Minicourse on "A mathematical perspective on the structure of matter"

Date: May 14 - 18, 2012, 10:00 a.m. - 12:00 noon
There will be no talk on Thursday, May 17, because of holiday
Venue: HIM lecture hall, Poppelsdorfer Allee 45

Speaker: Richard D. James (University of Minnesota)

Lecture 1: Objective Structures (OS)
Lecture 2: Objective molecular dynamics (OMD)
Lecture 3: Invariant solutions of the Boltzmann equation
Lecture 4: Special topics on objective structures



Lecture 1: Objective Structures (OS).  The building block of crystals is the Bravais lattice, integer coefficients on three linearly independent vectors.  However, about half the elements in the periodic table do not crystallize as Bravais lattices, the most common non-Bravais example being the hexagonal close packed (HCP) lattice.  However, HCP, and nearly all the others have the following property.  Imagine “sitting on” an atom of the lattice and viewing the crystal structure from this perspective.  Now sit on another atom of the lattice and “reorient yourself” by a suitable orthogonal transformation.  Then one sees precisely the same arrangement of atoms. This idea leads to a definition “objective structures” that includes not only HCP and most of the periodic table, but also the structures – nanotubes, graphene sheets, buckyballs – of main interest in nanotechnology.  In the first lecture I will state this definition mathematically and explore its elementary consequences, and its relation to discrete groups of isometries. 

Lecture 2: Objective molecular dynamics (OMD).  While the definition of OS is a unifying way to think of atomic structures, its most important application is not to structure, but to invariance.  The interaction of discrete isometry groups with the main equations of physics, from full quantum mechanics to continuum mechanics, leads to major simplifications.  I will explore this assertion mainly in the context of molecular dynamics, and show that it implies the existence of an invariant manifold of these equations.  It is a time-dependent manifold, but its dependence on time is explicit.  In the case of the translation group, it has dimension 6N, where N is an assignable positive integer.  The form of the manifold is independent of the expression for the atomic forces, and for this reason it relates to some of the main procedures used in experimental science to measure the properties of materials.  But, this manifold is not optimally exploited, and we suggest better experimental methods implied by theory.

Lecture 3: Invariant solutions of the Boltzmann equation.  The invariant manifold of OMD also implies a certain “statistics” of molecular motion, and has implications for statistical physics.  We explore this connection in the context of the Boltzmann equation.  We show that it leads to new explicit far-from-equilibrium solutions of this equation.  The Boltzmann equation inherits exactly the invariant manifold of OMD, but, of course, unlike MD, the solutions are not time-reversible on this manifold.  These solutions have some surprising properties.  For example, Boltzmann’s H-function (minus the entropy) for at least some of these solutions is log (T3/2/ρ) + const., a formula from equilibrium statistical mechanics that is usually interpreted as strictly applicable only to equilibrium, but in this case the temperature T(t) and density ρ(t) are rapidly changing functions of time.

Lecture 4: Special topics on objective structures (OS).   We give surveys of some current lines of research on OS.  a) ODFT: a strategy for density functional theory on OS, allowing one-atom-per-cell calculations of energy of the main structures of nanotechnology, as well as a method for the investigation of phenomena like “flexoelectricity”, b) Ox-ray crystallography: a possible method analogous to x-ray crystallography to determine the detailed atomic structure of nanostructures in which [plane waves/translation group] is replaced by [a different solution of Maxwell’s equations/different isometry group], c) Oself-assembly: a hypothesis that OS are the natural structures for self-assembly, a design strategy, and a stochastic theory for the time to assemble.

Acknowledgments: A. Banerjee, K. Dayal, G. Friesecke, D. Jüstel, H. van Lengerich, S. Müller