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Modeling of 1D spin glasses from first principles ofclassical mechanics

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#### Abstract:

We study the classical 1D Heisenberg spin glasses assuming that spins are spatial. The system of recurrence equations is derived by minimization of the nearestneighboring Hamiltonian in nodes of 1D lattice. We have proved that in each node of the lattice there is a probability that the solution of recurrence equations can bifurcate. This leads to the fact that, performing a consecutive node-by-node calculations on the n-th step instead of a single stable spin-chain we get a set of spin-chains which form Fibonacci subtree (graph). We have assessed the complexity of computation of one graph and have shown that it is ∝ 2 nKs, where n and Ks denote the subtree’s height (the length of spin-chain) and the Kolmogorov’s complexity of a string (the branch of subtree) respectively. It is shown that the statistical ensemble may be represented as a set of random graphs, where the computational complexity of each graph is NP hard. It is proved, that all strings of the ensemble have the same weights. The latter circumstance allows in the limit of statistical equilibrium with predetermined accuracy to reduce the NP hard problem to the P problem with complexity ∝ NKs, where N is the number of spin-chains in the ensemble. As it is shown by comparing statistical distributions of different parameters which are performed by using NP and P algorithms the coincidence of the corresponding curves is ideal. This allows to claim that it is possible to calculate all parameters and the corresponding distributions of the statistical ensemble from the first principles of classical mechanics without using any additional considerations. Finally, using formal similarity between of the ergodic dynamical system and the ensemble of spin-chains, it is proposed a new representation for the partition function in the form of one dimensional integral from the spin-chains’ energy distribution.

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Institute for Informatics and Automation Problems

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South Caucasus Computing and Technology Workshop