Complex Line Bundles over Simplicial Complexes and their Applications

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of Andr\'e Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension

Noninvasive Measurement of Dissipation in Colloidal Systems

According to Harada and Sasa [Phys. Rev. Lett. 95, 130602 (2005)], heat production generated in a non-equilibrium steady state can be inferred from measuring response and correlation functions. In many colloidal systems, however, it is a nontrivial task to determine response functions, whereas details about spatial steady state trajectories are easily accessible. Using a simple conditional averaging procedure, we show how this fact can be exploited to reliably evaluate average heat production. We test this method using Brownian dynamics simulations, and apply it to experimental data of an interacting driven colloidal system

Mobility and Diffusion of a Tagged Particle in a Driven Colloidal Suspension

We study numerically the influence of density and strain rate on the diffusion and mobility of a single tagged particle in a sheared colloidal suspension. We determine independently the time-dependent velocity autocorrelation functions and, through a novel method, the response functions with respect to a small force. While both the diffusion coefficient and the mobility depend on the strain rate the latter exhibits a rather weak dependency. Somewhat surprisingly, we find that the initial decay of response and correlation functions coincide, allowing for an interpretation in terms of an 'effective temperature'. Such a phenomenological effective temperature recovers the Einstein relation in nonequilibrium. We show that our data is well described by two expansions to lowest order in the strain rate.Comment: submitted to EP

SPH Simulations of Counterrotating Disk Formation in Spiral Galaxies

We present the results of Smoothed Particle Hydrodynamics (SPH) simulations of the formation of a massive counterrotating disk in a spiral galaxy. The current study revisits and extends (with SPH) previous work carried out with sticky particle gas dynamics, in which adiabatic gas infall and a retrograde gas-rich dwarf merger were tested as the two most likely processes for producing such a counterrotating disk. We report on experiments with a cold primary similar to our Galaxy, as well as a hot, compact primary modeled after NGC 4138. We have also conducted numerical experiments with varying amounts of prograde gas in the primary disk, and an alternative infall model (a spherical shell with retrograde angular momentum). The structure of the resulting counterrotating disks is dramatically different with SPH. The disks we produce are considerably thinner than the primary disks and those produced with sticky particles. The time-scales for counterrotating disk formation are shorter with SPH because the gas loses kinetic energy and angular momentum more rapidly. Spiral structure is evident in most of the disks, but an exponential radial profile is not a natural byproduct of these processes. The infalling gas shells that we tested produce counterrotating bulges and rings rather than disks. The presence of a considerable amount of preexisting prograde gas in the primary causes, at least in the absence of star formation, a rapid inflow of gas to the center and a subsequent hole in the counterrotating disk. In general, our SPH experiments yield stronger evidence to suggest that the accretion of massive counterrotating disks drives the evolution of the host galaxies towards earlier (S0/Sa) Hubble types.Comment: To appear in ApJ. 20 pages LaTex 2-column with 3 tables, 23 figures (GIF) available at this site. Complete gzipped postscript preprint with embedded figures available from http://tarkus.pha.jhu.edu/~thakar/cr3.html (3 Mb

Fluctuation-Dissipation Theorem in Nonequilibrium Steady States

In equilibrium, the fluctuation-dissipation theorem (FDT) expresses the response of an observable to a small perturbation by a correlation function of this variable with another one that is conjugate to the perturbation with respect to \emph{energy}. For a nonequilibrium steady state (NESS), the corresponding FDT is shown to involve in the correlation function a variable that is conjugate with respect to \emph{entropy}. By splitting up entropy production into one of the system and one of the medium, it is shown that for systems with a genuine equilibrium state the FDT of the NESS differs from its equilibrium form by an additive term involving \emph{total} entropy production. A related variant of the FDT not requiring explicit knowledge of the stationary state is particularly useful for coupled Langevin systems. The \emph{a priori} surprising freedom apparently involved in different forms of the FDT in a NESS is clarified.Comment: 6 pages; EPL, in pres

Quantum Invariants, Modular Forms, and Lattice Points II

We study the SU(2) Witten--Reshetikhin--Turaev invariant for the Seifert fibered homology spheres with M-exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 1/2 and 3/2. By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large-N limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in N ->\infinity, is related to the number of integral lattice points inside the M-dimensional tetrahedron

Shell Crossing Singularities in Quasi-Spherical Szekeres Models

We investigate the occurrence of shell crossing singularities in quasi-spherical Szekeres dust models with or without a cosmological constant. We study the conditions for shell crossing singularity both from physical and geometrical point of view and they are in agreement.Comment: 10 latex pages, RevTex style, no figure

Effective Confinement as Origin of the Equivalence of Kinetic Temperature and Fluctuation-Dissipation Ratio in a Dense Shear Driven Suspension

We study response and velocity autocorrelation functions for a tagged particle in a shear driven suspension governed by underdamped stochastic dynamics. We follow the idea of an effective confinement in dense suspensions and exploit a time-scale separation between particle reorganization and vibrational motion. This allows us to approximately derive the fluctuation-dissipation theorem in a "hybrid" form involving the kinetic temperature as an effective temperature and an additive correction term. We show numerically that even in a moderately dense suspension the latter is negligible. We discuss similarities and differences with a simple toy model, a single trapped particle in shear flow