Spin glass -like models

Random energy model The random energy model (REM) results from using a fitness distribution p(f) to assign fitnesses randomly to points in the landscape [ 14,59,60,70,71,81, 91,92], p(f) is the probability that a point in the sequence space has fitness fand is exactly analogous to affinity distribution p(Ka). Such landscapes have zero correlation (are very rugged), have many local fitness peaks, and result in very short adaptive walks. Very few of the local peaks are accessible by adaptive walks from any particular point. 

REM landscapes have been applied to the maturation of the immune system. When exposed to a new antigen, antibodies undergo on average only 6-10 point mutations in the course of achieving typically a 50-100-fold increase in affinity [70,71], Such short walks from random initial points to local peaks occur in rugged landscapes. The short walks and only modest affinity increase can be interpreted to indicate that the antibody affinity landscape is essentially uncorrelated [14,71,81,91], However, because there is such 

NK model The NK model is a simple landscape model that allows arbitrary degrees of correlation between 0 and 1 [4,92,93], The sequence space is an N-dimensional space 

The keys to the model are (i) The fitness of a sequence is the average of fitness contributions from each site 

The properties of p-spin landscapes have been studied in detail [34,79], One observation of importance is that p plays a role similar to that of K. The number of local peaks increases exponentially with p [76], This occurs because, as p increases, there are more conflicting constraints that cause the site values that maximize one J function to conflict with those site values that maximize other J functions. 

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Search on RNA secondary structure landscapes is distinctly different from search on the spin glass-like models. The difference is a result of the neutral networks that percolate the space. Note that, in practice, the sequences on neutral networks need not have exactly the same fitness, but fitnesses whose differences are below a threshold determined by the mutation rate and noise in the system. As with search on spin glass landscapes, this topic is quite extensive and is reviewed in several papers [39,67,69,113] as well as in Schuster s contribution to this collection, so I will only touch on a few key points. 

A geometrical view of all known orthogonalization procedures is taken to understand their distinctive features and the inter-connections between them. Curious new information is gained which is also found useful to understand the basis of certain cognitive phenomena, like discrimination and categorisation. A spin-glass like neural network model has been introduced to understand the cognitive phe-... 

F.4.3.1. Field Dependence. First, we will discuss the works devoted to the dependence of TRM values on the previously applied field, just after the removal of the field or a short time after (actually the calculations are valid for any time). A maximum in the TRM vs. H dependence, rather than a continuous increase to saturation (like for IRM), has been reported in the literature in analogy with what is observed in spin glasses. Two models have been presented, that take into account the TRM variation during the cutting-off of the field. In both cases a volume... 

The Schottky-like anomaly observed in the specific heat of the compounds discussed in this section can be derived phenomenologically using (a) the resonance-level model, (b) the spin glass behaviour, (c) the crystal field (Schottky) contribution or even (d) low-dimensional magnetic fluctuations. The cases where an HF behaviour is deduced from a large value will be discussed in sect. 9, in connection with the contribution to of the excited crystal field levels. It is clear that complementary techniques, such as NMR, AC susceptibility and electrical resistivity, can easily reveal the magnetic character of the microscopic interactions. In some of the HF compounds the ratio between the y term and the (( -> 0) = Xo value of the susceptibility, and between the / term and the coefficient of the resistivity. A, have values predicted by... 

To discuss this new type of ordered phase in spin glasses, one would like to have a microscopic model where the actual interactions and anisotropies are considered and the average over a realistic description of the site dilution disorder is performed. Clearly this is a difficult task and up to now no realistic model of a spin glass has been solved analytically. In addition, there exists another difficulty because a proper treatment of systems with quenched disorder like spin glasses involves averaging the free energy F rather than the partition function Z... 

The EA model contains the two essential ingredients for a spin glass - disorder and competition as discussed above - but it is a crude approximation to real spin glasses. Therefore at the beginning we have to discuss how well this model reproduces spin-glass properties. Up to now there is no analytic solution available for the short-range EA model. Even modern methods like real space renormalization yield inconclusive results (Kinzel and Fischer 1978, Tatsumi 1978). Hence numerical methods like the Monte Carlo simulation have been applied to answer this question the results of which will be briefly summarized now. Then the mean-field analysis of the EA model will be discussed, and finally the question... 

This feature is sketched in fig. 19. Thus, a vector spin system in a field H within the SK model exhibits a phase with spin-glass order transverse to the field, reminiscent of a spin-flop phase of a pure antiferromagnet. Just below the GT line, only weak irreversibilty occurs in the longitudinal component but there is a crossover to strong irreversibility in at a region which goes as like the AT line (indicated by the dashed line in fig. 19). 

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