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Kurt Broderix

Publications and source records attributed to Kurt Broderix.

15 recordsLinked to original sources
arXiv2004

Continuous integral kernels for unbounded Schroedinger semigroups and their spectral projections

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of the generating Schroedinger operator also act as Carleman operators with continuous integral kernels. Applications to Schroedinger operators with rather general random scalar potentials include a rigorous justification of an integral-kernel representation of their integrated density of states - a relation frequently used in the physics literature on disordered solids.

math-ph↗
arXiv2002

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.

cond-mat.soft↗
arXiv2001

Normal stresses at the gelation transition

A simple Rouse-type model, generalised to incorporate the effects of chemical crosslinks, is used to obtain a theoretical prediction for the critical behaviour of the normal-stress coefficients $Ψ_{1}$ and $Ψ_{2}$ at the gelation transition. While the exact calculation shows $Ψ_{2}\equiv 0$, a typical result for these types of models, an additional scaling ansatz is used to demonstrate that $Ψ_{1}$ diverges with a critical exponent $\ell = k+z$. Here, $k$ denotes the critical exponent of the shear viscosity and $z$ the exponent governing the divergence of the time scale in the Kohlrausch decay of the shear-stress relaxation function. For crosslinks distributed according to mean-field percolation, this scaling relation yields $\ell =3$, in a accordance with an exact expression for the first normal-stress coefficient based on a replica calculation. Alternatively, using three-dimensional percolation for the crosslink ensemble we find the value $\ell \approx 4.9$. Results on time-dependent normal-stress response are also presented.

cond-mat.stat-mech↗
arXiv2001

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 $k=ϕ-β$, where $ϕ$ is the (first) crossover exponent of random resistor networks, and $β$ is the critical exponent for the gel fraction. We also derive new results on the behaviour of normal stress coefficients.

cond-mat.stat-mech↗
arXiv2001

Energy landscape of a Lennard-Jones liquid: Statistics of stationary points

Molecular dynamics simulations are used to generate an ensemble of saddles of the potential energy of a Lennard-Jones liquid. Classifying all extrema by their potential energy u and number of unstable directions k, a well defined relation k(u) is revealed. The degree of instability of typical stationary points vanishes at a threshold potential energy, which lies above the energy of the lowest glassy minima of the system. The energies of the inherent states, as obtained by the Stillinger-Weber method, approach the threshold energy at a temperature close to the mode-coupling transition temperature Tc.

cond-mat.stat-mech↗
arXiv2000

Stress Relaxation of Near-Critical Gels

The time-dependent stress relaxation for a Rouse model of a crosslinked polymer melt is completely determined by the spectrum of eigenvalues of the connectivity matrix. The latter has been computed analytically for a mean-field distribution of crosslinks. It shows a Lifshitz tail for small eigenvalues and all concentrations below the percolation threshold, giving rise to a stretched exponential decay of the stress relaxation function in the sol phase. At the critical point the density of states is finite for small eigenvalues, resulting in a logarithmic divergence of the viscosity and an algebraic decay of the stress relaxation function. Numerical diagonalization of the connectivity matrix supports the analytical findings and has furthermore been applied to cluster statistics corresponding to random bond percolation in two and three dimensions.

cond-mat.soft↗
arXiv2000

Critical Dynamics of Gelation

Shear relaxation and dynamic density fluctuations are studied within a Rouse model, generalized to include the effects of permanent random crosslinks. We derive an exact correspondence between the static shear viscosity and the resistance of a random resistor network. This relation allows us to compute the static shear viscosity exactly for uncorrelated crosslinks. For more general percolation models, which are amenable to a scaling description, it yields the scaling relation $ k=ϕ-β$ for the critical exponent of the shear viscosity. Here $β$ is the thermal exponent for the gel fraction and $ϕ$ is the crossover exponent of the resistor network. The results on the shear viscosity are also used in deriving upper and lower bounds on the incoherent scattering function in the long-time limit, thereby corroborating previous results.

cond-mat.stat-mech↗
arXiv2000

Saddles on the potential energy landscape of a Lennard-Jones liquid

By means of molecular dynamics simulations, we study the stationary points of the potential energy in a Lennard-Jones liquid, giving a purely geometric characterization of the energy landscape of the system. We find a linear relation between the degree of instability of the stationary points and their potential energy, and we locate the energy where the instability vanishes. This threshold energy marks the border between saddle-dominated and minima-dominated regions of the energy landscape. The temperature where the potential energy of the Stillinger-Weber minima becomes equal to the threshold energy turns out to be very close to the mode-coupling transition temperature.

cond-mat.stat-mech↗
arXiv2000

Continuity properties of Schrödinger semigroups with magnetic fields

The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simon's landmark paper [Bull. Amer. Math. Soc. (N.S.) {\bf 7}, 447--526 (1982)] to non-zero vector potentials and more general configuration spaces.

math-ph↗
arXiv1999

Shear viscosity of a crosslinked polymer melt

We investigate the static shear viscosity on the sol side of the vulcanization transition within a minimal mesoscopic model for the Rouse-dynamics of a randomly crosslinked melt of phantom polymers. We derive an exact relation between the viscosity and the resistances measured in a corresponding random resistor network. This enables us to calculate the viscosity exactly for an ensemble of crosslinks without correlations. The viscosity diverges logarithmically as the critical point is approached. For a more realistic ensemble of crosslinks amenable to the scaling description of percolation, we prove the scaling relation $k=ϕ-β$ between the critical exponent $k$ of the viscosity, the thermal exponent $β$ associated with the gel fraction and the crossover exponent $ϕ$ of a random resistor network.

cond-mat.stat-mech↗
arXiv1997

Dynamical signatures of the vulcanization transition

Dynamical properties of vulcanized polymer networks are addressed via a Rouse-type model that incorporates the effect of permanent random crosslinks. The incoherent intermediate scattering function is computed in the sol and gel phases, and at the vulcanization transition between them. At any nonzero crosslink density within the sol phase Kohlrausch relaxation is found. The critical point is signalled by divergence of the longest time-scale, and at this point the scattering function decays algebraically, whereas within the gel phase it acquires a time-persistent part identified with the gel fraction.

cond-mat.soft↗
arXiv1995

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.

cond-mat↗
arXiv1995

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.

cond-mat↗
arXiv1995

The Fate of Lifshitz Tails in Magnetic Fields

We investigate the integrated density of states of the Schrödinger 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.

cond-mat↗