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Finding the Tools--and the Questions--to Understand Dynamics in Many Dimensions

Finding the Tools--and the Questions--to Understand Dynamics in Many Dimensions. R. Stephen Berry The University of Chicago TELLURIDE, APRIL 2007. We know the problem, at least in a diffuse way. But what should we do with this? Too much information!.

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Finding the Tools--and the Questions--to Understand Dynamics in Many Dimensions

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  1. Finding the Tools--and the Questions--to Understand Dynamics in ManyDimensions R. Stephen Berry The University of Chicago TELLURIDE, APRIL 2007

  2. We know the problem, at least in a diffuse way

  3. But what should we do with this? Too much information! • We must decide what questions are the most important, and then • Decide how little information we need to answer those questions

  4. One approach: distinguish glass-formers from structure-seekers • A useful start, but, as stated, only qualitative, and only a qualitative criterion distinguishes them––so far! • Glass-formers have sawtooth-like paths from min to saddle to next min up; structure-seekers have staircase-like pathways

  5. Old pictures: Ar19, a glass-former

  6. (KCl)32, a structure-seeker

  7. Some easy inferences • Short-range interparticle forces lead to glass-formers; long-range forces, to structure-seekers • Effective long-range forces, as in a polymer, also generate structure-seekers sometimes -- compare foldable proteins with random-sequence non-folders

  8. Can we invent a scale between extreme structure-seekers and extreme glass-formers? • Some exploration of the connection between range of interaction and these limits, but not yet the best to address this question; that’s next on our agenda

  9. Another Big Question: Can we do useful kinetics to describe behavior of these systems? • The issue: can we construct a Master Equation based on a statistical sample of the potential surface that can give us reliable eigenvalues (rate coefficients), especially for the important slow processes? • How to do it? Not a solved problem!

  10. Some progress: Some sampling methods are better than others • Assume Markovian well-to-well motion; justified for many systems • Use Transition State Theory (TST) to compute rate coefficients • But which minima and saddles should be in the sample?

  11. Two approaches: 1) various samplings & 2) autocorrelations • Jun Lu: invent a simple model that can be made more and more complex • Try different sampling methods and compare with full Master Equation • One or two methods seem good for getting the “slow” eigenvalues

  12. The simplest: a 10x10 net • The full net, and three samples: a 4x10 sample, “10x10 tri” and “10x10 inv”

  13. Relaxation from a uniform distribution in the top layer

  14. What are eigenvalue patterns? • a: full net; b: 4x10; c: tri; d: inv

  15. Now a bigger system: 37x70;compare three methods • Sampling pathways “by sequence” (highest is full system)--not great.

  16. Sampling by choosing “low barrier” pathways • Also not so good; slowest too slow!

  17. Sample choosing rough topography pathways • Clearly much better!

  18. Try Ar13 as a realistic test • Relaxation times vs. sample size, for several sampling methods; Rough Topography wins!

  19. Big, unanswered questions: • What is the minimum sample size to yield reliable slow eigenvalues? • How can we extend such sampling methods and tests to much larger systems, e.g. proteins and nanoscale particles?

  20. Another Big Question: How does local topography guide a system to a structure? • How does the distribution of energies of minima influence this? • How does the distribution of barrier energies influence this? • How does the distribution of asymmetries of barriers influence this?

  21. One question with a partial answer: How does range affect behavior? • Long range implies smoother topography and collective behavior • Short range implies bumpier topography and few-body motions

  22. Short range implies more basins, more complex surface; use extended disconnection diagrams •  = 4  = 5  = 6

  23. Shorter range implies more complex topography; GM=global min; DB=data base; Bh=barrier height

  24. Consider barrier asymmetries • The 13-atom Morse cluster, with = 4, 5, 6 • = 4 is the longest range and the most structure-seeking • Examine asymmetries using Ehighside/Elowside = Bh/Bl= Br where low Br means high asymmetry

  25. Asymmetry distributions, Br

  26. Correlate barriers Bhand Bl with bands of energies • Example: = 6; (red=low E; blue=high E)

  27. A pattern emerges • Deep saddles are the most asymmetric • High levels have many interlinks; low levels, fewer • For a system this small, bands of energy minima are clearly distinguishable

  28. More questions ahead • Can we make a quantifying scale between extremes of glass-forming and structure-seeking? • What role do multiple pathways play? What difference does it make if they are interconnected? • Should we focus on big basins or are the detailed bumps important?

  29. The people who did the recent work • Jun Lu • Chi Zhang • And Graham Cox

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