arXiv subjects

Eric Vanden-Eijnden

Publications and source records attributed to Eric Vanden-Eijnden.

At least 91 records · Page 5Linked to original sources

arXiv2017

A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)

Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.

math-ph↗

arXiv2016

Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory

This paper is concerned with the long-time dynamics of the nonlinear wave equation in one-space dimension, $$ u_{tt} - δ^2 u_{xx} +V'(u) =0 \qquad x\in [0,1] $$ where $δ>0$ is a parameter and $V(u)$ is a potential bounded from below and growing at least like $u^2$ as $|u|\to\infty$. Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure and when the potential is double-welled, for example when $V(u) = \tfrac14(1-u^2)^2$, there is a regime such that two small disjoint sets in the system's phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here we quantify this phenomenon by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter $δ$ in small enough. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger $δ$, however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency, indicating that successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.

math.PR↗

arXiv2016

A 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, like nanomagnetic bits, finding optimal protocols is challenging, requiring detailed information about the dynamical fluctuations of the controlled system. We provide a new, 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 simple 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.

cond-mat.stat-mech↗

arXiv2016

Optimized Markov State Models for Metastable Systems

A method is proposed to identify target states that optimize a metastability index amongst a set of trial states and use these target states as milestones (or core sets) to build Markov State Models (MSMs). If the optimized metastability index is small, this automatically guarantees the accuracy of the MSM, in the sense that the transitions between the target milestones is indeed approximately Markovian. The method is simple to implement and use, it does not require that the dynamics on the trial milestones be Markovian, and it also offers the possibility to partition the system's state-space by assigning every trial milestone to the target milestones it is most likely to visit next and to identify transition state regions. Here the method is tested on the Gly-Ala-Gly peptide, where it shown to correctly identify the expected metastable states in the dihedral angle space of the molecule without \textit{a~priori} information about these states. It is also applied to analyze the folding landscape of the Beta3s mini-protein, where it is shown to identify the folded basin as a connecting hub between an helix-rich region, which is entropically stabilized, and a beta-rich region, which is energetically stabilized and acts as a kinetic trap.

cond-mat.stat-mech↗

arXiv2016

Fluctuations in the heterogeneous multiscale methods for fast-slow systems

How heterogeneous multiscale methods (HMM) handle fluctuations acting on the slow variables in fast-slow systems is investigated. In particular, it is shown via analysis of central limit theorems (CLT) and large deviation principles (LDP) that the standard version of HMM artificially amplifies these fluctuations. A simple modification of HMM, termed parallel HMM, is introduced and is shown to remedy this problem, capturing fluctuations correctly both at the level of the CLT and the LDP. Similar type of arguments can also be used to justify that the tau-leaping method used in the context of Gillespie's stochastic simulation algorithm for Markov jump processes also captures the right CLT and LDP for these processes.

math.PR↗

arXiv2015

Large Deviations in Fast-Slow Systems

The incidence of rare events in fast-slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior -- such fluctuations are rare on short timescales but become ubiquitous eventually. This LDP involves an Hamilton-Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta -- in other words, the LDP for the slow variables in fast-slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE.

cond-mat.stat-mech↗

arXiv2015

Continuous-time Random Walks for the Numerical Solution of Stochastic Differential Equations

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the spatial increment of the approximation can be bounded uniformly in space, which guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension permits to analyze the complexity of the approximation. Using the theory of semigroups of linear operators on Banach spaces, we show that the approximation is (weakly) accurate in representing finite and infinite-time statistics, with an order of accuracy identical to that of its generator. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors.

math.PR↗

arXiv2014

Relevance of instantons in Burgers turbulence

Instanton calculations are performed in the context of stationary Burgers turbulence to estimate the tails of the probability density function (PDF) of velocity gradients. These results are then compared to those obtained from massive direct numerical simulations (DNS) of the randomly forced Burgers equation. The instanton predictions are shown to agree with the DNS in a wide range of regimes, including those that are far from the limiting cases previously considered in the literature. These results settle the controversy of the relevance of the instanton approach for the prediction of the velocity gradient PDF tail exponents. They also demonstrate the usefulness of the instanton formalism in Burgers turbulence, and suggest that this approach may be applicable in other contexts, such as 2D and 3D turbulence in compressible and incompressible flows.

physics.flu-dyn↗

arXiv2014

Stochastic Mode-Reduction in Models with Conservative Fast Sub-Systems

A stochastic mode reduction strategy is applied to multiscale models with a deterministic energy-conserving fast sub-system. Specifically, we consider situations where the slow variables are driven stochastically and interact with the fast sub-system in an energy-conserving fashion. Since the stochastic terms only affect the slow variables, the fast-subsystem evolves deterministically on a sphere of constant energy. However, in the full model the radius of the sphere slowly changes due to the coupling between the slow and fast dynamics. Therefore, the energy of the fast sub-system becomes an additional hidden slow variable that must be accounted for in order to apply the stochastic mode reduction technique to systems of this type.

math.PR↗

arXiv2014

Exact dynamical coarse-graining without time-scale separation

A family of collective variables is proposed to perform exact dynamical coarse-graining even in systems without time scale separation. More precisely, it is shown that these variables are not slow in general but they satisfy an overdamped Langevin equation that statistically preserves the sequence in which any regions in collective variable space are visited and permits to calculate exactly the mean first passage times from any such region to another. The role of the free energy and diffusion coefficient in this overdamped Langevin equation is discussed, along with the way they transform under any change of variable in collective variable space. These results apply both for systems with and without inertia, and they can be generalized to using several collective variables simultaneously. The view they offer on what makes collective variables and reaction coordinates optimal breaks from the standard notion that good collective variable must be slow variable, and it suggests new ways to interpret data from molecular dynamic simulations and experiments.

cond-mat.stat-mech↗

arXiv2014

Metropolis Integration Schemes for Self-Adjoint Diffusions

We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte-Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated on several examples including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictor-corrector schemes.

math.NA↗

arXiv2014

Data assimilation in the low noise regime with application to the Kuroshio

On-line data assimilation techniques such as ensemble Kalman filters and particle filters lose accuracy dramatically when presented with an unlikely observation. Such an observation may be caused by an unusually large measurement error or reflect a rare fluctuation in the dynamics of the system. Over a long enough span of time it becomes likely that one or several of these events will occur. Often they are signatures of the most interesting features of the underlying system and their prediction becomes the primary focus of the data assimilation procedure. The Kuroshio or Black Current that runs along the eastern coast of Japan is an example of such a system. It undergoes infrequent but dramatic changes of state between a small meander during which the current remains close to the coast of Japan, and a large meander during which it bulges away from the coast. Because of the important role that the Kuroshio plays in distributing heat and salinity in the surrounding region, prediction of these transitions is of acute interest. Here we focus on a regime in which both the stochastic forcing on the system and the observational noise are small. In this setting large deviation theory can be used to understand why standard filtering methods fail and guide the design of the more effective data assimilation techniques. Motivated by our analysis we propose several data assimilation strategies capable of efficiently handling rare events such as the transitions of the Kuroshio. These techniques are tested on a model of the Kuroshio and shown to perform much better than standard filtering methods.

physics.data-an↗

arXiv2014

Flows in Complex Networks: Theory, Algorithms, and Application to Lennard-Jones Cluster Rearrangement

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.

cond-mat.stat-mech↗

arXiv2013

Arclength parametrized Hamilton's equations for the calculation of instantons

A method is presented to compute minimizers (instantons) of action functionals using arclength parametrization of Hamilton's equations. This method can be interpreted as a local variant of the geometric minimum action method (gMAM) introduced to compute minimizers of the Freidlin-Wentzell action functional that arises in the context of large deviation theory for stochastic differential equations. The method is particularly well-suited to calculate expectations dominated by noise-induced excursions from deterministically stable fixpoints. Its simplicity and computational efficiency are illustrated here using several examples: a finite-dimensional stochastic dynamical system (an Ornstein-Uhlenbeck model) and two models based on stochastic partial differential equations: the $ϕ^4$-model and the stochastically driven Burgers equation.

cond-mat.stat-mech↗

arXiv2013

Infinite swapping replica exchange molecular dynamics leads to a simple simulation patch using mixture potentials

Replica exchange molecular dynamics (REMD) becomes more efficient as the frequency of swap between the temperatures is increased. Recently in [Plattner et al, J. Chem. Phys. 135, 134111 (2011)] a method was proposed to implement infinite swapping REMD in practice. Here we show that this method naturally leads to a reformulation in terms of molecular dynamics simulations over a mixture potential, which is both simple to implement in practice and provides a better, energy based understanding of how to choose the temperatures in REMD to optimize efficiency. It also opens the door to generalizations of REMD in which the swaps involve other parameters than the temperature.

cond-mat.stat-mech↗

arXiv2012

An observable for vacancy characterization and diffusion in crystals

To locate the position and characterize the dynamics of a vacancy in a crystal, we propose to represent it by the ground state density of a quantum probe quasi-particle for the Hamiltonian associated to the potential energy field generated by the atoms in the sample. In this description, the h^2/2mu coefficient of the kinetic energy term is a tunable parameter controlling the density localization in the regions of relevant minima of the potential energy field. Based on this description, we derive a set of collective variables that we use in rare event simulations to identify some of the vacancy diffusion paths in a 2D crystal. Our simulations reveal, in addition to the simple and expected nearest neighbor hopping path, a collective migration mechanism of the vacancy. This mechanism involves several lattice sites and produces a long range migration of the vacancy. Finally, we also observed a vacancy induced crystal reorientation process.

cond-mat.mtrl-sci↗

arXiv2012

Thermal Stability of the Magnetization in Perpendicularly Magnetized Thin Film Nanomagnets

Understanding the stability of thin film nanomagnets with perpendicular magnetic anisotropy (PMA) against thermally induced magnetization reversal is important when designing perpendicularly magnetized patterned media and magnetic random access memories. The leading-order dependence of magnetization reversal rates are governed by the energy barrier the system needs to surmount in order for reversal to proceed. In this paper we study the reversal dynamics of these systems and compute the relevant barriers using the string method of E, Vanden-Eijnden, and Ren. We find the reversal to be often spatially incoherent; that is, rather than the magnetization flipping as a rigid unit, reversal proceeds instead through a soliton-like domain wall sweeping through the system. We show that for square nanomagnetic elements the energy barrier increases with element size up to a critical length scale, beyond which the energy barrier is constant. For circular elements the energy barrier continues to increase indefinitely, albeit more slowly beyond a critical size. In both cases the energy barriers are smaller than those expected for coherent magnetization reversal.

cond-mat.mes-hall↗

arXiv2012

Force-clamp analysis techniques reveal stretched exponential unfolding kinetics in ubiquitin

Force-clamp spectroscopy reveals the unfolding and disulfide bond rupture times of single protein molecules as a function of the stretching force, point mutations and solvent conditions. The statistics of these times reveal whether the protein domains are independent of one another, the mechanical hierarchy in the polyprotein chain, and the functional form of the probability distribution from which they originate. It is therefore important to use robust statistical tests to decipher the correct theoretical model underlying the process. Here we develop multiple techniques to compare the well-established experimental data set on ubiquitin with existing theoretical models as a case study. We show that robustness against filtering, agreement with a maximum likelihood function that takes into account experimental artifacts, the Kuiper statistic test and alignment with synthetic data all identify the Weibull or stretched exponential distribution as the best fitting model. Our results are inconsistent with recently proposed models of Gaussian disorder in the energy landscape or noise in the applied force as explanations for the observed non-exponential kinetics. Since the physical model in the fit affects the characteristic unfolding time, these results have important implications on our understanding of the biological function of proteins.

physics.bio-ph↗