Mathematical Analysis of Temperature Accelerated Dynamics

Abstract

We give a mathematical framework for temperature accelerated dynamics (TAD), an algorithm proposed by M.R. S{\o}rensen and A.F. Voter to efficiently generate metastable stochastic dynamics. Using the notion of quasistationary distributions, we propose some modifications to TAD. Then considering the modified algorithm in an idealized setting, we show how TAD can be made mathematically rigorous.Comment: 28 pages, 2 figure

Jump Markov models and transition state theory: the quasi-stationary distribution approach

“…Finally, the temperature accelerated dynamics 5 consists in simulating exit events at high temperature, and to extrapolate them at low temperature using the Eyring-Kramers law (7). In this paper, for the sake of conciseness, we concentrate on the analysis of the parallel replica method, and we refer to the papers 51,52 for an analysis of hyperdynamics and temperature accelerated dynamics. See also the recent review 44 for a detailed presentation.…”

“…The QSD approach is also useful to analyze the two other accelerated dynamics: hyperdynamics 51 and temperature accelerated dynamics 52 . Typically, one expects better speed up with these algorithms than with parallel replica, but at the expense of larger errors and more stringent assumptions (typically energetic barriers, and small temperature regime), see 44 for a review paper.…”

“…Typically, one expects better speed up with these algorithms than with parallel replica, but at the expense of larger errors and more stringent assumptions (typically energetic barriers, and small temperature regime), see 44 for a review paper. Let us mention in particular that the mathematical analysis of the temperature accelerated dynamics algorithms requires to prove that the distribution for the next visited state predicted using the Eyring-Kramers formula (7) is correct, as explained in 52 . The next section is thus also motivated by the development of an error analysis for temperature accelerated dynamics.…”

Repartition of the Quasi-stationary Distribution and First Exit Point Density for a Double-Well Potential

“…In molecular dynamics, the quasi-stationary distribution ν h is used to quantify the metastability of the subdomain Ω of R d as follows: for a probability measure µ 0 supported in Ω, the domain Ω is said to be metastable for the initial condition µ 0 if, when X 0 ∼ µ 0 , the convergence in (2) is much quicker than the average exit time from Ω. When Ω is metastable, it is thus relevant to study the exit event (τ Ω , X τΩ ) of the process (1) from Ω starting from ν h , i.e. when X 0 ∼ ν h .…”

Markov state modeling and dynamical coarse-graining via discrete relaxation path sampling

“…We see in Table I that this estimate is never very accurate, for the reasons explained in Appendix D. We did not test other methods such as boxed MD, [64][65][66] temperatureaccelerated MD, 67,68 and hyperdynamics [69][70][71] for this example. Unlike DRPS the accuracy of these methods will depend on the position of the dividing surface and will only give reliable results when the trajectory commits to the other cell after hitting the dividing surface (i.e., if the distributions of first exit times of these cells are exponential).…”