Asymptotic Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point $S$. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at $H$. If $\mu<1$ then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the $O(\epsilon^{\mu/2})$ length scale near $H$. If $\mu>1$ it is generically asymptotic to a distribution on the $O(\epsilon^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$, in the $\epsilon\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$\epsilon$ exponential asymptotics of the mean first exit time is generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy requests to [email protected] or [email protected]

Noise-Activated Escape from a Sloshing Potential Well

We treat the noise-activated escape from a one-dimensional potential well of an overdamped particle, to which a periodic force of fixed frequency is applied. We determine the boundary layer behavior, and the physically relevant length scales, near the oscillating well top. We show how stochastic behavior near the well top generalizes the behavior first determined by Kramers, in the case without forcing. Both the case when the forcing dies away in the weak noise limit, and the case when it does not, are examined. We also discuss the relevance of various scaling regimes to recent optical trap experiments.Comment: 9 pages, no figures, REVTeX, expanded versio

The Effect of Focusing and Caustics on Exit Phenomena in Systems Lacking Detailed Balance

We study the trajectories followed by a particle subjected to weak noise when escaping from the domain of attraction of a stable fixed point. If detailed balance is absent, a _focus_ may occur along the most probable exit path, leading to a breakdown of symmetry (if present). The exit trajectory bifurcates, and the exit location distribution may become `skewed' (non-Gaussian). The weak-noise asymptotics of the mean escape time are strongly affected. Our methods extend to the study of skewed exit location distributions in stochastic models without symmetry.Comment: REVTEX macros (latest version). Two accompanying PS figures, one of which is large (over 600K unpacked

[3+2] Fragmentation of a Pentaphosphido Ligand by Cyanide

The activation of white phosphorus (P4) by transition‐metal complexes has been studied for several decades, but the functionalization and release of the resulting (organo)phosphorus ligands has rarely been achieved. Herein we describe the formation of rare diphosphan‐1‐ide anions from a P5 ligand by treatment with cyanide. Cobalt diorganopentaphosphido complexes have been synthesized by a stepwise reaction sequence involving a low‐valent diimine cobalt complex, white phosphorus, and diorganochlorophosphanes. The reactions of the complexes with tetraalkylammonium or potassium cyanide afford a cyclotriphosphido cobaltate anion 5 and 1‐cyanodiphosphan‐1‐ide anions [R2PPCN]− (6‐R). The molecular structure of a related product 7 suggests a novel reaction mechanism, where coordination of the cyanide anion to the cobalt center induces a ligand rearrangement. This is followed by nucleophilic attack of a second cyanide anion at a phosphorus atom and release of the P2 fragment

A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem

We consider the overdamped limit of two-dimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double well system are varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. In this paper we quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our analysis relies on the development of a new scaling theory, which yields `critical exponents' describing weak-noise behavior near the saddle, at the bifurcation point.Comment: LaTeX, 60 pages, 24 Postscript figures. Uses epsf macros to include the figures. A file in `uufiles' format containing the figures is separately available at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/figures.uu and a Postscript version of the whole paper (figures included) is available at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/paperF.p

The Geometry of Most Probable Trajectories in Noise-Driven Dynamical Systems

This paper presents a heuristic derivation of a geometric minimum action method that can be used to determine most-probable transition paths in noise-driven dynamical systems. Particular attention is focused on systems that violate detailed balance, and the role of the stochastic vorticity tensor is emphasized. The general method is explored through a detailed study of a two-dimensional quadratic shear flow which exhibits bifurcating most-probable transition pathways.Comment: 8 pages, 7 figure

The Escape Problem for Irreversible Systems

The problem of noise-induced escape from a metastable state arises in physics, chemistry, biology, systems engineering, and other areas. The problem is well understood when the underlying dynamics of the system obey detailed balance. When this assumption fails many of the results of classical transition-rate theory no longer apply, and no general method exists for computing the weak-noise asymptotics of fundamental quantities such as the mean escape time. In this paper we present a general technique for analysing the weak-noise limit of a wide range of stochastically perturbed continuous-time nonlinear dynamical systems. We simplify the original problem, which involves solving a partial differential equation, into one in which only ordinary differential equations need be solved. This allows us to resolve some old issues for the case when detailed balance holds. When it does not hold, we show how the formula for the mean escape time asymptotics depends on the dynamics of the system along the most probable escape path. We also present new results on short-time behavior and discuss the possibility of focusing along the escape path.Comment: 24 pages, APS revtex macros (version 2.1) now available from PBB via `get oldrevtex.sty

Review of Microfluidic Devices and Imaging Techniques for Fluid Flow Study in Porous Geomaterials

Understanding transport phenomena and governing mechanisms of different physical and chemical processes in porous media has been a critical research area for decades. Correlating fluid flow behaviour at the micro-scale with macro-scale parameters, such as relative permeability and capillary pressure, is key to understanding the processes governing subsurface systems, and this in turn allows us to improve the accuracy of modelling and simulations of transport phenomena at a large scale. Over the last two decades, there have been significant developments in our understanding of pore-scale processes and modelling of complex underground systems. Microfluidic devices (micromodels) and imaging techniques, as facilitators to link experimental observations to simulation, have greatly contributed to these achievements. Although several reviews exist covering separately advances in one of these two areas, we present here a detailed review integrating recent advances and applications in both micromodels and imaging techniques. This includes a comprehensive analysis of critical aspects of fabrication techniques of micromodels, and the most recent advances such as embedding fibre optic sensors in micromodels for research applications. To complete the analysis of visualization techniques, we have thoroughly reviewed the most applicable imaging techniques in the area of geoscience and geo-energy. Moreover, the integration of microfluidic devices and imaging techniques was highlighted as appropriate. In this review, we focus particularly on four prominent yet very wide application areas, namely &ldquo;fluid flow in porous media&rdquo;, &ldquo;flow in heterogeneous rocks and fractures&rdquo;, &ldquo;reactive transport, solute and colloid transport&rdquo;, and finally &ldquo;porous media characterization&rdquo;. In summary, this review provides an in-depth analysis of micromodels and imaging techniques that can help to guide future research in the in-situ visualization of fluid flow in porous media