# Computation Statistical Properties of Disordered Spin Systems from the First Principles of Classical Mechanics

## Keywords:

Disordered system, Heisenberg spin glass, Ergodic ensemble, NP hard problem, , P algorithm## Abstract

We study the classical 1D spin glasses in the framework of Heisenberg model. Based on the Hamilton equations we obtained the system of recurrence equations, which allows performing node-by-node calculations of a spin-chain. It is shown that the calculations from the first principles of classical mechanics lead to NP hard problem, that however, in the limit of the statistical equilibrium can be calculated by P algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.

## References

K. Binder and A. Young, “Spin glasses, Experimental facts, theoretical concepts and open questions”, Rev. Mod. Phys., vol. 58, pp. 801-976, 1986.

M. Mézard, G. Parisi and M. Virasoro, “Spin Glass Theory and Beyond (World Scientific)”, Singapore 1987.

A. Young, Spin Glasses and Random Fields, World Scientific, Singapore, 1998.

R. Fisch and A. Harris, “Spin-glass model in continuous dimensionality”, Phys. Rev. Lett., vol. 47, page 620, 1981.

C. Ancona-Torres, D. Silevitch, G. Aeppli and T. Rosenbaum, “Quantum and classical glass transitions in LiHoxY1-xF4”, Phys. Rev. Lett., vol. 101, no. 5, 057201, 2008.

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

Y. Tu, J. Tersoff and G. Grinstein, “Properties of a Continuous-Random-Network model for amorphous systems”, Phys. Rev. Let., vol. 81 ,page 2490, 1998.

K. Chary and G. Govil, “NMR in biological systems, from molecules to human”, Springer, vol. 6, page 511, 2008.

E. Baake, M. Baake and H. Wagner, “Ising quantum chain is a equivalent to a model of biological evolution”, Phys. Rev. Let., vol. 78, page 559, 1997.

A. S. Gevorkyan and H. G. Abajyan, “A new parallel algorithm for simulation of spin glasses on scales of space-time periods of external fields with consideration of relaxation effects”, Phys. of Particles and Nuclei Letters, vol. 9, no. 6-7, page 530, 2012.

F. Liers, M. Palassini, A. K. Hartmann and M. Jünger,“Ground state of the Bethe lattice spin glass and running time of an exact optimization algorithm”,Phys. Rev. B, 68, 094406, 2003.

J. C. Angles, D′Auriac, M. Preissmann and A. Sebo Leibniz-Imag, “Optimal cuts in graphs and statistical mechanics”, Mathl. Comput. Modeling, 26, no. S-10, page 11, 1997.

C. Papadimitriou, Computational Complexity, (1st ed.). Addison-Wesley. Section 2.7: Nondeterministic machines, 1993.

H. R. Lewis and C. Papadimitriou, Elements of the Theory of Computation, (1st ed.). Prentice-Hall. Section 4.6: Nondeterministic Turing Machines, 1981.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, “Equation of State Calculations by Fast Computing Machines”, J. Chem. Phys., vol. 21, no. 6, page 1087, 1953.

B. Hayes, “Can’t get no satisfaction”, Am. Scientist, vol. 85, page 108, 1997.

R. Monasson et all, Diameter of the world wide web, Nature, London, 400, 133, 1999.

Special issue of Theor. Comput. Sci. 265, edited by O. Dubois, R. Monasson, B. Selman, and R. Zecchina, 2001.

M. J. Alava, P. M. Duxbury, C. F. Moukarzel and H. Rieger, Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz, Academic Press, New York, 18, 2001.

A. K. Hartmann and H. Rieger, Optimization Algorithms in Physics, While-VCH, Berlin, 2001.

C. J. Thompson, “Phase Transitions and Critical Phenomena”, Academic Press, vol. 1, pp. 177-226, 1972.

H. Goldstein, Classical Mechanics, Reading, MA: Addison-Wesley, 2nd ed., 1980.

A. N. Kolmogorov, “Logical basis for information theory and probability theory”, IEEE Transactions on Information Theory, vol. 14, no. 5, pp. 662–664, 1968.

M. Li and P. Vit´anyi, An introduction to Kolmogorov Complexity and its Applications, New York, Springer-Verlag, 1997.

S. Y. Park and A. K. Bera, “Maximum entropy autoregressive conditional heteroskedasticity model”, Journal of Econometrics, Elsevier, vol. 50, no. 2, pp. 219–230, 2009.

G. D. Birkhoff, “What is the ergodic theorem?”, The American Mathematical Monthly, vol. 49, no. 4, pp. 222–226, 1942.

V. I. Arnol’d and A. Avez, Ergodic Problems of Classical Mechanics, New York, W.A. Benjamin, 1968.

E. A. Ayryan, A. S. Gevorkyan and V. V. Sahakyan, “New algorithm for simulation of 3D classical spin glasses under the influence of external electromagnetic fields”, Physics of Particles and Nuclei Letters, vol. 2, no. 3, pp. 380–384, 2015.

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*Mathematical Problems of Computer Science*,

*44*, 22–32. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/179

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