8 research outputs found
Nanoscale Dynamics of Phase Flipping in Water near its Hypothesized Liquid-Liquid Critical Point
Achieving a coherent understanding of the many thermodynamic and dynamic
anomalies of water is among the most important unsolved puzzles in physics,
chemistry, and biology. One hypothesized explanation imagines the existence of
a line of first order phase transitions separating two liquid phases and
terminating at a novel "liquid-liquid" critical point in a region of low
temperature () and high pressure (). Here we analyze a common model of water, the ST2 model, and find
that the entire system flips between liquid states of high and low density.
Further, we find that in the critical region crystallites melt on a time scale
of nanoseconds. We perform a finite-size scaling analysis that accurately
locates both the liquid-liquid coexistence line and its associated
liquid-liquid critical point.Comment: 22 pages, 5 figure
Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"
We review recent results on superintegrable quantum systems in a
two-dimensional Euclidean space with the following properties. They are
integrable because they allow the separation of variables in Cartesian
coordinates and hence allow a specific integral of motion that is a second
order polynomial in the momenta. Moreover, they are superintegrable because
they allow an additional integral of order . Two types of such
superintegrable potentials exist. The first type consists of "standard
potentials" that satisfy linear differential equations. The second type
consists of "exotic potentials" that satisfy nonlinear equations. For , 4
and 5 these equations have the Painlev\'e property. We conjecture that this is
true for all . The two integrals X and Y commute with the Hamiltonian,
but not with each other. Together they generate a polynomial algebra (for any
) of integrals of motion. We show how this algebra can be used to calculate
the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume
"Integrability, Supersymmetry and Coherent States", a volume in honour of
Professor V\'eronique Hussin. arXiv admin note: text overlap with
arXiv:1703.0975