Towards finite-dimensional gelation

We consider the gelation of particles which are permanently connected by random crosslinks, drawn from an ensemble of finite-dimensional continuum percolation. To average over the randomness, we apply the replica trick, and interpret the replicated and crosslink-averaged model as an effective molecular fluid. A Mayer-cluster expansion for moments of the local static density fluctuations is set up. The simplest non-trivial contribution to this series leads back to mean-field theory. The central quantity of mean-field theory is the distribution of localization lengths, which we compute for all connectivities. The highly crosslinked gel is characterized by a one-to-one correspondence of connectivity and localization length. Taking into account higher contributions in the Mayer-cluster expansion, systematic corrections to mean-field can be included. The sol-gel transition shifts to a higher number of crosslinks per particle, as more compact structures are favored. The critical behavior of the model remains unchanged as long as finite truncations of the cluster expansion are considered. To complete the picture, we also discuss various geometrical properties of the crosslink network, e.g. connectivity correlations, and relate the studied crosslink ensemble to a wider class of ensembles, including the Deam-Edwards distribution.Comment: 18 pages, 4 figures, version to be published in EPJ

Lowest Landau level broadened by a Gaussian random potential with an arbitrary correlation length: An efficient continued-fraction approach

For an electron in the plane subjected to a perpendicular constant magnetic field and a homogeneous Gaussian random potential with a Gau{ss}ian covariance function we approximate the averaged density of states restricted to the lowest Landau level. To this end, we extrapolate the first 9 coefficients of the underlying continued fraction consistently with the coefficients' high-order asymptotics. We thus achieve the first reliable extension of Wegner's exact result [Z. Phys. B {\bf 51}, 279 (1983)] for the delta-correlated case to the physically more relevant case of a non-zero correlation length.Comment: 9 pages ReVTeX, three figure

Thermal Equilibrium with the Wiener Potential: Testing the Replica Variational Approximation

We consider the statistical mechanics of a classical particle in a one-dimensional box subjected to a random potential which constitutes a Wiener process on the coordinate axis. The distribution of the free energy and all correlation functions of the Gibbs states may be calculated exactly as a function of the box length and temperature. This allows for a detailed test of results obtained by the replica variational approximation scheme. We show that this scheme provides a reasonable estimate of the averaged free energy. Furthermore our results shed more light on the validity of the concept of approximate ultrametricity which is a central assumption of the replica variational method.Comment: 6 pages, 1 file LaTeX2e generating 2 eps-files for 2 figures automaticall

Anomalous stress relaxation in random macromolecular networks

Within the framework of a simple Rouse-type model we present exact analytical results for dynamical critical behaviour on the sol side of the gelation transition. The stress-relaxation function is shown to exhibit a stretched-exponential long-time decay. The divergence of the static shear viscosity is governed by the critical exponent $k=\phi -\beta$, where $\phi$ is the (first) crossover exponent of random resistor networks, and $\beta$ is the critical exponent for the gel fraction. We also derive new results on the behaviour of normal stress coefficients.Comment: 13 pages, 6 figures; contribution to the proceedings of the Minerva International Workshop on Frontiers In The Physics Of Complex Systems (25-28 March 2001) - to appear in a special issue of Physica

The Fate of Lifshitz Tails in Magnetic Fields

We investigate the integrated density of states of the Schr\"odinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.Comment: 19 pages LaTe

Energy Landscape and Overlap Distribution of Binary Lennard-Jones Glasses

We study the distribution of overlaps of glassy minima, taking proper care of residual symmetries of the system. Ensembles of locally stable, low lying glassy states are efficiently generated by rapid cooling from the liquid phase which has been equilibrated at a temperature $T_{run}$. Varying $T_{run}$, we observe a transition from a regime where a broad range of states are sampled to a regime where the system is almost always trapped in a metastable glassy state. We do not observe any structure in the distribution of overlaps of glassy minima, but find only very weak correlations, comparable in size to those of two liquid configurations.Comment: 7 pages, 5 figures, uses europhys-style. Minor notational changes, typos correcte