What Happens on Many- Dimensional Landscapes? What’s Important about the Landscape? R. Stephen Berry The University of Chicago Workshop on “The Complexity of Dynamics and Kinetics in Many Dimensions Telluride, 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 K/sec, roughly 5 to 10 vibrational periods. Yet rare gas clusters “get stuck” in amorphous structures even at cooling rates of 10 9 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 Ar 19.

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, M 13 : 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!