Probable Transition Path Research Articles

Convergence analysis for minimum action methods coupled with a finite difference method

Abstract The minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform mesh, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $\theta $-method for SDEs with small noise in terms of large deviations. Numerical experiments are reported to verify the theoretical results.

The most probable transition paths of stochastic dynamical systems: a sufficient and necessary characterisation

The most probable transition paths (MPTPs) of a stochastic dynamical system are the global minimisers of the Onsager–Machlup action functional and can be described by a necessary but not sufficient condition, the Euler–Lagrange (EL) equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devoted to showing a sufficient and necessary characterisation for the MPTPs of stochastic dynamical systems with Brownian noise. We prove that, under appropriate conditions, the MPTPs are completely determined by a first-order ordinary differential equation. The equivalence is established by showing that the Onsager–Machlup action functional of the original system can be derived from the corresponding Markovian bridge process. For linear stochastic systems and the nonlinear Hongler’s model, the first-order differential equations determining the MPTPs are shown analytically to imply the EL equations of the Onsager–Machlup functional. For general nonlinear systems, the determining first-order differential equations can be approximated, in a short time or for the small noise case. Some numerical experiments are presented to illustrate our results.

Vapor-liquid phase transition in fluctuating hydrodynamics: The most probable transition path and its computation

In this paper, we consider transition events between metastable states in fluid systems modeled by two-phase Navier-Stokes equations with thermal noise. The vapor-liquid phase transition which occurs during the vapor condensation is such an example. We propose a numerical method to compute the transition pathway. We first approximate the continuum hydrodynamics model by the smoothed particle hydrodynamics (SPH). In the SPH formulation, the dynamics of the system take the form of Langevin equations. This allows us to derive conditions that characterize the most probable transition pathway. Then we compute the most probable transition pathway by solving these conditions in the space of mass and momentum density fields using the string method. The force, which is needed to evolve the string in the space of the density fields, is computed by discretizing the deterministic model using the finite difference method. The numerical method is applied to study the condensation of vapor in a periodic cell and the Cassie-Wenzel transition on pillared solid substrates.

Most probable transition paths in piecewise-smooth stochastic differential equations

We develop a path integral framework for determining most probable paths for a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin–Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an n-dimensional system with a switching manifold in the drift that forms an (n−1)−dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use Γ−convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin–Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Abstract. We study the impact of applying stochastic forcing to the Ghil–Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and α-stable Lévy – α∈(0,2) – noise laws, examine the statistics of transition times, and estimate the most probable transition paths. While the Gaussian noise case – used here as a reference – has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, both in terms of mathematical theory and heuristics, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to −α, in apparent agreement with rigorous results obtained for additive noise in a related – yet different – reaction–diffusion equation and in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space, and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition paths cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance.

Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force

The effects of stochastic perturbations and periodic excitations on the eutrophicated lake ecosystem are explored. Unlike the existing work in detecting early warning signals, this paper presents the most probable transition paths to characterize the regime shifts. The most probable transition paths are obtained by minimizing the Freidlin–Wentzell (FW) action functional and Onsager–Machlup (OM) action functional, respectively. The most probable path shows the movement trend of the lake eutrophication system under noise excitation, and describes the global transition behavior of the system. Under the excitation of Gaussian noise, the results show that the stability of the eutrophic state and the oligotrophic state has different results from two perspectives of potential well and the most probable transition paths. Under the excitation of Gaussian white noise and periodic force, we find that the transition occurs near the nearest distance between the stable periodic solution and the unstable periodic solution.

Most probable transitions from metastable to oscillatory regimes in a carbon cycle system

Global climate changes are related to the ocean’s store of carbon. We study a carbonate system of the upper ocean, which has metastable and oscillatory regimes, under small random fluctuations. We calculate the most probable transition path via a geometric minimum action method in the context of the large deviation theory. By examining the most probable transition paths from metastable to oscillatory regimes for various external carbon input rates, we find two different transition patterns, which gives us an early warning sign for the dramatic change in the carbonate state of the ocean.

Landscape and kinetic path quantify critical transitions in epithelial-mesenchymal transition

Epithelial-mesenchymal transition (EMT), a basic developmental process that might promote cancer metastasis, has been studied from various perspectives. Recently, the early warning theory has been used to anticipate critical transitions in EMT from mathematical modeling. However, the underlying mechanisms of EMT involving complex molecular networks remain to be clarified. Especially, how to quantify the global stability and stochastic transition dynamics of EMT and what the underlying mechanism for early warning theory in EMT is remain to be fully clarified. To address these issues, we constructed a comprehensive gene regulatory network model for EMT and quantified the corresponding potential landscape. The landscape for EMT displays multiple stable attractors, which correspond to E, M, and some other intermediate states. Based on the path-integral approach, we identified the most probable transition paths of EMT, which are supported by experimental data. Correspondingly, the results of transition actions demonstrated that intermediate states can accelerate EMT, consistent with recent studies. By integrating the landscape and path with early warning concept, we identified the potential barrier height from the landscape as a global and more accurate measure for early warning signals to predict critical transitions in EMT. The landscape results also provide an intuitive and quantitative explanation for the early warning theory. Overall, the landscape and path results advance our mechanistic understanding of dynamical transitions and roles of intermediate states in EMT, and the potential barrier height provides a new, to our knowledge, measure for critical transitions and quantitative explanations for the early warning theory.

Estimating the most probable transition time for stochastic dynamical systems

This work is devoted to the investigation of the most probable transition time between metastable states for stochastic dynamical systems with non-vanishing Brownian noise. Instead of minimizing the Onsager–Machlup action functional, we examine the maximum probability that the solution process of the system stays in a neighbourhood (or a tube) of a transition path, in order to characterize the most probable transition path. We first establish the exponential decay lower bound and a power law decay upper bound for the maximum of this probability. Based on these estimates, we further derive the lower and upper bounds for the most probable transition time, under suitable conditions. Finally, we illustrate our results in simple stochastic dynamical systems, and highlight the relation with some relevant works.

Dominant Reaction Pathways by Quantum Computing.

Characterizing thermally activated transitions in high-dimensional rugged energy surfaces is a very challenging task for classical computers. Here, we develop a quantum annealing scheme to solve this problem. First, the task of finding the most probable transition paths in configuration space is reduced to a shortest-path problem defined on a suitable weighted graph. Next, this optimization problem is mapped into finding the ground state of a generalized Ising model. A finite-size scaling analysis suggests this task may be solvable efficiently by a quantum annealing machine. Our approach leverages on the quantized nature of qubits to describe transitions between different system's configurations. Since it does not involve any lattice space discretization, it paves the way towards future biophysical applications of quantum computing based on realistic all-atom models.

Vapor-Liquid Phase Transition in Fluctuating Hydrodynamics: The Most Probable Transition Path and its Computation

Vapor-Liquid Phase Transition in Fluctuating Hydrodynamics: The Most Probable Transition Path and its Computation

Gamma-Limit of the Onsager--Machlup Functional on the Space of Curves

The Onsager--Machlup (OM) and Freidlin--Wentzell (FW) functionals are both widely used in seeking the most probable transition path between two states for a diffusion process. We study the relation between these two functionals on the space of curves. We prove that the $\Gamma$-limit of the OM functional on the space of curves is the geometric form of the FW functional in a proper time scale $T=T(\epsilon)$ as $\epsilon\to 0$. For other time scales, the limit of the OM functional is infinite in general. We then introduce the concept of renormalization for the OM functional and prove that the $\Gamma$-limit of the renormalized OM functional is the geometric FW functional in any time scale on the space of curves.

The graph limit of the minimizer of the Onsager-Machlup functional and its computation

The Onsager-Machlup (OM) functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise. However, it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to infinity. This hinders the interpretation of the results obtained by minimizing the OM functional. We provide a new perspective on this issue. Under mild conditions, we show that although the infimum of the OM functional becomes unbounded when T goes to infinity, the sequence of minimizers does contain convergent subsequences on the space of curves. The graph limit of this minimizing subsequence is an extremal of the abbreviated action functional, which is related to the OM functional via the Maupertuis principle with an optimal energy. We further propose an energy-climbing geometric minimization algorithm (EGMA) which identifies the optimal energy and the graph limit of the transition path simultaneously. This algorithm is successfully applied to several typical examples in rare event studies. Some interesting comparisons with the Freidlin-Wentzell action functional are also made.

The maximum likelihood climate change for global warming under the influence of greenhouse effect and Lévy noise.

An abrupt climatic transition could be triggered by a single extreme event, and an α-stable non-Gaussian Lévy noise is regarded as a type of noise to generate such extreme events. In contrast with the classic Gaussian noise, a comprehensive approach of the most probable transition path for systems under α-stable Lévy noise is still lacking. We develop here a probabilistic framework, based on the nonlocal Fokker-Planck equation, to investigate the maximum likelihood climate change for an energy balance system under the influence of greenhouse effect and Lévy fluctuations. We find that a period of the cold climate state can be interrupted by a sharp shift to the warmer one due to larger noise jumps with low frequency. Additionally, the climate change for warming 1.5°C under an enhanced greenhouse effect generates a steplike growth process. These results provide important insights into the underlying mechanisms of abrupt climate transitions triggered by a Lévy process.

The Onsager–Machlup function as Lagrangian for the most probable path of a jump-diffusion process

This work is devoted to deriving the Onsager–Machlup function for a class of stochastic dynamical systems under (non-Gaussian) Lévy noise as well as (Gaussian) Brownian noise, and examining the corresponding most probable paths. This Onsager–Machlup function is the Lagrangian giving the most probable path connecting metastable states for jump-diffusion processes. This is done by applying the Girsanov transformation for measures induced by jump-diffusion processes. Moreover, we have found this Lagrangian function is consistent with the result in the special case of diffusion processes. Finally, we apply this new Onsager–Machlup function to investigate dynamical behaviors analytically and numerically in several examples. These include the transitions from one metastable state to another metastable state in a double-well system, with numerical experiments illustrating most probable transition paths for various noise parameters.

Characterization of the most probable transition paths of stochastic dynamical systems with stable Lévy noise

This work is devoted to the investigation of the most probable transition paths for stochastic dynamical systems with either symmetric -stable Lévy motion or Brownian motion. For stochastic dynamical systems with Brownian motion, minimizing an action functional is a general method to determine the most probable transition path. We have developed a method based on path integrals to obtain the most probable transition path of stochastic dynamical systems with either symmetric -stable Lévy motion () or Brownian motion. Furthermore, we have shown that the most probable path can be characterized by a deterministic dynamical system.

An Improved Adaptive Minimum Action Method for the Calculation of Transition Path in Non-Gradient Systems

The minimum action method (MAM) is to calculate the most probable transition path in randomly perturbed stochastic dynamics, based on the idea of action minimization in the path space. The accuracy of the numerical path between different metastable states usually suffers from the clustering near fixed points. The adaptive minimum action method (aMAM) solves this problem by relocating image points equally along arc-length with the help of moving mesh strategy. However, when the time interval is large, the images on the path may still be locally trapped around the transition state in a tangle, due to the singularity of the relationship between arc-length and time at the transition state. Additionally, in most non-gradient dynamics, the tangent direction of the path is not continuous at the transition state so that a geometric corner forms, which brings extra challenges for the aMAM. In this note, we improve the aMAM by proposing a better monitor function that does not contain the numerical approximation of derivatives, and taking use of a generalized scheme of the Euler-Lagrange equation to solve the minimization problem, so that both the path-tangling problem and the non-smoothness in parametrizing the curve do not exist. To further improve the accuracy, we apply the Weighted Essentially non-oscillatory (WENO) method for the interpolation to achieve better performance. Numerical examples are presented to demonstrate the advantages of our new method.

Minimum energy paths for conformational changes of viral capsids.

In this work we study conformational changes of viral capsids using techniques of large deviations theory for stochastic differential equations. The viral capsid is a model of a complex system in which many units-the proteins forming the capsomers-interact by weak forces to form a structure with exceptional mechanical resistance. The destabilization of such a structure is interesting both, per se, since it is related either to infection or maturation processes and because it yields insights into the stability of complex structures in which the constitutive elements interact by weak attractive forces. We focus here on a simplified model of a dodecahedral viral capsid and assume that the capsomers are rigid plaquettes with one degree of freedom each. We compute the most probable transition path from the closed capsid to the final configuration using minimum energy paths and discuss the stability of intermediate states.

Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond.

Motivated by the famous Waddington's epigenetic landscape metaphor in developmental biology, biophysicists and applied mathematicians made different proposals to construct the landscape for multi-stable complex systems. We aim to summarize and elucidate the relationships among these theories from a mathematical point of view. We systematically investigate and compare three different but closely related realizations in the recent literature: the Wang's potential landscape theory from steady state distribution of stochastic differential equations (SDEs), the Freidlin-Wentzell quasi-potential from the large deviation theory, and the construction through SDE decomposition and A-type integral. We revisit that the quasi-potential is the zero noise limit of the potential landscape, and the potential function in the third proposal coincides with the quasi-potential. We compare the difference between local and global quasi-potential through the viewpoint of exchange of limit order for time and noise amplitude. We argue that local quasi-potentials are responsible for getting transition rates between neighboring stable states, while the global quasi-potential mainly characterizes the residence time of the states as the system reaches stationarity. The difference between these two is prominent when the transitivity property is broken. The most probable transition path by minimizing the Onsager-Machlup or Freidlin-Wentzell action functional is also discussed. As a consequence of the established connections among different proposals, we arrive at the novel result which guarantees the existence of SDE decomposition while denies its uniqueness in general cases. It is, therefore, clarified that the A-type integral is more appropriate to be applied to the decomposed SDEs rather than its primitive form as believed by previous researchers. Our results contribute to a deeper understanding of landscape theories for biological systems.

Model the nonlinear instability of wall-bounded shear flows as a rare event: a study on two-dimensional Poiseuille flow

In this work, we study the nonlinear instability of two-dimensional (2D) wall-bounded shear flows from the large deviation point of view. The main idea is to consider the Navier–Stokes equations perturbed by small noise in force and then examine the noise-induced transitions between the two coexisting stable solutions due to the subcritical bifurcation. When the amplitude of the noise goes to zero, the Freidlin–Wentzell (F–W) theory of large deviations defines the most probable transition path in the phase space, which is the minimizer of the F–W action functional and characterizes the development of the nonlinear instability subject to small random perturbations. Based on such a transition path we can define a critical Reynolds number for the nonlinear instability in the probabilistic sense. Then the action-based stability theory is applied to study the 2D Poiseuille flow in a short channel.