Geometric discretization of the Koenigs nets

We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darboux-type transformations of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. We also investigate the geometry of the discrete Koenigs transformation. In particular we characterize the Koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice.Comment: 17 pages, 2 figures; some spelling and typing errors correcte

Hopping in a Supercooled Lennard-Jones Liquid: Metabasins, Waiting Time Distribution, and Diffusion

We investigate the jump motion among potential energy minima of a Lennard-Jones model glass former by extensive computer simulation. From the time series of minima energies, it becomes clear that the energy landscape is organized in superstructures, called metabasins. We show that diffusion can be pictured as a random walk among metabasins, and that the whole temperature dependence resides in the distribution of waiting times. The waiting time distribution exhibits algebraic decays: $\tau^{-1/2}$ for very short times and $\tau^{-\alpha}$ for longer times, where $\alpha\approx2$ near $T_c$. We demonstrate that solely the waiting times in the very stable basins account for the temperature dependence of the diffusion constant.Comment: to be published in Phys. Rev.

What does the potential energy landscape tell us about the dynamics of supercooled liquids and glasses?

For a model glass-former we demonstrate via computer simulations how macroscopic dynamic quantities can be inferred from a PEL analysis. The essential step is to consider whole superstructures of many PEL minima, called metabasins, rather than single minima. We show that two types of metabasins exist: some allowing for quasi-free motion on the PEL (liquid-like), the others acting as traps (solid-like). The activated, multi-step escapes from the latter metabasins are found to dictate the slowing down of dynamics upon cooling over a much broader temperature range than is currently assumed

Finite-Size Effects in a Supercooled Liquid

We study the influence of the system size on various static and dynamic properties of a supercooled binary Lennard-Jones liquid via computer simulations. In this way, we demonstrate that the treatment of systems as small as N=65 particles yields relevant results for the understanding of bulk properties. Especially, we find that a system of N=130 particles behaves basically as two non-interacting systems of half the size.Comment: Proceedings of the III Workshop on Non Equilibrium Phenomena in Supercooled Fluids, Glasses and Amorphous Materials, Sep 2002, Pis

Time scale for the onset of Fickian diffusion in supercooled liquids

We propose a quantitative measure of a time scale on which Fickian diffusion sets in for supercooled liquids and use Brownian Dynamics computer simulations to determine the temperature dependence of this onset time in a Lennard-Jones binary mixture. The time for the onset of Fickian diffusion ranges between 6.5 and 31 times the $\alpha$ relaxation time (the $\alpha$ relaxation time is the characteristic relaxation time of the incoherent intermediate scattering function). The onset time increases faster with decreasing temperature than the $\alpha$ relaxation time. Mean squared displacement at the onset time increases with decreasing temperature

Slow dynamics of a confined supercooled binary mixture II: Q space analysis

We report the analysis in the wavevector space of the density correlator of a Lennard Jones binary mixture confined in a disordered matrix of soft spheres upon supercooling. In spite of the strong confining medium the behavior of the mixture is consistent with the Mode Coupling Theory predictions for bulk supercooled liquids. The relaxation times extracted from the fit of the density correlator to the stretched exponential function follow a unique power law behavior as a function of wavevector and temperature. The von Schweidler scaling properties are valid for an extended wavevector range around the peak of the structure factor. The parameters extracted in the present work are compared with the bulk values obtained in literature.Comment: 8 pages with 8 figures. RevTeX. Accepted for publication in Phys. Rev.

Time evolution of dynamic propensity in a model glass former. The interplay between structure and dynamics

By means of the isoconfigurational method we calculate the change in the propensity for motion that the structure of a glass-forming system experiences during its relaxation dynamics. The relaxation of such a system has been demonstrated to evolve by means of rapid crossings between metabasins of its potential energy surface (a metabasin being a group of mutually similar, closely related structures which differ markedly from other metabasins), as collectively relaxing units (d-clusters) take place. We now show that the spatial distribution of propensity in the system does not change significantly until one of these d-clusters takes place. However, the occurrence of a d-cluster clearly de-correlates the propensity of the particles, thus ending up with the dynamical influence of the structural features proper of the local metabasin. We also show an important match between particles that participate in d-clusters and that which show high changes in their propensity.Comment: 7 pages, 8 figures, articl

Integrable dynamics of Toda-type on the square and triangular lattices

In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. In this paper we construct other examples of integrable dynamics of Toda-type on the square lattice, as well as on the triangular lattice, as nonlinear symmetries of the discrete Laplace equations on the square and triangular lattices. We also construct the $\tau$ - function formulations and the Darboux-B\"acklund transformations of these novel dynamics.Comment: 22 pages, 4 figure

A geometric interpretation of the spectral parameter for surfaces of constant mean curvature

Considering the kinematics of the moving frame associated with a constant mean curvature surface immersed in S^3 we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S^3. The application of the Sym formula to this linear problem yields constant mean curvature surfaces in E^3. Independently, we show that the Sym formula itself can be derived by an appropriate limiting process R -> infinity.Comment: 12 page