No abstract

The quasi-potential is a key function in the Large Deviation Theory. It characterizes the difficulty of the escape from the neighborhood of an attractor of a stochastic non-gradient dynamical system due to the influence of small white noise. It also gives an estimate of the invariant probability distribution in the neighborhood of the attractor up to the exponential order. We present a new family of methods for computing the quasi-potential on a regular mesh named the Ordered Line Integral Methods (OLIMs). In comparison with the first proposed quasi-potential finder based on the Ordered Upwind Method (OUM) (Cameron, 2012), the new methods are 1.5 to 4 times faster, can produce error two to three orders of magnitude smaller, and may exhibit faster convergence. Similar to the OUM, OLIMs employ the dynamical programming principle. Contrary to it, they (i) have an optimized strategy for the use of computationally expensive triangle updates leading to a notable speed-up, and (ii) directly solve local minimization problems using quadrature rules instead of solving the corresponding Hamilton-Jacobi-type equation by the first order finite difference upwind scheme. The OLIM with the right-hand quadrature rule is equivalent to OUM. The use of higher order quadrature rules in local minimization problems dramatically boosts up the accuracy of OLIMs. We offer a detailed discussion on the origin of numerical errors in OLIMs and propose rules-of-thumb for the choice of the important parameter, the update factor, in the OUM and OLIMs. Our results are supported by extensive numerical tests on two challenging 2D examples.

A set of analytical and computational tools based on transition path theory (TPT) is proposed to analyze flows in complex networks. Specifically, TPT is used to study the statistical properties of the reactive trajectories by which transitions occur between specific groups of nodes on the network. Sampling tools are built upon the outputs of TPT that allow to generate these reactive trajectories directly, or even transition paths that travel from one group of nodes to the other without making any detour and carry the same probability current as the reactive trajectories. These objects permit to characterize the mechanism of the transitions, for example by quantifying the width of the tubes by which these transitions occur, the location and distribution of their dynamical bottlenecks, etc. These tools are applied to a network modeling the dynamics of the Lennard-Jones cluster with 38 atoms (LJ38) and used to understand the mechanism by which this cluster rearranges itself between its two most likely states at various temperatures.transition path theory and self-assembly and protein folding and glassy dynamics and Markov State Models arXiv:1402.1736v1 [cond-mat.stat-mech]

Nongradient SDEs with small white noise often arise when modeling biological and ecological time-irreversible processes. If the governing SDE were gradient, the maximum likelihood transition paths, transition rates, expected exit times, and the invariant probability distribution would be given in terms of its potential function. The quasipotential plays a similar role for nongradient SDEs. Unfortunately, the quasipotential is the solution of a functional minimization problem that can be obtained analytically only in some special cases. We propose a Dijkstra-like solver for computing the quasipotential on regular rectangular meshes in 3D. This solver results from a promotion and an upgrade of the previously introduced ordered line integral method with the midpoint quadrature rule for 2D SDEs. The key innovations that have allowed us to keep the CPU times reasonable while maintaining good accuracy are (i) a new hierarchical update strategy, (ii) the use of Karush-Kuhn-Tucker theory for rejecting unnecessary simplex updates, and (iii) pruning the number of admissible simplexes and a fast search for them. An extensive numerical study is conducted on a series of linear and nonlinear examples where the quasipotential is analytically available or can be found at transition states by other methods. In particular, the proposed solver is applied to Tao's examples where the transition states are hyperbolic periodic orbits, and to a genetic switch model by Lv et al. (2014). The C source code implementing the proposed algorithm is available at M. Cameron's web page.

The concept of metastability has caused a lot of interest in recent years. The spectral decomposition of the generator matrix of a stochastic network exposes all of the transition processes in the system. The assumption of the existence of a low lying group of eigenvalues separated by a spectral gap, leading to factorization of the dynamics, has become a popular theme. We consider stochastic networks representing potential energy landscapes where the states and the edges correspond to local minima and transition states respectively, and the pairwise transition rates are given by the Arrhenuis formula. Using the minimal spanning tree, we construct the asymptotics for eigenvalues and eigenvectors of the generator matrix starting from the low lying group. This construction gives rise to an efficient algorithm for computing the asymptotic spectrum suitable for large and complex networks. We apply it to Wales's Lennard-Jones-38 network with 71887 states and 119853 edges where the underlying potential energy landscape has a double-funnel structure. Our results demonstrate that the concept of metastability should be applied with care to this system. For the full network, there is no significant spectral gap separating the eigenvalue corresponding to the exit from the wider and shallower icosahedral funnel at any reasonable temperature range. However, if the observation time is limited, the expected spectral gap appears.1 This criterion for the states being connected by an edge can be relaxed. More generally, we connect states i and j by an edge (i, j) if and only if one can find a Minimum Energy Path φij(α), α ∈ [0, 1] with the following properties: (i) φij(0) = x1 and φij(1) = xj, where xi and xj are the local minima corresponding to the states i and j; (ii) φij passes through no other local minima other than its endpoints xi and xj; (iii) the only critical points that φij passes through are saddles; (iv) the maximal value of the potential along φij is achieved at a Morse index one saddle. Then the number Vij is the maximal potential value along φij. A number of interesting phenomena regarding the Minimum Energy Paths is discussed in [8].