What Happens on Many-Dimensional Landscapes?
What’s Important about the Landscape?R. Stephen Berry
The University of ChicagoWorkshop on “The Complexity of Dynamics
and Kinetics in Many DimensionsTelluride, Colorado, 20 April 2011
Real Complexity: What Happens When a System Explores a Landscape of Tens or Hundreds of Dimensions—or More? • How is the topography of that landscape
related to:• The interparticle forces?• The dynamics of passage among the local
minima?• The kinetics of relaxation, annealing or
thermal excitation and phase change?
The Trigger of Our Interest • Studying alkali halide clusters, we found that
there are vastly more locally-stable amorphous structures than crystalline, rocksalt structures.
• Nevertheless, simulated cooling from the liquid state inevitably yielded rocksalt, unless the cooling rate was faster than 1013 K/sec, roughly 5 to 10 vibrational periods.
• Yet rare gas clusters “get stuck” in amorphous structures even at cooling rates of 109 K/sec.
This led to examination of topographies
• Start with a low minimum, ideally the bottom of a big basin
• Find a saddle taking the system to a higher minimum
• Continue, generating sequences of stationary points, with energies of the minima increasing monotonically
• Examine these monotonic sequences for both alkali halides and rare gases
The Sample Sequences for Ar19 .
The Sample Sequences for (KCl)32
The Differences• The Argon cluster’s topography shows very
few large asymmetries in the min-saddle-min triples; the energy changes are small
• The alkali halide has several very asymmetric triples, with large drops in energy from upper to lower minimum
• Very few atoms move in each step, in the rare gas; many atoms move at every step in the KCl cluster.
The Differences, cont.
• The Argon-Argon interaction is through a Lennard-Jones potential, relatively short-range
• The alkali halide interaction is Coulombic, the longest range interaction possible
• What happens “nearby” has no influence on what goes on “far away” in the argon cluster
• What happens “nearby” in the alkali halide cluster is important throughout the structure
The Differences, cont.
• This tells us that short-range forces are associated with glass-formers and long-range forces, with structure-seekers.
• This is a helpful, qualitative picture that we can use as a starting point.
• We can test this with proteins to see how general the idea is.
The First “Protein” Test : the 46-Bead “BNL” Model
• This has at least 8 “folded” structures, e.g. these
Sample Monotonic Sequences of the 46-Bead “BNL” Model
A Real Protein, BPTI, and a Randomized Peptide of the Same Residues
The Qualitative Idea Seems Okay
• Then can we give it more precision and more quantitative meaning?
• We can ask how the range of interparticle interactions affects the topography of the energy landscape!
• Then perhaps we can say something more precise about the relation of or transition between structure-seeking and glass-forming
Two Exemplary Cases
• The homogeneous Morse cluster, in which the range parameter can vary; in real diatomics, goes between about 3 and 7.
• The shielded Coulomb binary cluster, with (KCl)32 as the long-range extremum
• We learn that long-range forces make for smoother landscapes and fewer minima; short-range forces, for more complex surfaces
The Morse Example, M13 : Disconnection Diagrams, but with
Distance Indicated Horizontally• = 4 (smooth) = 5 = 6 (rough)
The Barrier Asymmetries Correlate with Energies of the Lower Minima
• = 6 ; red = lowest deeper E; blue = highest
The Barrier Asymmetries Correlate with Energies of the Lower Minima
• Thus, the deeper you go, the more asymmetric are the barriers, i.e. the more you gain in stabiilzing the system by going over the next barrier to a lower minimum
• Likewise, the deeper you go, the harder it is to climb back out
• Moreover, the shorter the interaction range, the greater is this effect
The Shielded Coulomb Model
• The potentials for various shielding parameters
The Caloric Curve: Full Coulomb Case• The shielding parameter = 0
The Caloric Curve: Shielded Coulomb Case
• The shielding parameter = 0.250
The Caloric Curve: Shielded Coulomb Case
• The shielding parameter = 0.350
The Caloric Curve: Shielded Coulomb Case
• The shielding parameter = 0.400
The Caloric Curve: Shielded Coulomb Case
• The shielding parameter = 0.450
The Caloric Curve: Most Shielded Coulomb Case
• The shielding parameter = 0.500
The Changes: What Are They?
• There are two very distinctive changes• The heat capacity has a region of negative
values (in microcanonical ensembles) for low values of the shielding, but not high values
• The global minimum structure is rocksalt for low values of the shielding, but is a hollow shell structure for high values and short-range potentials
Where Does the Change Come?• The negative heat capacity disappears and the
lowest-energy structure changes when goes to and beyond 0.40 (––)
The Change of Character: Where Does It Occur?
• This happens when the “Halfway Point” of the attractive curve, halfway from asymptote to minimum, is reached when the internuclear distance of the two ions is ~3Å
• This, in turn, is about 1.5 times the equilibrium distance between nearest neighbors
• Briefly, long-range behavior persists up to rather short-range potentials
The Next Steps: Open Questions• What are the important general
characteristics of a high-dimensional topography?
• Which among those characteristics tell us how to sample the monotonic sequences on the surface to construct statistical-sample master equations that will give reliable eigenvalues, i.e. rate coefficients, for the important slow processes? (We know a bit about this, but not nearly enough.)
Open Questions About Pathways• Can we find a way to count the pathways
between two structures—or to a specific destination structure from a random starting point?
• What role do multiple pathways play? Does it matter whether they are interconnected? Can we tell from observation whether they are?
• How important are the local minima and saddles along a pathway? What role do they have?
The People • Jun Lu Chi Zhang Graham Cox Shoji
Takada
• Jason Green Chengju Wang Julius Jellinek
Thank you!