Spin-glass

In the case of the threshold rules defined in this section, we must consider sequential iterations of deterministic rules. Also, the choice of spins that may change state is not random but is fixed by some random permutation of the sites on the lattice. Such rules may be shown to correspond to spin glasses in the zero-temperature limit. 

Although the exact cycle decomposition of a finite-size CA depends on the set of elementary divisors of hij (see section 5.4), it is convenient to classify topologies according to their cyclic equivalence classes Q . Any Pi x) = 

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Virasaro M, Mezard M and Paris G 1987 Spin Glass Theory and Beyond Singapore World Scientific)... 

Bryngelson J D and Wolynes P G 1987 Spin glasses and the statistical mechanics of protein folding Proc. Natl Acad. Sci. (USA) 84 7524-8... 

Anderson P W 1978 The concept of frustration in spin glasses J. Less-Common Metais 62 291-4... 

Bryngelson J D and P G Wolynes 1987. Spin Glasses and the Statistical Mechanics of Protein Folding. Proceedings of the National Academy of Sciences LISA 84 7524-7528. 

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

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

AuFe like CuMn is one of the well known prototype systems showing magnetic spin glass behaviour at low temperatures. It may be recognized that by the above method we are... 

R. Clad, R. Kuentzler and W. Pfeiler, Atomic short-range order and spin-glass behaviour in concentrated... 

An expanded discussion of spin glasses appears in section 7.1.7... 

Much of what is currently known about spin glasses can be traced back to the pioneering work of Edwards and Anderson [edw75]. Their idea was to try to capture the essential properties of real spin glasses with a simple Jsiug-like Hamiltonian of the form ... 

While the locations of the spins are not random - indeed, the spins populate sites of a regular lattice - the interactions themselves are completely random. Frustration, too, has been retained. Thus, arguably, two of the three fundamental properties of real spin glass systems are satisfied. What remains to be seen, of course, is the extent to which this simplified model retains the overall physics. 

For high temperatures, the spin-glass system behaves essentially the way conventional Ising-spin systems behave namely, a variety of different configurations are accessible, each with some finite probability. It is only at low enough tempera tures that a unique spin-glass phase - characterized chiefly by the appearance of a continuum of equilibrium states - first appears. 

Fig, 7,2 Spin-glass overlap probability Pspin-giaes(5) versus overlap q (equation 7,57) 6 x) is the Dirac-Delta function. 

Fig. 7.3 A representation of the (conjectured) ultrametric distribution of spin-glass equilibrium states. The leaves of the tree at bottom are identified with the states overlaps between states are measured by the number of levels it takes to trace the states back to their common roots . For the three states a, 0 and 7, for example, we have that qot y = q y = q and = 92 > 9l-...

There are many excellent reviews of spin glass systems. We recommend the following three sources the review article by Binder and Young [binder86], the book by Chowdhury [chowd86], and the collection of papers edited by Mezard, et.aJ. [mez87]. 

Derrida and Flyvberg [flyvb88] and Derrida and Bessis [derrida88] have examined the overlap between two attractors by looking at the probability that two randomly chosen initial state.s evolve toward the same attractor. These studies find that the distribution is very similar to that found in certain spin-glass models. 

As mentioned above, Hopfield s original approach to this problem was to introduce an energy function reminiscent of a spin-glass Hamiltonian ... 

Finally, if there are a large number of stored patterns, there may enough mutual interference to give rise to local minima, or inetastable states - sometimes also called spin glass states to emphasize the similarity between the formalisms of Hopfield nets and spin glasses - that are not correlated with any subset of the set of stored patterns [amitSSb]. 

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