Large time dynamics and aging of a polymer chain in a random potential

We study the out-of-equilibrium large time dynamics of a gaussian polymer chain in a quenched random potential. The dynamics studied is a simple Langevin dynamics commonly referred to as the Rouse model. The equations for the two-time correlation and response function are derived within the gaussian variational approximation. In order to implement this approximation faithfully, we employ the supersymmetric representation of the Martin-Siggia-Rose dynamical action. For a short ranged correlated random potential the equations are solved analytically in the limit of large times using certain assumptions concerning the asymptotic behavior. Two possible dynamical behaviors are identified depending upon the time separation- a stationary regime and an aging regime. In the stationary regime time translation invariance holds and so is the fluctuation dissipation theorem. The aging regime which occurs for large time separations of the two-time correlation functions is characterized by history dependence and the breakdown of certain equilibrium relations. The large time limit of the equations yields equations among the order parameters that are similar to the equations obtained in the statics using replicas. In particular the aging solution corresponds to the broken replica solution. But there is a difference in one equation that leads to important consequences for the solution. The stationary regime corresponds to the motion of the polymer inside a local minimum of the random potential, whereas in the aging regime the polymer hops between different minima. As a byproduct we also solve exactly the dynamics of a chain in a random potential with quadratic correlations.Comment: 21 pages, RevTeX

Interpolation of the Josephson interaction in highly anisotropic superconductors from a solution of the two dimensional sine-Gordon equation

In this paper we solve numerically the two dimensional elliptic sine-Gordon equation with appropriate boundary conditions. These boundary conditions are chosen to correspond to the Josephson interaction between two adjacent pancakes belonging to the same flux-line in a highly anisotropic superconductor. An extrapolation is obtained between the regimes of low and high separation of the pancakes. The resulting formula is a better candidate for use in numerical simulations than previously derived formulas.Comment: 18 pages, 9 figure

Molecular Dynamics of pancake vortices with realistic interactions: Observing the vortex lattice melting transition

In this paper we describe a version of London Langevin molecular dynamics simulations that allows for investigations of the vortex lattice melting transition in the highly anisotropic high-temperature superconductor material Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. We include the full electromagnetic interaction as well as the Josephson interaction among pancake vortices. We also implement periodic boundary conditions in all directions, including the z-axis along which the magnetic field is applied. We show how to implement flux cutting and reconnection as an analog to permutations in the multilevel Monte Carlo scheme and demonstrate that this process leads to flux entanglement that proliferates in the vortex liquid phase. The first-order melting transition of the vortex lattice is observed to be in excellent agreement with previous multilevel Monte Carlo simulations.Comment: 4 figure

Replica field theory for a polymer in random media

In this paper we revisit the problem of a (non self-avoiding) polymer chain in a random medium which was previously investigated by Edwards and Muthukumar (EM). As noticed by Cates and Ball (CB) there is a discrepancy between the predictions of the replica calculation of EM and the expectation that in an infinite medium the quenched and annealed results should coincide (for a chain that is free to move) and a long polymer should always collapse. CB argued that only in a finite volume one might see a ``localization transition'' (or crossover) from a stretched to a collapsed chain in three spatial dimensions. Here we carry out the replica calculation in the presence of an additional confining harmonic potential that mimics the effect of a finite volume. Using a variational scheme with five variational parameters we derive analytically for d<4 the result R~(g |ln \mu|)^{-1/(4-d)} ~(g lnV)^{-1/(4-d)}, where R is the radius of gyration, g is the strength of the disorder, \mu is the spring constant associated with the confining potential and V is the associated effective volume of the system. Thus the EM result is recovered with their constant replaced by ln(V) as argued by CB. We see that in the strict infinite volume limit the polymer always collapses, but for finite volume a transition from a stretched to a collapsed form might be observed as a function of the strength of the disorder. For d<2 and for large V>V'~exp[g^(2/(2-d))L^((4-d)/(2-d))] the annealed results are recovered and R~(Lg)^(1/(d-2)), where L is the length of the polymer. Hence the polymer also collapses in the large L limit. The 1-step replica symmetry breaking solution is crucial for obtaining the above results.Comment: Revtex, 32 page

Differential systems associated with tableaux over Lie algebras

We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiveness for the associated systems and their prolongations is addressed; examples are discussed.Comment: 16 pages; to appear in: "Symmetries and Overdetermined Systems of Partial Differential Equations" (M. Eastwood and W. Miller, Jr., eds.), IMA Volumes in Mathematics and Its Applications, Springer-Verlag, New Yor

Urban Catholic Elementary Schools: What are the Governance Models?

The closure of nearly half of Catholic elementary schools in the United States since the 1960s has led to the development of many innovative initiatives to stabilize, strengthen, and sustain urban Catholic elementary education. Improving school governance models has been a common agenda of these efforts. This study examined the governance models in use by urban Catholic elementary schools across the United States. Seven major governance models for urban Catholic elementary schools were identified and studied using structured interviews and document analysis. An eighth model, faith-inspired charter schools, is presented as one alternative to a Catholic school. The variety of governance models demonstrates innovation in response to the plight of urban Catholic elementary schools across the country. Common trends across the models are discussed. In short, traditional governance approaches are giving way to more strategic, data-supported models that have the potential t

Localization of a polymer in random media: Relation to the localization of a quantum particle

In this paper we consider in detail the connection between the problem of a polymer in a random medium and that of a quantum particle in a random potential. We are interested in a system of finite volume where the polymer is known to be {\it localized} inside a low minimum of the potential. We show how the end-to-end distance of a polymer which is free to move can be obtained from the density of states of the quantum particle using extreme value statistics. We give a physical interpretation to the recently discovered one-step replica-symmetry-breaking solution for the polymer (Phys. Rev. E{\bf 61}, 1729 (2000)) in terms of the statistics of localized tail states. Numerical solutions of the variational equations for chains of different length are performed and compared with quenched averages computed directly by using the eigenfunctions and eigenenergies of the Schr\"odinger equation for a particle in a one-dimensional random potential. The quantities investigated are the radius of gyration of a free gaussian chain, its mean square distance from the origin and the end-to-end distance of a tethered chain. The probability distribution for the position of the chain is also investigated. The glassiness of the system is explained and is estimated from the variance of the measured quantities.Comment: RevTex, 44 pages, 13 figure

Directed polymers on a Cayley tree with spatially correlated disorder

In this paper we consider directed walks on a tree with a fixed branching ratio K at a finite temperature T. We consider the case where each site (or link) is assigned a random energy uncorrelated in time, but correlated in the transverse direction i.e. within the shell. In this paper we take the transverse distance to be the hierarchical ultrametric distance, but other possibilities are discussed. We compute the free energy for the case of quenched disorder and show that there is a fundamental difference between the case of short range spatial correlations of the disorder which behaves similarly to the non-correlated case considered previously by Derrida and Spohn and the case of long range correlations which has a totally different overlap distribution which approaches a single delta function about q=1 for large L, where L is the length of the walk. In the latter case the free energy is not extensive in L for the intermediate and also relevant range of L values, although in the true thermodynamic limit extensivity is restored. We identify a crossover temperature which grows with L, and whenever T<T_c(L) the system is always in the low temperature phase. Thus in the case of long-ranged correlation as opposed to the short-ranged case a phase transition is absent.Comment: 23 pages, 1 figure, standard latex. J. Phys. A, accepted for publicatio

The Stellar Populations and Evolution of Lyman Break Galaxies

Using deep near-IR and optical observations of the HDF-N from the HST NICMOS and WFPC2 and from the ground, we examine the spectral energy distributions (SEDs) of Lyman break galaxies (LBGs) at 2.0 < z < 3.5. The UV-to-optical rest-frame SEDs of the galaxies are much bluer than those of present-day spiral and elliptical galaxies, and are generally similar to those of local starburst galaxies with modest amounts of reddening. We use stellar population synthesis models to study the properties of the stars that dominate the light from LBGs. Under the assumption that the star-formation rate is continuous or decreasing with time, the best-fitting models provide a lower bound on the LBG mass estimates. LBGs with ``L*'' UV luminosities are estimated to have minimum stellar masses ~ 10^10 solar masses, or roughly 1/10th that of a present-day L* galaxy. By considering the effects of a second component of maximally-old stars, we set an upper bound on the stellar masses that is ~ 3-8 times the minimum estimate. We find only loose constraints on the individual galaxy ages, extinction, metallicities, initial mass functions, and prior star-formation histories. We find no galaxies whose SEDs are consistent with young (< 10^8 yr), dust-free objects, which suggests that LBGs are not dominated by ``first generation'' stars, and that such objects are rare at these redshifts. We also find that the typical ages for the observed star-formation events are significantly younger than the time interval covered by this redshift range (~ 1.5 Gyr). From this, and from the relative absence of candidates for quiescent, non-star-forming galaxies at these redshifts in the NICMOS data, we suggest that star formation in LBGs may be recurrent, with short duty cycles and a timescale between star-formation events of < 1 Gyr. [Abridged]Comment: LaTeX, 37 pages, 21 figures. Accepted for publication in the Astrophysical Journa