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Spin Glasses and Complexity: Lecture 2

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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  1. 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

  2. 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

  3. Spin Glasses – a prototype disordered system? Dilute magnetic alloy, e.g., CuMn Frustration

  4. Crystal Glass Ferromagnet Ground States Spin Glass

  5. 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).

  6. 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 …

  7. 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 …

  8. 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.

  9. 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)

  10. Construct a ``cooling schedule’’

  11. 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)

  12. Dynamics corresponds to an energy function

  13. Protein Conformational Dynamics Myoglobin D.L. Stein, ed., Spin Glasses and Biology (World Scientific, Singapore, 1992)

  14. Fluctuations important for biological processes (e.g., ligand diffusion) • Recombination experiments imply many conformational substates A. Osterman et al., Nature404, 205 (2000)

  15. Spin Glass Model of Protein Conformational Substates D.L. Stein, Proc. Natl. Acad. Sci. USA 82, 3670 (1985)

  16. Protein Folding • Levinthal paradox • ``Principle of minimal frustration’’ J.D. Bryngelson and P.G. Wolynes, Proc. Natl. Acad. Sci. USA84, 7524 (1987)

  17. 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)

  18. 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!

  19. The Geometry of ``Information Propagation’’

  20. 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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