The tremendous success of statistical mechanics in describing macroscopic properties of condensed matter systems often rests on the translational invariance of these systems. Examples include the heat capacity of crystalline solids, the paramagnetic to ferromagnetic phase transition, and superconductivity in metals. On the other hand, in many real systems deviations from translational symmetry are not just dirt effects but the manifestation of genuinely new properties like competing interactions, frustration, highly degenerated ground states, non-ergodic dynamics and hysteretic response.
Spin glasses are model systems in which magnetic moments interact via random couplings thereby combining disorder and frustration. Their properties as obtained within analytical and numerical studies show many similarities with real disordered magnets and structural glasses. Moreover, the tools and concepts developed for their theoretical description have found interesting applications in interdisciplinary fields as information theory, combinatorial optimization, error-correcting codes, algorithmic complexity, and game theory.
Artificial neural networks
A. Engel and C. van den Broeck,"Statistical mechanics of learning",
Cambridge University Press, 2001
(first 42 pages, 183 Kb, gzipped postscript)
The amazing abilities for information processing shown by biological neural networks are far beyond the capabilities of individual neurons and are therefore truely collective properties. As such they should be at least partly amenable to a statistical mechanics analysis. In fact, using concepts and techniques from the statistical mechanics of disordered systems it is possible to mathematically characterize some of these abilities of neural networks like storage, classification and learning from examples. In particular the process of generalization from some examples to the underlying rule is a very interesting problem in machine learning. It is also an attractive alternative to explicit programming in situations where a task ought to be done by a computer but it is difficult to design an algorithm. The textbook discusses and analyses various aspects of learning from examples in artificial neural networks from a statistical mechanics point of view.
Replica theory of Lévy Spin glasses
K. Janzen, A. Hartmann, A. Engel, J. Stat. Mech. P04006
Mean-field models of spin glasses with Gaussian distribution of couplings can be thoroughly analyzed by analytical techniques such as the replica and cavity method, respectively, for which the central limit theorem is of crucial importance. On the other hand, experimental spin glasses, in which magnetic ions are randomly distributed in a nonmagnetic host lattice, show large variations in the interaction strengths between the magnetic impurities. Consequently they are best modelled by coupling distributions with long tails as exhibited by Lévy distributions. For these, however, the central limit theorem does not apply. A mean field model of a spin glass with Lévy distributed couplings shows many interesting features and interpolates between the fully connected Sherrington Kirkpatrick model and the finite-connectivity Viana-Bray model. The results of a replica treatment using an imaginary temperature at intermediate steps of the analysis only partly agrees with the findings from a cavity analysis building on the assumption of a Gaussian distribution of local magnetic fields.
Large deviations in Erdös-Rényi random graphs
A. Engel, R. Monasson, A. Hartmann, J. Stat. Phys. 117 (2004) 387
Properties of random graphs are central to many problems in complex optimization and algorithmic complexity. Moreover, they provide models of trophic nets in ecosystems, traffic networks,and the internet. It has been clear since the 1960s that large random graphs can to a large extend be characterized by their typical properties which were therefore thoroughly investigated. However, recently it became clear that many important features as, e.g., the sensitivity of the net to failure of single nodes or the tendency to congestions depend on specific rare properties or configurations. Although only occuring with a probability exponentially small in the size of the graph these events may dominated the overall behaviour of the network. A systematic study of large deviation properties of Erdös-Rényi graphs can be accomplished by exploiting the mapping between random graphs and the Potts model and by using a variant of the cavity method from the statistical mechanics of disordered systems.