Phase Transitions

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 Friedguts 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 lt 3.9 easy, satisfiable (probably P)
• 3.9 lt M/N lt 4.27 hard, but still satisfiable
• 4.27 lt M/N unsatisfiable, exponential cost
• Recent advances, applying message-passing, pushed
  boundary of solubility in the hard-SAT region
  from N 300 to N 107.
• 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)