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Phase Transitions – like death and taxes? Why we should care; what to do about it.

Phase Transitions – like death and taxes? Why we should care; what to do about it. Scott Kirkpatrick, Hebrew University, Jerusalem With thanks to: Uri Gordon, Erik Aurell, Johannes Schneider,…. Phase transitions are inevitable in Avogadro-scale engineering.

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Phase Transitions – like death and taxes? Why we should care; what to do about it.

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  1. Phase Transitions – like death and taxes?Why we should care; what to do about it. Scott Kirkpatrick, Hebrew University, Jerusalem With thanks to: Uri Gordon, Erik Aurell, Johannes Schneider,…

  2. Phase transitions are inevitable in Avogadro-scale engineering. • System properties go from “good” to “not good enough” at the extremes of their parameter space • What happens in between these phases? • We have limited tools for understanding such transitions. • Physics of disordered materials: • Sharp vs. smeared – Harris criterion • Glass transitions • Combinatorics on large scales • Sharp property crossovers – Friedgut’s theorem • We should care because computing is HARD at phase boundaries.

  3. SAT and 3-SAT as classic test cases • Parameters: N variables, M constraining clauses, M/N constant • For 3-SAT, phase diagram is known: • M/N < 3.9 “easy,” satisfiable (probably P) • 3.9 < M/N < 4.27 “hard,” but still satisfiable • 4.27… < M/N unsatisfiable, exponential cost • Recent advances, applying message-passing, pushed boundary of solubility in the “hard-SAT” region from N = 300 to N = 10^7. • General technique – soften the variables into beliefs or surveys

  4. Surveys and Beliefs for the SAT problem • Beliefs – probabilities that the spin is up or down • Avg. over satisfying configurations, as estimated by the local tree • Surveys – probabilities that the spin is up or down • In all sat configurations. • This leaves a third possibility – spins that do both at different times. • Equations for both are nearly identical. We can define hybrid methods, and they prove useful. • Use beliefs or surveys to guide decimation – this solves problem.

  5. Propagating surveys or beliefs in a “cavity”

  6. Depth of decimation characterizes BP, SP and mixed-P

  7. Study how SP, BP, m-P evolve by analyzing their hydrodynamics • Use movies of N = 100,000 • Look at three cases, SP, BP, and the hybrid-P: • http://www.cs.huji.ac.il/~kirk/SP.avi • http://www.cs.huji.ac.il/~kirk/BP.avi • http://www.cs.huji.ac.il/~kirk/mixedP.avi • (caution, large files)

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