Reconstruction of manifolds in noncommutative geometry

We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.Comment: 67 pages, no figures, Latex; major changes, a new Appendix

Water Evaporation: A Transition Path Sampling Study

We use transition path sampling to study evaporation in the SPC/E model of liquid water. Based on thousands of evaporation trajectories, we characterize the members of the transition state ensemble (TSE), which exhibit a liquid-vapor interface with predominantly negative mean curvature at the site of evaporation. We also find that after evaporation is complete, the distributions of translational and angular momenta of the evaporated water are Maxwellian with a temperature equal to that of the liquid. To characterize the evaporation trajectories in their entirety, we find that it suffices to project them onto just two coordinates: the distance of the evaporating molecule to the instantaneous liquid-vapor interface, and the velocity of the water along the average interface normal. In this projected space, we find that the TSE is well-captured by a simple model of ballistic escape from a deep potential well, with no additional barrier to evaporation beyond the cohesive strength of the liquid. Equivalently, they are consistent with a near-unity probability for a water molecule impinging upon a liquid droplet to condense. These results agree with previous simulations and with some, but not all, recent experiments.Comment: 33 pages, 11 figures. Added appendix on time reversibility, fixed error in Eqs (7)-(10

Fourier analysis on the affine group, quantization and noncompact Connes geometries

We find the Stratonovich-Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.Comment: 37 pages, Latex, uses TikZ package to draw 3 figures. Two new subsections, main results unchange

Orbifolds are not commutative geometries

In this note we show that the crucial orientation condition for commutative geometries fails for the natural spectral triple of an orbifold M/G.Comment: 6 pages, Latex, no figure

Quantifying density fluctuations in volumes of all shapes and sizes using indirect umbrella sampling

Water density fluctuations are an important statistical mechanical observable that is related to many-body correlations, as well as hydrophobic hydration and interactions. Local water density fluctuations at a solid-water surface have also been proposed as a measure of its hydrophobicity. These fluctuations can be quantified by calculating the probability, $P_v(N)$, of observing $N$ waters in a probe volume of interest $v$. When $v$ is large, calculating $P_v(N)$ using molecular dynamics simulations is challenging, as the probability of observing very few waters is exponentially small, and the standard procedure for overcoming this problem (umbrella sampling in $N$) leads to undesirable impulsive forces. Patel et al. [J. Phys. Chem. B, 114, 1632 (2010)] have recently developed an indirect umbrella sampling (INDUS) method, that samples a coarse-grained particle number to obtain $P_v(N)$ in cuboidal volumes. Here, we present and demonstrate an extension of that approach to other basic shapes, like spheres and cylinders, as well as to collections of such volumes. We further describe the implementation of INDUS in the NPT ensemble and calculate $P_v(N)$ distributions over a broad range of pressures. Our method may be of particular interest in characterizing the hydrophobicity of interfaces of proteins, nanotubes and related systems.Comment: 11 pages, 6 figure