Convex polytopes and quantization of symplectic manifolds

Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered.

Dynamical hierarchy in transition states: Why and how does a system climb over the mountain?

How a reacting system climbs through a transition state during the course of a reaction has been an intriguing subject for decades. Here we present and quantify a technique to identify and characterize local invariances about the transition state of an N-particle Hamiltonian system, using Lie canonical perturbation theory combined with microcanonical molecular dynamics simulation. We show that at least three distinct energy regimes of dynamical behavior occur in the region of the transition state, distinguished by the extent of their local dynamical invariance and regularity. Isomerization of a six-atom Lennard–Jones cluster illustrates this: up to energies high enough to make the system manifestly chaotic, approximate invariants of motion associated with a reaction coordinate in phase space imply a many-body dividing hypersurface in phase space that is free of recrossings even in a sea of chaos. The method makes it possible to visualize the stable and unstable invariant manifolds leading to and from the transition state, i.e., the reaction path in phase space, and how this regularity turns to chaos with increasing total energy of the system. This, in turn, illuminates a new type of phase space bottleneck in the region of a transition state that emerges as the total energy and mode coupling increase, which keeps a reacting system increasingly trapped in that region.

Multicomponent diffusion in polymeric liquids.

It is shown how the phase-space kinetic theory of polymeric liquid mixtures leads to a set of extended Maxwell-Stefan equations describing multicomponent diffusion. This expression reduces to standard results for dilute solutions and for undiluted polymers. The polymer molecules are modeled as flexible bead-spring structures. To obtain the Maxwell-Stefan equations, the usual expression for the hydrodynamic drag force on a bead, used in previous kinetic theories, must be replaced by a new expression that accounts explicitly for bead-bead interactions between different molecules.

A sampling problem in molecular dynamics simulations of macromolecules.

Correlations in low-frequency atomic displacements predicted by molecular dynamics simulations on the order of 1 ns are undersampled for the time scales currently accessible by the technique. This is shown with three different representations of the fluctuations in a macromolecule: the reciprocal space of crystallography using diffuse x-ray scattering data, real three-dimensional Cartesian space using covariance matrices of the atomic displacements, and the 3N-dimensional configuration space of the protein using dimensionally reduced projections to visualize the extent to which phase space is sampled.

Neural mechanisms underlying binocular fusion and stereopsis: Position vs. phase

The visual system utilizes binocular disparity to discriminate the relative depth of objects in space. Since the striate cortex is the first site along the central visual pathways at which signals from the left and right eyes converge onto a single neuron, encoding of binocular disparity is thought to begin in this region. There are two possible mechanisms for encoding binocular disparity through simple cells in the striate cortex: a difference in receptive field (RF) position between the two eyes (RF position disparity) and a difference in RF profile between the two eyes (RF phase disparity). Although there have been studies supporting each of the two encoding mechanisms, both mechanisms have not been examined in a single study. Therefore, the relative roles of the two mechanisms have not been determined. To address this issue, we have mapped left and right eye RFs of simple cells in the cat’s striate cortex using binary m-sequence noise, and then we have estimated RF position and phase disparities. We find that RF position disparities are generally limited to small values that are not sufficient to encode large binocular disparities. In contrast, RF phase disparities cover a wide range of binocular disparities and exhibit dependencies on orientation and spatial frequency in a manner expected for a mechanism that encodes binocular disparity. These results indicate that binocular disparity is mainly encoded through RF phase disparity. However, RF position disparity may play a significant role for cells with high spatial frequency selectivity, which are constrained to small RF phase disparities.