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!