc. Nat. Acad. Sci. USA, 18 (3): pp. 263-266 (1932).

G.D. Birkhoff, “What is ergodic theorem?", American Mathematical Monthly, vol. 49, no. 4, pp. 222-226, 1931.

S. Flügge, Practical quantum mechanics I, Springer-Verlag, Berlin-Heidelberg-New York, 1971.

M.R. Spiegle, “Theory and problems of probability and stochastics", New-York, McGraw-Hill, pp. 114-115, 1992.

I. Ibragimov and Yu. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing Groningen, The Netherlands, 1971.

J.P. Nolan, “Stable distributions: models for heavy tailed data (2009-02-21). en:wikipedia:org=Stable=distribution.

H.G. Katzgraber, A.K. Hartmann and A.P. Young, “New insights from one-dimensional spin glasses", ArXiv: 0803.3417v1 [cond-mat.dis-nn], 2008.

## Downloads

## Published

## How to Cite

*Mathematical Problems of Computer Science*,

*35*, 86–98. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/290

## Issue

## Section

## License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.