Solitons in the noisy Burgers equation

We investigate numerically the coupled diffusion-advective type field equations originating from the canonical phase space approach to the noisy Burgers equation or the equivalent Kardar-Parisi-Zhang equation in one spatial dimension. The equations support stable right hand and left hand solitons and in the low viscosity limit a long-lived soliton pair excitation. We find that two identical pair excitations scatter transparently subject to a size dependent phase shift and that identical solitons scatter on a static soliton transparently without a phase shift. The soliton pair excitation and the scattering configurations are interpreted in terms of growing step and nucleation events in the interface growth profile. In the asymmetrical case the soliton scattering modes are unstable presumably toward multi soliton production and extended diffusive modes, signalling the general non-integrability of the coupled field equations. Finally, we have shown that growing steps perform anomalous random walk with dynamic exponent z=3/2 and that the nucleation of a tip is stochastically suppressed with respect to plateau formation.Comment: 11 pages Revtex file, including 15 postscript-figure

Canonical phase space approach to the noisy Burgers equation

Presenting a general phase approach to stochastic processes we analyze in particular the Fokker-Planck equation for the noisy Burgers equation and discuss the time dependent and stationary probability distributions. In one dimension we derive the long-time skew distribution approaching the symmetric stationary Gaussian distribution. In the short time regime we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer-replica model and asymmetric exclusion model results.Comment: 4 pages, Revtex file, submitted to Phys. Rev. Lett. a reference has been added and a few typos correcte

Nonequilibrium dynamics of a growing interface

A growing interface subject to noise is described by the Kardar-Parisi-Zhang equation or, equivalently, the noisy Burgers equation. In one dimension this equation is analyzed by means of a weak noise canonical phase space approach applied to the associated Fokker-Planck equation. The growth morphology is characterized by a gas of nonlinear soliton modes with superimposed linear diffusive modes. We also discuss the ensuing scaling properties.Comment: 14 pages, 11 figures, conference proceeding; a few corrections have been adde

Patterns in the Kardar-Parisi-Zhang equation

We review a recent asymptotic weak noise approach to the Kardar-Parisi-Zhang equation for the kinetic growth of an interface in higher dimensions. The weak noise approach provides a many body picture of a growing interface in terms of a network of localized growth modes. Scaling in 1d is associated with a gapless domain wall mode. The method also provides an independent argument for the existence of an upper critical dimension.Comment: 8 pages revtex, 4 eps figure

Random Resistor-Diode Networks and the Crossover from Isotropic to Directed Percolation

By employing the methods of renormalized field theory we show that the percolation behavior of random resistor-diode networks near the multicritical line belongs to the universality class of isotropic percolation. We construct a mesoscopic model from the general epidemic process by including a relevant isotropy-breaking perturbation. We present a two-loop calculation of the crossover exponent $\phi$. Upon blending the $\epsilon$-expansion result with the exact value $\phi =1$ for one dimension by a rational approximation, we obtain for two dimensions $\phi = 1.29\pm 0.05$. This value is in agreement with the recent simulations of a two-dimensional random diode network by Inui, Kakuno, Tretyakov, Komatsu, and Kameoka, who found an order parameter exponent $\beta$ different from those of isotropic and directed percolation. Furthermore, we reconsider the theory of the full crossover from isotropic to directed percolation by Frey, T\"{a}uber, and Schwabl and clear up some minor shortcomings.Comment: 24 pages, 2 figure

The excited-state structure, vibrations, lifetimes, and nonradiative dynamics of jet-cooled 1-methylcytosine

We have investigated the S0 → S1 UV vibronic spectrum and time-resolved S1 state dynamics of jet-cooled amino-keto 1-methylcytosine (1MCyt) using two-color resonant two-photon ionization, UV/UV holeburning and depletion spectroscopies, as well as nanosecond and picosecond timeresolved pump/delayed ionization measurements. The experimental study is complemented with spin-component-scaled second-order coupled-cluster and multistate complete active space second order perturbation ab initio calculations. Above the weak electronic origin of 1MCyt at 31 852 cm−1 about 20 intense vibronic bands are observed. These are interpreted as methyl group torsional transitions coupled to out-of-plane ring vibrations, in agreement with the methyl group rotation and out-of-plane distortions upon 1ππ∗ excitation predicted by the calculations. The methyl torsion and ν′1 (butterfly) vibrations are strongly coupled, in the S1 state. The S0 → S1 vibronic spectrum breaks off at a vibrational excess energy Eexc ∼ 500 cm−1, indicating that a barrier in front of the ethylene-type S1 S0 conical intersection is exceeded, which is calculated to lie at Eexc = 366 cm−1. The S1 S0 internal conversion rate constant increases from kIC = 2 · 109 s−1 near the S1(v = 0) level to 1 · 1011 s−1 at Eexc = 516 cm−1. The 1ππ∗ state of 1MCyt also relaxes into the lower-lying triplet T1 (3ππ∗) state by intersystem crossing (ISC); the calculated spin-orbit coupling (SOC) value is 2.4 cm−1. The ISC rate constant is 10–100 times lower than kIC; it increases from kISC = 2 · 108 s−1 near S1(v = 0) to kISC = 2 · 109 s−1 at Eexc = 516 cm−1. The T1 state energy is determined from the onset of the time-delayed photoionization efficiency curve as 25 600 ± 500 cm−1. The T2 (3nπ∗) state lies >1500 cm−1 above S1(v = 0), so S1 T2 ISC cannot occur, despite the large SOC parameter of 10.6 cm−1. An upper limit to the adiabatic ionization energy of 1MCyt is determined as 8.41 ± 0.02 eV. Compared to cytosine, methyl substitution at N1 lowers the adiabatic ionization energy by ≥0.32 eV and leads to a much higher density of vibronic bands in the S0 → S1 spectrum. The effect of methylation on the radiationless decay to S0 and ISC to T1 is small, as shown by the similar break-off of the spectrum and the similar computed mechanismsThis research has been supported by the Schweiz. Nationalfonds (Grant Nos. 121993 and 132540), the Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) from Catalonia (Spain) (Grant No. 2014SGR1202), the Ministerio de Economía y Competividad (MINECO) from Spain (Grant No. CTQ2015-69363-P), and the National Natural Science Foundation of China (Grant No. 21303007

Canonical phase space approach to the noisy Burgers equation: Probability distributions

We present a canonical phase space approach to stochastic systems described by Langevin equations driven by white noise. Mapping the associated Fokker-Planck equation to a Hamilton-Jacobi equation in the nonperturbative weak noise limit we invoke a {\em principle of least action} for the determination of the probability distributions. We apply the scheme to the noisy Burgers and KPZ equations and discuss the time-dependent and stationary probability distributions. In one dimension we derive the long-time skew distribution approaching the symmetric stationary Gaussian distribution. In the short-time region we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer-replica and asymmetric exclusion model results. We also comment on the distribution in higher dimensions.Comment: 18 pages Revtex file, including 8 eps-figures, submitted to Phys. Rev.

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.Comment: 27 pages, Revtex file, 17 figures in ps format attached, submitted to Phys. Rev.

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 $z$.Comment: 30 pages, Revtex file, 14 figures, to be submitted to Phys. Rev.

Instances and connectors : issues for a second generation process language

This work is supported by UK EPSRC grants GR/L34433 and GR/L32699Over the past decade a variety of process languages have been defined, used and evaluated. It is now possible to consider second generation languages based on this experience. Rather than develop a second generation wish list this position paper explores two issues: instances and connectors. Instances relate to the relationship between a process model as a description and the, possibly multiple, enacting instances which are created from it. Connectors refers to the issue of concurrency control and achieving a higher level of abstraction in how parts of a model interact. We believe that these issues are key to developing systems which can effectively support business processes, and that they have not received sufficient attention within the process modelling community. Through exploring these issues we also illustrate our approach to designing a second generation process language.Postprin