arXiv subjects

Georg Foltin

Publications and source records attributed to Georg Foltin.

8 recordsLinked to original sources

arXiv2004

The morphology of nodal lines-random waves versus percolation

In this paper we investigate the properties of nodal structures in random wave fields, and in particular we scrutinize their recently proposed connection with short-range percolation models. We propose a measure which shows the difference between monochromatic random waves, which are characterized by long-range correlations, and Gaussian fields with short-range correlations, which are naturally assumed to be better modelled by percolation theory. We also study the relevance of the quantities which we compute to the probability that nodal lines are in the vicinity of a given reference line.

nlin.CD↗

arXiv2003

The distribution of extremal points of Gaussian scalar fields

We consider the signed density of the extremal points of (two-dimensional) scalar fields with a Gaussian distribution. We assign a positive unit charge to the maxima and minima of the function and a negative one to its saddles. At first, we compute the average density for a field in half-space with Dirichlet boundary conditions. Then we calculate the charge-charge correlation function (without boundary). We apply the general results to random waves and random surfaces. Furthermore, we find a generating functional for the two-point function. Its Legendre transform is the integral over the scalar curvature of a 4-dimensional Riemannian manifold.

math-ph↗

arXiv2003

Counting nodal domains

We consider the nodal domains of Gaussian random waves in two dimensions. We present a method to calculate the distribution of the number of nodal domains and the average connectivity with the help of auxiliary Potts-spins. An analytical approach could be helpful to decide whether the pattern of nodal domains belongs to the universality class of short-ranged percolation. This is not completely evident since we find (weak) long-ranged correlations between distant avoided nodal intersections.

nlin.CD↗

arXiv2003

Signed zeros of Gaussian vector fields-density, correlation functions and curvature

We calculate correlation functions of the (signed) density of zeros of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann-Cartan or Riemannian manifold. As an application, we discuss one- and two-point functions. The zeros of a two-dimensional Gaussian vector field model the distribution of topological defects in the high-temperature phase of two-dimensional systems with orientational degrees of freedom, such as superfluid films, thin superconductors and liquid crystals.

cond-mat.stat-mech↗

arXiv2001

The crumpling transition of membranes driven by quantum fluctuations in a D=epsilon expansion

We consider a D-dimensional fluid membrane in a D+1-dimensional embedding space, subject to quantum fluctuations. The corresponding action is invariant under coordinate transformations and depends only on the shape of the membrane and its variation, neglecting tangential degrees of freedom. We calculate the resulting field theory to one loop order in a D=epsilon expansion and find a quantum transition even at T=0.

cond-mat.soft↗

arXiv2001

An alternative field theory for the Kosterlitz-Thouless transition

We extend a Gaussian model for the internal electrical potential of a two-dimensional Coulomb gas by a non-Gaussian measure term, which singles out the physically relevant configurations of the potential. The resulting Hamiltonian, expressed as a functional of the internal potential, has a surprising large-scale limit: The additional term simply counts the number of maxima and minima of the potential. The model allows for a transparent derivation of the divergence of the correlation length upon lowering the temperature down to the Kosterlitz-Thouless transition point.

cond-mat.stat-mech↗

arXiv2000

Spreading and annihilating particles: surprises from mean-field theory

We study the pairwise annihilation process $A+A\to$ inert of a number of random walkers, which originally are localized in a small region in space. The size of the colony and the typical distance between particles increases with time and, consequently, the reaction rate goes down. In the long time limit the spatial density profile becomes scale invariant. The mean-field approximation of this scenario bears some surprises. It predicts an upper critical dimension $d_c=2$, with logarithmic corrections at the critical dimension and nontrivial scaling behavior for d<2. Based on an exact solution of the one dimensional system we conjecture that the mean-field {\it exponents} are in fact correct even below the upper critical dimension, down to d=1, while the corresponding scaling function that describes the spatial density profile, changes for d<2 due to the stochastic fluctuations.

cond-mat.stat-mech↗

arXiv2000

Orientational order on curved surfaces - the high temperature region

We study orientational order, subject to thermal fluctuations, on a fixed curved surface. We derive, in particular, the average density of zeros of Gaussian distributed vector fields on a closed Riemannian manifold. Results are compared with the density of disclination charges obtained from a Coulomb gas model. Our model describes the disordered state of two dimensional objects with orientational degrees of freedom, such as vector ordering in Langmuir monolayers and lipid bilayers above the hexatic to fluid transition.

cond-mat.soft↗