Why Water Freezes (A probabilistic view of phase transitions)
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1 Why Water Freezes (A probabilistic view of phase transitions) Mark Huber Dept. of Mathematics and Institute of Statistics and Decision Sciences Duke University
2 Mom's Old Fashioned Ice Water 1: Get water, cubes of ice 2: Place ice in water 3: Wait until water is cold 4: Drink and enjoy!
3 Mixing liquids 10 o C 20 o C 15 o C
4 Not so with ice water 15 o C 0 o C 0 o C
5 Thermometer calibration First to notice this: Anders Celsius ( ) (Freezing point independent of latitude) 100 o C Freezing 0 o C Boiling
6 Phase Transition Temperature 100 C 0 C Takes extra energy to complete phase transition from ice to water Adding energy at constant rate
7 Properties Ice Long Range Behaviour Molecules in lockstep Keeps its shape Water Short Range Behaviour Molecules loosely bound Loses shape almost instantly
8 A short history of physics Newton ( )? Einstein ( )
9 A slightly longer history of physics Newton ( ) Laplace ( ) Einstein ( ) "... I have sought to establish that the phenomena "The most of nature important can be reduced in the questions last analysis of life to actions are, for at a distance the most between part, molecule really only and molecule, and that the consideration problems of probability." of these actions must serve as the basis of the mathematical theory of these phenomena."
10 A slightly longer history of physics Newton ( ) Laplace ( ) Einstein ( ) "... I have sought to establish that the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule, and that the consideration of these actions must serve as the basis of the mathematical theory of these phenomena."
11 The original: the Ising model Simple probabilistic model First to exhibit a phase transition (1924) Originally used for magnets currently used to describe alloys
12 The Hard Core Gas Model Configuration x: a placement of molecules No two molecules can be adjacent to each other x = x Z
13 Hard Core has phase transition! Distribution looks "smooth" in λ When λ small, short range interactions Past critical point, long range ("checkerboard")
14 Moving up to the 1950's... Computers speed increasing exponentially Physicists begin simulating Ising, Hard Core gas model (Monte Carlo Markov Chain approach) Algorithms for combinatorial optimization problems become feasible
15 Monte Carlo Markov chain Goal: generate samples from distribution Start at an arbitrary configuration Make "lots" of small random changes to the configuration Hope that final configuration has distribution close to the target Example: shuffling cards The Problem: How many is "lots"?
16 Upper bounding "lots" Use bounding chain to experimentally bound "lots" (H. '96) Start at all configurations simultaneously Make "lots" of small random changes to all these configurations at same time Merge configurations as they run into each other Time needed to merge into one configuration is an upper bound on "lots" (Doeblin '33, Aldous '82)
17 The Results Time needed for Markov chain to mix undergoes a phase transition at the same λ Runtime λ
18 Combinatorial Optimization The Traveling Salesman Problem
19 Independent Sets of a graph MAX independent set problem: Find the largest set of vertices such that no two are adj. Both TSP and MAX IND SET are NP-complete
20 Simulated Annealing Sneak up on answer: Small λ gives small ind sets, MC quick Large λ gives large ind sets, MC slower x = x Z The Problem: Below phase transition, MC quick Above phase transition, MC very slow
21 Even sampling hard Let be maximum degree of a graph Dyer, Frieze, Jerrum ('98) showed: if you can sample from π for any graph with parameters 25 then NP = RP
22 Markov chain for Independent Sets Choose a site uniformly at random If a neighbor is occupied leave the site unoccupied Otherwise With probability Make the site occcupied With probability Make the site unoccupied
23 Graph structure determins phase transition 1 Dimensional No phase transition 2 Dimensional Phase transition Expander graphs Worst case
24 The Vicious Circle MCMC Samples
25 Breaking the cycle of dependence: perfect samplers Perfect sampling algorithms can sample from: x = weight x Z without the need to know Z
26 Idea: modify graph to make problem easy (H., Fill '98) Step 1: Remove all the edges in the graph Step 2: Randomly include each node in independent set with probability 1
27 Add edges back one at a time Step 3: Keep adding edges until get conflict
28 Dealing with conflict Remove edges of complex Recolor nodes of complex
29 Conclusions Phase transitions pop up everywhere The physics behind ice also make the traveling salesman problem difficult What they have in common are long range interactions can't just look locally Markov chains (that were supposed to simulate the models) also have phase transitions Perfect sampling techniques can indicate where these phase transitions lie
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