Probable Escape Path Research Articles

A dynamical systems approach for most probable escape paths over periodic boundaries

Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler–Lagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The Onsager–Machlup (OM) functional, which is treated as a perturbation of the Friedlin–Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the paper is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.

Change of the Most Probable Escape Path in a Fast-Slow Insect Outbreak System Under Different Noise Amplitude Ratios

Abstract Noise-induced escape from the domain of attraction of a stable state in a fast-slow insect outbreak system is investigated. According to Dannenberg's theory (et al., 2014, “Steering Most Probable Escape Paths by Varying Relative Noise Intensities,” Phys. Rev. Lett., 113(2), p. 020601), only noise amplitude ratio μ will lead to the change of the most probable escape path (MPEP). Therefore, the research emphasis of this paper is to extend their study and discuss the variation of the MPEP in more detail. First, for the case of μ = 1, the MPEP almost moves along the critical manifold. Via projecting the full system onto the critical manifold, a reduced system is obtained, and the action of the MPEP in the full system can be partly evaluated by that in the reduced system. In order to test the accuracy of the computed MPEP, based on the iterative action minimizing method (IAMM), a new relaxation method, which can reduce the central processing unit time, is then presented. Then, as μ converges to zero, an improved analytical method is given, through which a better approximation for the MPEP at the turning point is obtained. And then, when the value of μ is moderate, the MPEP will peel off the critical manifold. To determine the changing point on the critical manifold, an effective numerical algorithm is presented. In brief, a complete investigation on the structural change of the MPEP in a fast-slow insect outbreak system under different noise ratios is given, and the results of the numerical simulation match well with the analytical ones.

Dispersion of the prehistory distribution for non-gradient systems

The most probable escape path can reveal the optimal fluctuation with overwhelming probability for vanishing noise during escape. However, it fails to offer information for the feature of the nearby paths, while the dispersion of the prehistory distribution does. For gradient systems, the dispersion can be obtained via a relaxation method which takes the advantage of the time reversibility of the fluctuation-dissipation relation. For non-gradient systems, due to the breaking of the time-reversal symmetry, the traditional relaxation method cannot be applied. In this paper, we investigate the dispersion of the exit phenomena in the Maier–Stein system for three sets of parameters. For the gradient case, the traditional relaxation method is extended to the 2D situation. For the non-gradient case, we propose a revised version of the relaxation method which relies on the computation of quasipotential. The results are compared with those of Monte Carlo simulation which shows the efficiency of the algorithms.

Probability evolution method for exit location distribution

The exit problem in the framework of the large deviation theory has been a hot topic in the past few decades. The most probable escape path in the weak-noise limit has been clarified by the Freidlin–Wentzell action functional. However, noise in real physical systems cannot be arbitrarily small while noise with finite strength may induce nontrivial phenomena, such as noise-induced shift and noise-induced saddle-point avoidance. Traditional Monte Carlo simulation of noise-induced escape will take exponentially large time as noise approaches zero. The majority of the time is wasted on the uninteresting wandering around the attractors. In this paper, a new method is proposed to decrease the escape simulation time by an exponentially large factor by introducing a series of interfaces and by applying the reinjection on them. This method can be used to calculate the exit location distribution. It is verified by examining two classical examples and is compared with theoretical predictions. The results show that the method performs well for weak noise while may induce certain deviations for large noise. Finally, some possible ways to improve our method are discussed.

Noise induced transitions and topological study of a periodically driven system

Noise induced transitions of an overdamped periodically driven oscillator are investigated theoretically and numerically in the limit of weak noise due to the Freidlin-Wentzell large deviation theory. Heteroclinic trajectories are found to approach the unstable orbit with fluctuational force tending to zeros. The global minimizer of the action functional corresponds to the most probable escape path and it shows a good agreement with statistical results. We then study the origins of singularities from a topological point of view by considering structures of the Lagrangian manifold and action surface. The switching line and cusp point turn out to have physical significance since they may impact the prehistory distributions, making the optimal path invalid.

FLUCTUATIONAL ESCAPE FROM CHAOTIC ATTRACTORS IN MULTISTABLE SYSTEMS

Recent progress towards an understanding of fluctuational escape from chaotic attractors (CAs) is reviewed and discussed in the contexts of both continuous systems and maps. It is shown that, like the simpler case of escape from a regular attractor, a unique most probable escape path (MPEP) is followed from a CA to the boundary of its basin of attraction. This remains true even where the boundary structure is fractal. The importance of the boundary conditions on the attractor is emphasized. It seems that a generic feature of the escape path is that it passes via certain unstable periodic orbits. The problems still remaining to be solved are identified and considered.

No resonant tunneling in standard scalar quantum field theory

We investigate the nature of resonant tunneling in Quantum Field Theory. Following the pioneering work of Banks, Bender and Wu, we describe quantum field theory in terms of infinite dimensional quantum mechanics and utilize the ``Most probable escape path'' (MPEP) as the class of paths which dominate the path integral in the classically forbidden region. Considering a 1+1 dimensional field theory example we show that there are five conditions that any associated bound state in the classically allowed region must satisfy if resonant tunnelling is to occur, and we then proceed to show that it is impossible to satisfy all five conditions simultaneously.

Fluctuational transitions across different kinds of fractal basin boundaries

We study fluctuational transitions in discrete and continuous dynamical systems that have two coexisting attractors in phase space, separated by a fractal basin boundary which may be either locally disconnected or locally connected. Theoretical and numerical evidence is given to show that, in each case, the transition occurs via a unique accessible point on the boundary, both in discrete systems and in flows. The complicated structure of the escape paths inside the locally disconnected fractal basin boundary is determined by a hierarchy of homoclinic points. The interrelation between the mechanism of transitions and the hierarchy is illustrated by consideration of fluctuational transitions in dynamical systems demonstrating "fractal-fractal" basin boundary metamorphosis at some value of a control parameter. The most probable escape path from an attractor, which can be either regular or chaotic, is found for each type of boundary using both statistical analysis of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Fluctuational transitions through a fractal basin boundary.

Fluctuational transitions between two coexisting chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the transition mechanism is determined by a hierarchy of homoclinic points. The most probable escape path from a chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Wave nucleation rate in excitable systems in the low noise limit.

Motivated by recent experiments on intracellular calcium dynamics, we study the general issue of fluctuation-induced nucleation of waves in excitable media. We utilize a stochastic Fitzhugh-Nagumo model for this study, a spatially extended nonpotential pair of equations driven by thermal (i.e., white) noise. The nucleation rate is determined by finding the most probable escape path via minimization of an action related to the deviation of the fields from their deterministic trajectories. Our results pave the way both for studies of more realistic models of calcium dynamics as well as of nucleation phenomena in other nonequilibrium pattern-forming processes.

Multidimensional WKB approximation and the lifetime calculation

A Wentzel-Kramers-Brillouin (WKB) approach for calculating the lifetime of the ground state of two coupled oscillators with the most probable escape path along one of the coordinate axes is suggested. The WKB approximation of the wave function in the neighborhood of this path is obtained by scaling the corresponding variable. An analytic formula for the lifetime is derived and numerically verified. The method is applied to the Henon-Heiles potentials. It is shown that the WKB method can be, in contrast to the numerical ones, easily extended to the problems of an arbitrary number of spatial dimensions.

CHARACTERISTIC TYPES OF EVOLUTION OF NOISE-INDUCED ESCAPE FLUX AT SHORT TIME SCALES

Unlike the quasi-stationary escape flux, the flux on the time scales preceding the quasi-stationary stage may significantly depend on the initial state of the system. We analyze three characteristic initial stages: (i) the stable state of the noise-free system, i.e. the bottom of the potential well; (ii) the non-bottom state with given coordinate and velocity; (iii) a thermalized state. We prove rigorously and and demonstrate in simulations that, on the time scale of a period of eigenoscillation, the flux grows stepwise for cases (i) and (iii), and in an oscillatory manner for case (ii). Different steps/oscillations correspond to different topologies of the most probable escape path.

State after quantum tunneling with gravity

The Wheeler-DeWitt equation is investigated and used to examine a state after a quantum tunneling with gravity. To make arguments definite we treat a discretized version of the Wheeler-DeWitt equation and adopt the WKB method. We expand an Euclidean wave function around an instanton, by using a deviation equation of a vector field tangent to a congruence of instantons. The instanton around which we expand the wave function corresponds to a so-called most probable escape path (MPEP). It is shown that, when the wave function is analytically continued, the corresponding state of physical perturbations is equivalent to the vacuum state determined by positive-frequency mode functions which satisfy appropriate boundary conditions. Thus a quantum field theory is effective to investigate a state after a quantum tunneling with gravity. The effective Lagrangian describing the field theory is obtained by simply reducing the original Lagrangian to a subspace spanned by the physical perturbations. The result of this paper does not depend on the operator ordering and can be applied to all physical perturbations, including gravitational perturbations, around a general MPEP.

Hamilton–Jacobi Theory of Colored Noise Problem and Probability Hole in the Bistable System

The Hamilton–Jacobi theory in nonpotential systems developed by Maier and Stein is applied to colored noise-driven systems. The behavior of the most probable escape path (MPEP) is discussed. Singularity occurs for the MPEP in the large correlation time limit, which is closely related to the well-known probability hole problem. The solution along the MPEP is derived. The behavior near the fixed point is analyzed, and a critical value is found, which is in agreement with that obtained by other authors.

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.

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 asymptotic behavior of fundamental quantities such as the mean escape time. In this paper we present a general technique for analyzing 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 asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path. We also present results on short-time behavior and discuss the possibility of [ital focusing] along the escape path.

Kramers time in bistable potentials

Diffusion in a general bistable potential governed by the Fokker-Planck equation is studied for large t. A simple variational calculation yields a lower bound for the Kramer's time. The method (using a supersymmetric pair of Hamiltonians) applies immediately to diffusion in a potential with a large number of minima and to bistable potentials in higher dimension characterised by a most probable escape path.

Thermally induced escape: The principle of minimum available noise energy.

The average time required for thermally induced escape from a basin of attraction increases exponentially with inverse temperature in proportion to exp(${E}_{A}$/kT) in the limit of low temperature. A minimum principle states that the activation energy ${E}_{A}$ is the minimum available noise energy required to execute a state-space trajectory which takes the system from the attractor of the noise-free system to the boundary of its basin of attraction and that the minimizing trajectory is the most probable low-temperature escape path. This principle is applied to the problem of thermally induced escape from two attractors of the dc-biased Josephson junction, the zero-voltage state and the voltage state, to determine activation energies and most probable escape paths. These two escape problems exemplify the classical case of escape from a potential well and the more general case of escape from an attractor of a nonequilibrium system. Monte Carlo simulations are used to verify the accuracy of the activation energies and most probable escape paths derived from the minimum principle.

Quantum-field transition rates at finite temperatures

The Euclidean solutions (instantons) discovered in field theories seem to be interpretable as tunnelings between vacuums with different winding numbers. The associated transition rate for such tunnelings has been estimated. We study the effect of temperature on the transition rate between quantum-field configurations in thermodynamic equilibrium. We do this by extending the most probable escape path (MPEP) WKB vacuum-tunneling formalism of Bitar and Chang to finite temperatures. Our approach employs elementary results from quantum statistical mechanics. We believe the method offers an easily calculable approximation for quantum-field transition rates at finite temperatures.

An example of quantum tunneling in molecular physics

A few years ago, Banks, Bender and Wu studied the tunneling phenomena involving many degrees of freedoml). They reduced the original question to a variational problem for finding an optimal change of configuration, referred to as the most probable escape path (MPEP) in phase space, which links the initial and the final states. Recently, Coleman gave an interesting interpretation to these MPEP as Euclidean solutions to a trivially related problem2). These methods have since been applied to non-abelian gauge theories for understanding the vacuum tunneling phenomena, and their relation to the Euclidean instanton solutions3). In the following, I wish to provide a simple tunneling process in molecular physics which may be relevant in some chemical or biomolecular systems.