Minimum action method for the Kardar-Parisi-Zhang equation.

We apply a numerical minimum action method derived from the Wentzell-Freidlin theory of large deviations to the Kardar-Parisi-Zhang equation for the height profile of a growing interface. In one dimension we find that the transition pathway between different height configurations is determined by the nucleation and subsequent propagation of facets or steps, corresponding to moving domain walls or growth modes in the underlying noise-driven Burgers equation. This transition scenario is in accordance with recent analytical studies of the one-dimensional Kardar-Parisi-Zhang equation in the asymptotic weak noise limit. We also briefly discuss transitions in two dimensions.

Single DNA denaturation and bubble dynamics.

While the Watson-Crick double-strand is the thermodynamically stable state of DNA in a wide range of temperature and salt conditions, even at physiological conditions local denaturation bubbles may open up spontaneously due to thermal activation. By raising the ambient temperature, titration, or by external forces in single molecule setups bubbles proliferate until full denaturation of the DNA occurs. Based on the Poland-Scheraga model we investigate both the equilibrium transition of DNA denaturation and the dynamics of the denaturation bubbles with respect to recent single DNA chain experiments for situations below, at, and above the denaturation transition. We also propose a new single molecule setup based on DNA constructs with two bubble zones to measure the bubble coalescence and extract the physical parameters relevant to DNA breathing. Finally we consider the interplay between denaturation bubbles and selectively single-stranded DNA binding proteins.

DNA bubble dynamics as a quantum Coulomb problem.

We study the dynamics of denaturation bubbles in double-stranded DNA. Demonstrating that the associated Fokker-Planck equation is equivalent to a Coulomb problem, we derive expressions for the bubble survival distribution W(t). Below Tm, W(t) is associated with the continuum of scattering states of the repulsive Coulomb potential. At Tm, the Coulomb potential vanishes and W(t) assumes a power-law tail with nontrivial dynamic exponents: the critical exponent of the entropy loss factor may cause a finite mean lifetime. Above Tm (attractive potential), the long-time dynamics is controlled by the lowest bound state. Correlations and finite size effects are discussed.

Dynamics of DNA breathing: weak noise analysis, finite time singularity, and mapping onto the quantum Coulomb problem.

We study the dynamics of denaturation bubbles in double-stranded DNA on the basis of the Poland-Scheraga model. We show that long time distributions for the survival of DNA bubbles and the size autocorrelation function can be derived from an asymptotic weak noise approach. In particular, below the melting temperature the bubble closure corresponds to a noisy finite time singularity. We demonstrate that the associated Fokker-Planck equation is equivalent to a quantum Coulomb problem. Below the melting temperature, the bubble lifetime is associated with the continuum of scattering states of the repulsive Coulomb potential; at the melting temperature, the Coulomb potential vanishes and the underlying first exit dynamics exhibits a long time power law tail; above the melting temperature, corresponding to an attractive Coulomb potential, the long time dynamics is controlled by the lowest bound state. Correlations and finite size effects are discussed.

Kardar-Parisi-Zhang equation in the weak noise limit: pattern formation and upper critical dimension.

We extend the previously developed nonperturbative weak noise scheme, applied to the noisy Burgers equation in one dimension, to the Kardar-Parisi-Zhang equation for a growing interface in arbitrary dimensions. By means of the Cole-Hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes. In the Cole-Hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear Schrödinger equation, subsequently boosted to finite velocity by a Galilei transformation. We discuss the dynamics of the pattern formation and, briefly, the superimposed linear modes. Implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector. In the dipole sector we obtain the Hurst exponent H=(3-d)/(4-d) or dynamic exponent Zdip(4-d)/(3-d) for the random walk of growth modes. Below d=3 the dipole growth modes show anomalous diffusion, above d=3 the dipole growth modes freeze. Finally, applying Derrick's theorem based on constrained minimization we show that the upper critical dimension is d=4 in the sense that growth modes cease to exist above this dimension.

Localized growth modes, dynamic textures, and upper critical dimension for the Kardar-Parisi-Zhang equation in the weak-noise limit.

A weak-noise scheme is applied to the Kardar-Parisi-Zhang equation for a growing interface in all dimensions. It is shown that the solutions can be interpreted in terms of a growth morphology of a dynamically evolving texture of localized growth modes with superimposed diffusive modes. By applying Derrick's theorem, it is conjectured that the upper critical dimension is four.

Domain wall propagation and nucleation in a metastable two-level system.

We present a dynamical description and analysis of nonequilibrium transitions in the noisy one-dimensional Ginzburg-Landau equation for an extensive system based on a weak noise canonical phase space formulation of the Freidlin-Wentzel or Martin-Siggia-Rose methods. We derive propagating nonlinear domain wall or soliton solutions of the resulting canonical field equations with superimposed diffusive modes. The transition pathways are characterized by the nucleation and subsequent propagation of domain walls. We discuss the general switching scenario in terms of a dilute gas of propagating domain walls and evaluate the Arrhenius factor in terms of the associated action. We find excellent agreement with recent numerical optimization studies.

Exact solution of a linear molecular motor model driven by two-step fluctuations and subject to protein friction.

We investigate by analytical means the stochastic equations of motion of a linear molecular motor model based on the concept of protein friction. Solving the coupled Langevin equations originally proposed by Mogilner et al. [Phys. Lett. A 237, 297 (1998)], and averaging over both the two-step internal conformational fluctuations and the thermal noise, we present explicit, analytical expressions for the average motion and the velocity-force relationship. Our results allow for a direct interpretation of details of this motor model which are not readily accessible from numerical solutions. In particular, we find that the model is able to predict physiologically reasonable values for the load-free motor velocity and the motor mobility.

Damped finite-time singularity driven by noise.

We consider the combined influence of linear damping and noise on a dynamical finite-time singularity model for a single degree of freedom. We find that the noise effectively resolves the finite-time singularity and replaces it by a first-passage-time distribution or absorbing state distribution with a peak at the singularity and a long time tail. The damping introduces a characteristic cross-over time. In the early time regime the probability distribution and first-passage-time distribution show a power law behavior with scaling exponent depending on the ratio of the nonlinear coupling strength to the noise strength. In the late time regime the behavior is controlled by the damping. The study might be of relevance in the context of hydrodynamics on a nanometer scale, in material physics, and in biophysics.

Correlations, soliton modes, and non-Hermitian linear mode transmutation in the one-dimensional noisy Burgers equation.

Using the previously developed canonical phase space approach applied to the noisy Burgers equation in one dimension, we discuss in detail the growth morphology in terms of nonlinear soliton modes and superimposed linear modes. We moreover analyze the non-Hermitian character of the linear mode spectrum and the associated dynamical pinning, and mode transmutation from diffusive to propagating behavior induced by the solitons. We discuss the anomalous diffusion of growth modes, switching and pathways, correlations in the multisoliton sector, and in detail the correlations and scaling properties in the two-soliton sector.

Power laws and stretched exponentials in a noisy finite-time-singularity model.

We discuss the influence of white noise on a generic dynamical finite-time-singularity model for a single degree of freedom. We find that the noise effectively resolves the finite-time-singularity and replaces it by a first-passage-time or absorbing state distribution with a peak at the singularity and a long time tail exhibiting power law or stretched exponential behavior. The study might be of relevance in the context of hydrodynamics on a nanometer scale, in material physics, and in biophysics.

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. Finally, we show that growing steps perform an anomalous random walk with dynamic exponent z=3/2.