finding the tools--and the questions--to understand dynamics in many dimensions
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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!. - PowerPoint PPT PresentationTRANSCRIPT
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Finding the Tools--and the Questions--to
Understand Dynamics in Many Dimensions
R. Stephen BerryThe University of
ChicagoTELLURIDE, APRIL 2007
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We know the problem, at least in a diffuse way
No. of atoms # of Minima Rank 1 Saddles
4 1 15 1 26 2 37 4 128 8 429 21 165
10 64 63511 170 242412 515 860713 1509 28,756
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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
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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
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Old pictures: Ar19, a glass-former
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(KCl)32, a structure-seeker
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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
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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
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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!
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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?
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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
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The simplest: a 10x10 net• The full net, and three samples: a
4x10 sample, “10x10 tri” and “10x10 inv”
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Relaxation from a uniform distribution in the top
layer
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What are eigenvalue patterns?
• a: full net; b: 4x10; c: tri; d: inv
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Now a bigger system: 37x70;
compare three methods • Sampling pathways “by sequence” (highest is full system)--not great.
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Sampling by choosing “low barrier” pathways
• Also not so good; slowest too slow!
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Sample choosing rough topography pathways
• Clearly much better!
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Try Ar13 as a realistic test• Relaxation times vs. sample size, for
several sampling methods; Rough Topography wins!
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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?
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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?
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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
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Short range implies more basins, more complex surface;
use extended disconnection diagrams
= 4 = 5 = 6
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Shorter range implies more complex
topography; GM=global min; DB=data base; Bh=barrier height
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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
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Asymmetry distributions, Br
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Correlate barriers Bh and Bl with bands of energies• Example: = 6; (red=low E;
blue=high E)
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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
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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?
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The people who did the recent work
• Jun Lu
• Chi Zhang
• And Graham Cox