arXiv subjects

Hans C. Fogedby

Publications and source records attributed to Hans C. Fogedby.

45 records · Page 3Linked to original sources

arXiv1998

Aspects of the Noisy Burgers Equation

The noisy Burgers equation describing for example the growth of an interface subject to noise is one of the simplest model governing an intrinsically nonequilibrium problem. In one dimension this equation is analyzed by means of the Martin-Siggia-Rose technique. In a canonical formulation the morphology and scaling behavior are accessed by a principle of least action in the weak noise limit. The growth morphology is characterized by a dilute gas of nonlinear soliton modes with gapless dispersion law with exponent z=3/2 and a superposed gas of diffusive modes with a gap. The scaling exponents and a heuristic expression for the scaling function follow from a spectral representation.

cond-mat.stat-mech↗

arXiv1998

Levy flights in quenched random force fields

Levy flights, characterized by the microscopic step index f, are for f<2 (the case of rare events) considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f-2 in the short range case and the critical fall-off exponent 2f-2 in the long range case. By means of a dynamic renormalization group analysis based on the momentum shell integration method, we determine flows, fixed point, and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient. Unlike the case of ordinary Brownian motion in a quenched force field characterized by a single critical dimension or fall-off exponent d=2, two critical dimensions appear in the Levy case. A critical dimension (or fall-off exponent) d=f below which the diffusion coefficient exhibits anomalous scaling behavior, i.e, algebraic spatial behavior and long time tails, and a critical dimension (or fall-off exponent) d=2f-2 below which the force correlations characterized by a non trivial fixed point become relevant. As a general result we find in all cases that the dynamic exponent z, characterizing the mean square displacement, locks onto the Levy index f, independent of dimension and independent of the presence of weak quenched disorder.

cond-mat.stat-mech↗

arXiv1997

Soliton approach to the noisy Burgers equation: Steepest descent method

The noisy Burgers equation in one spatial dimension is analyzed by means of the Martin-Siggia-Rose technique in functional form. In a canonical formulation the morphology and scaling behavior are accessed by mean of a principle of least action in the asymptotic non-perturbative weak noise limit. The ensuing coupled saddle point field equations for the local slope and noise fields, replacing the noisy Burgers equation, are solved yielding nonlinear localized soliton solutions and extended linear diffusive mode solutions, describing the morphology of a growing interface. The canonical formalism and the principle of least action also associate momentum, energy, and action with a soliton-diffusive mode configuration and thus provides a selection criterion for the noise-induced fluctuations. In a ``quantum mechanical'' representation of the path integral the noise fluctuations, corresponding to different paths in the path integral, are interpreted as ``quantum fluctuations'' and the growth morphology represented by a Landau-type quasi-particle gas of ``quantum solitons'' with gapless dispersion and ``quantum diffusive modes'' with a gap in the spectrum. Finally, the scaling properties are dicussed from a heuristic point of view in terms of a``quantum spectral representation'' for the slope correlations. The dynamic eponent z=3/2 is given by the gapless soliton dispersion law, whereas the roughness exponent zeta =1/2 follows from a regularity property of the form factor in the spectral representation. A heuristic expression for the scaling function is given by spectral representation and has a form similar to the probability distribution for Levy flights with index $z$.

cond-mat.stat-mech↗

arXiv1997

Solitons and diffusive modes in the noiseless Burgers equation: Stability analysis

The noiseless Burgers equation in one spatial dimension is analyzed from the point of view of a diffusive evolution equation in terms of nonlinear soliton modes and linear diffusive modes. The transient evolution of the profile is interpreted as a gas of right hand solitons connected by ramp solutions with superposed linear diffusive modes. This picture is supported by a linear stability analysis of the soliton mode. The spectrum and phase shift of the diffusive modes are determined. In the presence of the soliton the diffusive modes develop a gap in the spectrum and are phase-shifted in accordance with Levinson's theorem. The spectrum also exhibits a zero-frequency translation or Goldstone mode associated with the broken translational symmetry.

cond-mat.stat-mech↗

arXiv1997

Morphology and scaling in the noisy Burgers equation: Soliton approach to the strong coupling fixed point

The morphology and scaling properties of the noisy Burgers equation in one dimension are treated by means of a nonlinear soliton approach based on the Martin-Siggia-Rose technique. In a canonical formulation the strong coupling fixed point is accessed by means of a principle of least action in the asymptotic nonperturbative weak noise limit. The strong coupling scaling behaviour and the growth morphology are described by a gas of nonlinear soliton modes with a gapless dispersion law and a superposed gas of linear diffusive modes with a gap. The dynamic exponent is determined by the gapless soliton dispersion law, whereas the roughness exponent and a heuristic expression for the scaling function are given by the form factor in a spectral representation of the interface slope correlation function. The scaling function has the form of a Levy flight distribution.

cond-mat.stat-mech↗

arXiv1997

Monte Carlo Study of Correlations Near the Ground State of the Triangular Antiferromagnetic Ising Model

We study the spin-spin correlation function in or near the T=0 ground state of the antiferromagnetic Ising model on a triangular lattice. At zero temperature its modulation on the sublattices gives rise to two Bragg peaks in the structure factor, and a known expression for the algebraic decay of correlations enables us to examine the form of the diffusive scattering. We do so by means of a comparison between exact results and data calculated using standard Monte Carlo techniques. At non-zero temperatures the finite correlation length alters this form, and we account for the change by proposing a generalisation of the zero temperature pair correlation function. The size dependence of our simulation data is investigated through a novel finite-size scaling analysis where t = exp(-2/T) is used as the temperature parameter.

cond-mat.stat-mech↗

arXiv1995

Continuum limit, Galilean invariance, and solitons in the quantum equivalent of the noisy Burgers equation

A continuum limit of the non-Hermitian spin-1/2 chain, conjectured recently to belong to the universality class of the noisy Burgers or, equivalently, Kardar-Parisi-Zhang equation, is obtained and analyzed. The Galilean invariance of the Burgers equation is explicitly realized in the operator algebra. In the quasi-classical limit we find nonlinear soliton excitations exhibiting the $ω\propto k^z$ dispersion relation with dynamical exponent $z=3/2$.

cond-mat↗

arXiv1994

Langevin equations for continuous time Lévy flights

We consider the combined effects of a power law Lévy step distribution characterized by the step index $f$ and a power law waiting time distribution characterized by the time index $g$ on the long time behavior of a random walker. The main point of our analysis is a formulation in terms of coupled Langevin equations which allows in a natural way for the inclusion of external force fields. In the anomalous case for $f<2$ and $g<1$ the dynamic exponent $z$ locks onto the ratio $f/g$. Drawing on recent results on Lévy flights in the presence of a random force field we also find that this result is {\em independent} of the presence of weak quenched disorder. For $d$ below the critical dimension $d_c=2f-2$ the disorder is {\em relevant}, corresponding to a non trivial fixed point for the force correlation function.

cond-mat↗

arXiv1994

Lévy flights in random environments

We consider Lévy flights characterized by the step index $f$ in a quenched random force field. By means of a dynamic renormalization group analysis we find that the dynamic exponent $z$ for $f<2$ locks onto $f$, independent of dimension and {\em independent} of the presence of weak quenched disorder. The critical dimension, however, depends on the step index $f$ for $f<2$ and is given by $d_c=2f-2$. For $d<d_c$ the disorder is {\em relevant}, corresponding to a non trivial fixed point for the force correlation function.

cond-mat↗