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Spin Glasses and Complexity: Lecture 2. Brief review of yesterday’s lecture. Spin glass energy and broken symmetry. Applications. - Combinatorial optimization and traveling salesman. - Simulated annealing. - Hopfield-Tank neural network computation.

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## Spin Glasses and Complexity: Lecture 2

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**Spin Glasses and Complexity: Lecture 2**• Brief review of yesterday’s lecture • Spin glass energy and broken symmetry • Applications - Combinatorial optimization and traveling salesman - Simulated annealing - Hopfield-Tank neural network computation - Protein conformational dynamics and folding • Geometry of interactions and the infinite-range model**Homogeneous systems possess symmetries that greatly simplify**mathematical analysis and physical understanding--- Bloch’s theorem, broken symmetry, order parameter, Goldstone modes, … Examples: crystals, ferromagnets, superconductors and superfluids, liquid crystals, ferroelectrics, … But for glasses, spin glasses, and other systems with quenched disorder; many new ideas and concepts have been proposed, but so far no universal ones**Spin Glasses – a prototype disordered system?**Dilute magnetic alloy, e.g., CuMn Frustration**Crystal**Glass Ferromagnet Ground States Spin Glass**The Edwards-Anderson (EA) Ising Model**Site in Zd Coupling realization The couplings are i.i.d. random variables: Nearest neighbor spins only Site in Zd S.F. Edwards and P.W. Anderson, J. Phys. F 5, 965 (1975).**Broken symmetry in the spin glass**EA ’75: A low-temperature spin glass phase should be described by presence of temporal order (freezing) along with absence of spatial disorder. But there are some surprises in store …**L.E. Wenger and P.H. Keesom, Phys. Rev. B13, 4053 (1976)**V. Cannella and J.A. Mydosh, Phys. Rev. B6, 4220 (1972). The most fundamental questions remain unanswered: • Is there a phase transition? • What is the nature of low-temperature phase (broken symmetry, order parameter)? • How does one account for the anomalous dynamical behavior (slow relaxation, memory, aging…)? Important not only for physics, but may lend important concepts to other areas …**Quenched disorder**• Frustration • Combinatorially huge possible number of configurations, or states, or outcomes • Many statistically equivalent `ground’ states (more or less equally good optimal solutions)? • Slow equilibration • Memory, aging … (NP-complete) Applications to combinatorial optimization (graph theory) problems, neural networks, biological evolution, protein dynamics and folding, … Example – the traveling salesman problem • N=5 12 tours • N=10 181,440 tours • N=50 Forget it.**Simulated annealing**• Cost function (plays role of energy function) • Quenched disorder • Frustration • Combinatorially huge possible number of configurations, or states, or outcomes • Many statistically equivalent `ground’ states (more or less equally good optimal solutions) - TSP: length of a tour - ``Placement’’ in computer design - k-SAT Many of these resemble spin glass Hamiltonian! • Add a ``temperature’’, and treat problem like a statistical mechanical problem Metropolis algorithm S. Kirkpatrick, C.D. Gelatt, Jr., and M.P. Vecchi, Science 220, 671 (1983) M. Mézard, G. Parisi, and R. Zecchina, Science 297, 812 (2002)**where**is the potential of neuroni. Neural circuit computation • Circuit element (``neuron’’) can be in one of two states (on/off: 0/1, spin up/spin down) • Dynamics of ``neurons’’ given by J.J. Hopfield and D.W. Tank, Science 233, 625 (1986) W.S. McCullough and W.H. Pitts, Bull. Math. Biophys. 5, 115 (1943)**Protein Conformational Dynamics**Myoglobin D.L. Stein, ed., Spin Glasses and Biology (World Scientific, Singapore, 1992)**Fluctuations important for biological processes (e.g.,**ligand diffusion) • Recombination experiments imply many conformational substates A. Osterman et al., Nature404, 205 (2000)**Spin Glass Model of Protein Conformational Substates**D.L. Stein, Proc. Natl. Acad. Sci. USA 82, 3670 (1985)**Protein Folding**• Levinthal paradox • ``Principle of minimal frustration’’ J.D. Bryngelson and P.G. Wolynes, Proc. Natl. Acad. Sci. USA84, 7524 (1987)**Folding landscapes as a ``rough funnel’’**Used to develop algorithms for structure prediction (J. Pillardy et al., PNAS 98, 2329 2001); designing ``knowledge-based potentials for fold recognition; etc. C.L. Brooks III, J.N. Onuchic, and D.J. Wales, Science293, 612 (2001)**Back to spin glasses proper …**By now, it’s (hopefully) clear that understanding the behavior of these systems is important not only for condensed matter physics and statistical mechanics, but for many other fields as well… … so we will now turn to examine what we know about them. Unfortunately, understanding their nature has been very difficult --- theoretically, experimentally, and numerically!**The Sherrington-Kirkpatrick (SK) Model**``Infinite-range’’ model – no geometry left! ``Mean-field’’ model; infinite-dimensional model. Phase transition with Tc=1. What is the thermodynamic structure of the low-temperature phase? Broken replica symmetry --- one of the biggest surprises of all. Stay tuned … D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett.35, 1792 (1975).

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