878 research outputs found

    Smoluchowski-Kramers approximation in the case of variable friction

    Full text link
    We consider the small mass asymptotics (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a modification of the Smoluchowski-Kramers approximation. Some applications of the Smoluchowski-Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered.Comment: already publishe

    On second order elliptic equations with a small parameter

    Full text link
    The Neumann problem with a small parameter (1ϵL0+L1)uϵ(x)=f(x)forxG,.uϵγϵ(x)G=0(\dfrac{1}{\epsilon}L_0+L_1)u^\epsilon(x)=f(x) \text{for} x\in G, .\dfrac{\partial u^\epsilon}{\partial \gamma^\epsilon}(x)|_{\partial G}=0 is considered in this paper. The operators L0L_0 and L1L_1 are self-adjoint second order operators. We assume that L0L_0 has a non-negative characteristic form and L1L_1 is strictly elliptic. The reflection is with respect to inward co-normal unit vector γϵ(x)\gamma^\epsilon(x). The behavior of limϵ0uϵ(x)\lim\limits_{\epsilon\downarrow 0}u^\epsilon(x) is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.Comment: 28 pages, 1 figure, revised versio

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

    Full text link
    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 XtX_{t} in R\mathbb{R} that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator DvDuD_{v}D_{u}, where vv and uu are two strictly increasing functions, vv is right continuous and uu is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is ϵDvDu\epsilon D_{v}D_{u} where 0<ϵ10<\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 DvDuD_{v}D_{u}. We apply our results to the problem of wave front propagation for these type of reaction-diffusion equations.Comment: 23 page

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

    Full text link
    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

    Full text link
    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

    Full text link
    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

    Numerical computation of rare events via large deviation theory

    Get PDF
    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

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

    Full text link
    According to the Smoluchowski-Kramers approximation, the solution of the equation μq¨tμ=b(qtμ)q˙tμ+Σ(qtμ)W˙t,q0μ=q,q˙0μ=p{\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 q˙t=b(qt)+Σ(qt)W˙t,q0=q\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
    corecore