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

A nonperturbative weak noise scheme is applied to the Kardar-Parisi-Zhang equation for a growing interface in all dimensions. It is shown that the growth morphology can be interpreted in terms of a dynamically evolving texture of localized growth modes with superimposed diffusive modes. Applying Derrick's theorem it is conjectured that the upper critical dimension is four.Comment: 10 pages in revtex and 2 figures in eps, a few typos correcte

Pathways of activated escape in periodically modulated systems

We investigate dynamics of activated escape in periodically modulated systems. The trajectories followed in escape form diffusion broadened tubes, which are periodically repeated in time. We show that these tubes can be directly observed and find their shape. Quantitatively, the tubes are characterized by the distribution of trajectories that, after escape, pass through a given point in phase space for a given modulation phase. This distribution may display several peaks separated by the modulation period. Analytical results agree with the results of simulations of a model Brownian particle in a modulated potential

On stochasticity in nearly-elastic systems

Nearly-elastic model systems with one or two degrees of freedom are considered: the system is undergoing a small loss of energy in each collision with the "wall". We show that instabilities in this purely deterministic system lead to stochasticity of its long-time behavior. Various ways to give a rigorous meaning to the last statement are considered. All of them, if applicable, lead to the same stochasticity which is described explicitly. So that the stochasticity of the long-time behavior is an intrinsic property of the deterministic systems.Comment: 35 pages, 12 figures, already online at Stochastics and Dynamic

Large Deviations Principle for a Large Class of One-Dimensional Markov Processes

We study the large deviations principle for one dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process $X_{t}$ in $\mathbb{R}$ that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator $D_{v}D_{u}$, where $v$ and $u$ are two strictly increasing functions, $v$ is right continuous and $u$ is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is $\epsilon D_{v}D_{u}$ where $0<\epsilon\ll 1$. This result generalizes the classical large deviations results for a large class of one dimensional "classical" stochastic processes. Moreover, we consider reaction-diffusion equations governed by a generalized operator $D_{v}D_{u}$. We apply our results to the problem of wave front propagation for these type of reaction-diffusion equations.Comment: 23 page

Numerical computation of rare events via large deviation theory

An overview of rare events algorithms based on large deviation theory (LDT) is presented. It covers a range of numerical schemes to compute the large deviation minimizer in various setups, and discusses best practices, common pitfalls, and implementation trade-offs. Generalizations, extensions, and improvements of the minimum action methods are proposed. These algorithms are tested on example problems which illustrate several common difficulties which arise e.g. when the forcing is degenerate or multiplicative, or the systems are infinite-dimensional. Generalizations to processes driven by non-Gaussian noises or random initial data and parameters are also discussed, along with the connection between the LDT-based approach reviewed here and other methods, such as stochastic field theory and optimal control. Finally, the integration of this approach in importance sampling methods using e.g. genealogical algorithms is explored

Activated escape of periodically modulated systems

The rate of noise-induced escape from a metastable state of a periodically modulated overdamped system is found for an arbitrary modulation amplitude $A$. The instantaneous escape rate displays peaks that vary with the modulation from Gaussian to strongly asymmetric. The prefactor $\nu$ in the period-averaged escape rate depends on $A$ nonmonotonically. Near the bifurcation amplitude $A_c$ it scales as $\nu\propto (A_c-A)^{\zeta}$. We identify three scaling regimes, with $\zeta = 1/4, -1$, and 1/2

Resonant symmetry lifting in a parametrically modulated oscillator

We study a parametrically modulated oscillator that has two stable states of vibrations at half the modulation frequency $\omega_F$. Fluctuations of the oscillator lead to interstate switching. A comparatively weak additional field can strongly affect the switching rates, because it changes the switching activation energies. The change is linear in the field amplitude. When the additional field frequency $\omega_d$ is $\omega_F/2$, the field makes the populations of the vibrational states different thus lifting the states symmetry. If $\omega_d$ differs from $\omega_F/2$, the field modulates the state populations at the difference frequency, leading to fluctuation-mediated wave mixing. For an underdamped oscillator, the change of the activation energy displays characteristic resonant peaks as a function of frequency

A Note on the Smoluchowski-Kramers Approximation for the Langevin Equation with Reflection

According to the Smoluchowski-Kramers approximation, the solution of the equation ${\mu}\ddot{q}^{\mu}_t=b(q^{\mu}_t)-\dot{q}^{\mu}_t+{\Sigma}(q^{\mu}_t)\dot{W}_t, q^{\mu}_0=q, \dot{q}^{\mu}_0=p$ converges to the solution of the equation $\dot{q}_t=b(q_t)+{\Sigma}(q_t)\dot{W}_t, q_0=q$ as {\mu}->0. We consider here a similar result for the Langevin process with elastic reflection on the boundary.Comment: 14 pages, 2 figure

Poisson-noise induced escape from a metastable state

We provide a complete solution of the problems of the probability distribution and the escape rate in Poisson-noise driven systems. It includes both the exponents and the prefactors. The analysis refers to an overdamped particle in a potential well. The results apply for an arbitrary average rate of noise pulses, from slow pulse rates, where the noise acts on the system as strongly non-Gaussian, to high pulse rates, where the noise acts as effectively Gaussian