Minimum Action Method Research Articles

Unveiling the Phase Diagram and Reaction Paths of the Active Model B with the Deep Minimum Action Method.

Nonequilibrium phase transitions are notably difficult to analyze because their mechanisms depend on the system's dynamics in a complex way due to the lack of time-reversal symmetry. To complicate matters, the system's steady-state distribution is unknown in general. Here, the phase diagram of the active Model B is computed with a deep neural network implementation of the geometric minimum action method (gMAM). This approach unveils the unconventional reaction paths and nucleation mechanism in dimensions 1, 2, and 3, by which the system switches between the homogeneous and inhomogeneous phases in the binodal region. Our main findings are (i)the mean time to escape the phase-separated state is (exponentially) extensive in the system size L, but it increases nonmonotonically with L in dimension 1; (ii)the mean time to escape the homogeneous state is always finite, in line with the recent work of Cates and Nardini [Phys. Rev. Lett. 130, 098203 (2023)PRLTAO0031-900710.1103/PhysRevLett.130.098203]; (iii)at fixed L, the active term increases the stability of the homogeneous phase, eventually destroying the phase separation in the binodal for large but finite systems. Our results are particularly relevant for active matter systems in which the number of constituents hardly goes beyond 10^{7} and where finite-size effects matter.

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 likely transition path for a class of distribution-dependent stochastic systems

Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, the action functional does not involve the solution of the skeleton equation which describes the unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.

Stochastic transitions: Paths over higher energy barriers can dominate in the early stages.

The time evolution of many physical, chemical, and biological systems can be modeled by stochastic transitions between the minima of the potential energy surface describing the system of interest. We show that in cases where there are two (or more) possible pathways that the system can take, the time available for the transition to occur is crucially important. The well-known results of the reaction rate theory for determining the rates of transitions apply in the long-time limit. However, at short times, the system can, instead, choose to pass over higher energy barriers with a much higher probability, as long as the distance to travel in phase space is shorter. We construct two simple models to illustrate this general phenomenon. We also apply a version of the geometric minimum action method algorithm of Vanden-Eijnden and Heymann [J. Chem. Phys. 128, 061103 (2008)] to determine the most likely path at both short and long times.

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.

A Minimum Action Method for Dynamical Systems with Constant Time Delays

In this work, we construct a minimum action method for dynamical systems with constant time delays. The minimum action method (MAM) plays an important role in seeking the most probable transition pathway induced by small noise. There exist two formulations of the minimum action method: one is the geometric formulation based on the Maupertuis principle, and the other one is the temporal formulation. The geometric formulation relies on the conservation of Hamiltonian corresponding to the Freidlin--Wentzell action functional. For systems with time delays, the Hamiltonian does not conserve due to the explicit dependence on the time delay, which implies that the geometric MAM is not applicable. We work with the temporal formulation of MAM for problems with time delays. By defining an auxiliary path, we remove the optimization with respect to time through the optimal linear time scaling. The pointwise correspondence between the auxiliary path and the delayed transition path is dealt with by a penalty term included into the action functional. The action functional is then discretized by the finite element method, and strategies for $h$-adaptive mesh refinement have been developed. Numerical examples have been presented to demonstrate the effectiveness of our algorithm.

Stochastic dynamics of an active particle escaping from a potential well.

Active matter systems are driven out of equilibrium by the energy directly supplied at the level of constituent active particles that are self-propelled. We consider a model for an active particle in a potential well, characterized by an active velocity with a constant magnitude but a random orientation subject to white noises. We are interested in the escape of the active particle from the potential well in multiple-dimensional space. We investigate two distinct optimal paths, namely, the shortest arrival-time path and the most probable path, by using the analytical and numerical techniques from optimal control and rare event modeling. In particular, we elucidate the relationship between these optimal paths and the reachable set using the Hamiltonian dynamics for the shortest arrival-time path and the geometric minimum action method for the most probable path, respectively. Numerical results are presented by applying these techniques to a two-dimensional double-well potential.

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions.

We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su(2;C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external time-dependent magnetic field that specifies the su(2;C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.

Numerical computation of rare events via large deviation theory.

An overview of rare event 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 tradeoffs. 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.

Quasi-Potential Calculation and Minimum Action Method for Limit Cycle

We study the noise-induced escape from a stable limit cycle of a non-gradient dynamical system driven by a small additive noise. The fact that the optimal transition path in this case is infinitely long imposes a severe numerical challenge to resolve it in the minimum action method. We first consider the landscape of the quasi-potential near the limit cycle, which characterizes the minimal cost of the noise to drive the system far away form the limit cycle. We derive and compute the quadratic approximation of this quasi-potential near the limit cycle in the form of a positive definite solution to a matrix-valued periodic Riccati differential equation on the limit cycle. We then combine this local approximation in the neighborhood of the limit cycle with the minimum action method applied outside of the neighborhood. The neighborhood size is selected to be compatible with the path discretization error. By several numerical examples, we show that this strategy effectively improves the minimum action method to compute the spiral optimal escape path from limit cycles in various systems.

Computation of minimum action paths of the stochastic nonlinear Schrödinger equation with dissipation

Using the geometric minimum action method, we compute minimizers of the Freidlin–Wentzell functional for the dissipative linear and nonlinear Schrödinger equation. For the particular case of transitions between solitary waves of different amplitudes, we discuss the relationship of the minimizer of the PDE model to the minimizer of a finite-dimensional reduction.

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.

Geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems.

Optimal control of nanomagnets has become an urgent problem for the field of spintronics as technological tools approach thermodynamically determined limits of efficiency. In complex, fluctuating systems, such as nanomagnetic bits, finding optimal protocols is challenging, requiring detailed information about the dynamical fluctuations of the controlled system. We provide a physically transparent derivation of a metric tensor for which the length of a protocol is proportional to its dissipation. This perspective simplifies nonequilibrium optimization problems by recasting them in a geometric language. We then describe a numerical method, an instance of geometric minimum action methods, that enables computation of geodesics even when the number of control parameters is large. We apply these methods to two models of nanomagnetic bits: a Landau-Lifshitz-Gilbert description of a single magnetic spin controlled by two orthogonal magnetic fields, and a two-dimensional Ising model in which the field is spatially controlled. These calculations reveal nontrivial protocols for bit erasure and reversal, providing important, experimentally testable predictions for ultra-low-power computing.

A dynamic-solver–consistent minimum action method: With an application to 2D Navier–Stokes equations

This paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method (MAM) for a spatially extended system when employing the large deviation principle (LDP) to study the effects of small random perturbations. A dynamic solver is used to approximate the unperturbed system, and a minimum action method is used to approximate the LDP, which corresponds to solving an Euler–Lagrange equation related to but more complicated than the unperturbed system. We will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the LDP, based on which we propose to define both the dynamic solver and the MAM on the same approximation space for spatial discretization. The semi-discrete LDP can then be regarded as the exact LDP of the semi-discrete unperturbed system, which is a finite-dimensional ODE system. We achieve this methodology for the two-dimensional Navier–Stokes equations using a divergence-free approximation space. The method developed can be used to study the nonlinear instability of wall-bounded parallel shear flows, and be generalized straightforwardly to three-dimensional cases. Numerical experiments are presented.

Transitions in genetic toggle switches driven by dynamic disorder in rate coefficients.

In biochemical systems, intrinsic noise may drive the system switch from one stable state to another. We investigate how kinetic switching between stable states in a bistable network is influenced by dynamic disorder, i.e., fluctuations in the rate coefficients. Using the geometric minimum action method, we first investigate the optimal transition paths and the corresponding minimum actions based on a genetic toggle switch model in which reaction coefficients draw from a discrete probability distribution. For the continuous probability distribution of the rate coefficient, we then consider two models of dynamic disorder in which reaction coefficients undergo different stochastic processes with the same stationary distribution. In one, the kinetic parameters follow a discrete Markov process and in the other they follow continuous Langevin dynamics. We find that regulation of the parameters modulating the dynamic disorder, as has been demonstrated to occur through allosteric control in bistable networks in the immune system, can be crucial in shaping the statistics of optimal transition paths, transition probabilities, and the stationary probability distribution of the network.

Minimum-action paths for wave-number selection in nonequilibrium systems.

The problem of wave-number selections in nonequilibrium pattern-forming systems in the presence of noise is investigated. The minimum-action method is proposed to study the noise-induced transitions between the different spatiotemporal states by generalizing the traditional theory previously applied in low-dimensional dynamical systems. The scheme is shown as an example in the stabilized Kuramoto-Sivashinsky equation. The present method allows us to conveniently find the unique noise selected state, in contrast to previous work using direct simulations of the stochastic partial differential equation, where the constraints of the simulation only allow a narrow band to be determined.

Liquid-vapor transition on patterned solid surfaces in a shear flow.

Liquids on a solid surface patterned with microstructures can exhibit the Cassie-Baxter (Cassie) state and the wetted Wenzel state. The transitions between the two states and the effects of surface topography, surface chemistry as well as the geometry of the microstructures on the transitions have been extensively studied in earlier work. However, most of these work focused on the study of the free energy landscape and the energy barriers. In the current work, we consider the transitions in the presence of a shear flow. We compute the minimum action path between the Wenzel and Cassie states using the minimum action method [W. E, W. Ren, and E. Vanden-Eijnden, Commun. Pure Appl. Math. 57, 637 (2004)]. Numerical results are obtained for transitions on a surface patterned with straight pillars. It is found that the shear flow facilitates the transition from the Wenzel state to the Cassie state, while it inhibits the transition backwards. The Wenzel state becomes unstable when the shear rate reaches a certain critical value. Two different scenarios for the Wenzel-Cassie transition are observed. At low shear rate, the transition happens via nucleation of the vapor phase at the bottom of the groove followed by its growth. At high shear rate, in contrary, the nucleation of the vapor phase occurs at the top corner of a pillar. The vapor phase grows in the direction of the flow, and the system goes through an intermediate metastable state before reaching the Cassie state.

A Minimum Action Method with Optimal Linear Time Scaling

AbstractIn this work, we develop a minimum action method (MAM) with optimal linear time scaling, called tMAM for short. The main idea is to relax the integration time as a functional of the transition path through optimal linear time scaling such that a direct optimization of the integration time is not required. The Feidlin-Wentzell action functional is discretized by finite elements, based on whichh-type adaptivity is introduced to tMAM. The adaptive tMAM does not require reparametrization for the transition path. It can be applied to deal with quasi-potential: 1) When the minimal action path is subject to an infinite integration time due to critical points, tMAM with a uniform mesh converges algebraically at a lower rate than the optimal one. However, the adaptive tMAM can recover the optimal convergence rate. 2) When the minimal action path is subject to a finite integration time, tMAM with a uniform mesh converges at the optimal rate since the problem is not singular, and the optimal integration time can be obtained directly from the minimal action path. Numerical experiments have been implemented for both SODE and SPDE examples.

Noise-induced transition in barotropic flow over topography and application to Kuroshio

The minimum action method, developed by E et al. (2004) [1] based on the least action principle from the Wentzell–Freidlin theory of large deviations, is applied to barotropic flow over topography. Application is presented for Kuroshio. The optimal dynamical paths for transitions between the small and large meander states are obtained by minimizing the action functional with the initial and final states being constrained in the metastable sets. The action is minimized using the BFGS quasi-Newton method and the constraints are imposed using the penalty method with a barrier function. It is found that the transition from the small meander state to the large meander state proceeds by nucleation and subsequent propagation of small eddies near the corner of the Kyushu peninsula. The transition backwards follows roughly the same path though details of the flow are different. The transition states are identified by examining the action along the paths.

Computation of rare transitions in the barotropic quasi-geostrophic equations

We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier–Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager–Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherwise. We adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems.