Sampling rare event energy landscapes via birth-death augmented dynamics

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Sampling rare event energy landscapes via birth-death augmented dynamics

Benjamin Pampel, Simon Holbach, Lisa Hartung, and Omar Valsson

Phys. Rev. E 107, 024141 – Published 28 February 2023

Abstract

A common problem that affects simulations of complex systems within the computational physics and chemistry communities is the so-called sampling problem or rare event problem where proper sampling of energy landscapes is impeded by the presences of high kinetic barriers that hinder transitions between metastable states on typical simulation time scales. Many enhanced sampling methods have been developed to address this sampling problem and more efficiently sample rare event systems. An interesting idea, coming from the field of statistics, was introduced in a recent work [Lu, Lu, and Nolen, Accelerating Langevin sampling with birth-death, arXiv:1905.09863] in the form of a novel sampling algorithm that augments overdamped Langevin dynamics with a birth-death process. In this work, we expand on this idea and show that this birth-death sampling scheme can efficiently sample prototypical rare event energy landscapes, and that the speed of equilibration is independent of the barrier height. We amend a crucial shortcoming of the original algorithm that leads to incorrect sampling of barrier regions by introducing an alternative approximation of the birth-death term. We establish important theoretical properties of the modified algorithm and prove mathematically that the relevant convergence results still hold. We investigate via numerical simulations the effect of various parameters, and we investigate ways to reduce the computational effort of the sampling scheme. We show that the birth-death mechanism can be used to accelerate sampling in the more general case of underdamped Langevin dynamics that is more commonly used in simulating physical systems. Our results show that this birth-death scheme is a promising method for sampling rare event energy landscapes.

Received 1 September 2022
Accepted 21 December 2022

DOI:https://doi.org/10.1103/PhysRevE.107.024141

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Stochastic processes

Population dynamics

Fokker–Planck equation Langevin algorithm Langevin equation Large deviation & rare event statistics Phase space dynamics

Statistical Physics & Thermodynamics

Authors & Affiliations

Benjamin Pampel ¹, Simon Holbach ^2,*, Lisa Hartung ^2,†, and Omar Valsson ^1,3,‡

¹Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
²Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany
³Department of Chemistry, University of North Texas, Denton, Texas, USA

^*s.holbach@uni-mainz.de
^†lhartung@uni-mainz.de
^‡omar.valsson@unt.edu

Article Text

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Issue

Vol. 107, Iss. 2 — February 2023

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Images

Figure 1
Sketch of the method: (a) positions of particles (height irrelevant) and a smooth approximation of their ensemble particle distribution $ρ (x)$ (blue line) together with the desired equilibrium distribution $π (x)$ (orange line). At the position of the green particle the current ensemble particle distribution is significantly lower than the desired equilibrium distribution, while it is higher at the position of the red particle. (b) New ensemble particle distribution after the red particle of (a) has been killed and the green particle been duplicated. Effectively, this means that the red particle has performed a nonlocal move to the position of the green particle.
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Figure 2
Estimates of the energy landscape for the potential given in Eq. (58) obtained from sampling using histogramming for the different approximations to the birth-death term: (a) The original proposal $Λ^{0}$ . (b) The present proposal $Λ^{mu}$ . Both show results for different values of the kernel width $σ$ (colored) as well as the reference (black). The small inset shows a magnification of the barrier region. (c) The height of the barrier going from the left minimum to the right minimum estimated from the energy landscape as a function of the kernel width. The reference value is given as a black horizontal line. (d) Fraction of particles in the left state as a function of simulation time. The black horizontal line is the expected equilibrium value. Shown are results from two exemplary simulations with different kernel widths for both approximations ( $σ = 0.2$ as solid line, $σ = 0.5$ as lines with dashes and dots), as well as from a simulation without birth-death events (dashed line). Only the first 8000 steps of the simulations are displayed.
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Figure 3
(a) Estimates of the energy landscape for the potential given in Eq. (58) obtained from sampling using histogramming for different values $M$ of Langevin dynamics steps between attempted birth-death events. All but the data for $M = 10 000$ cannot be distinguished from the reference. (b) Percentage of accepted birth-death events of the total number of birth-death attempts. (c) Average fraction of particles in the left state. The error bars denote the standard deviation and the black horizontal line is the expected equilibrium value. The first 50000 steps were omitted when calculating the values. (d) Fraction of particles in the left state as a function of simulation time. The colored lines are from the same simulations as in (a), the black horizontal line is the expected equilibrium value. The inset shows a magnification of the first 12000 steps.
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Figure 4
(a) Estimates of the energy landscape obtained from sampling using histogramming for the potential given in Eq. (60) with different coefficients given in Table 1. The solid lines are from simulations with birth-death events, the dashed lines from Langevin dynamics simulations without birth-death events (i.e., independent particles), and the thin black lines are the reference from the potential. (b) Number of particles in the left state as a function of simulation time for the different potentials. Shown are only the first 5000 steps. The solid black line is the equilibrium value calculated from the potentials (same for all potentials).
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Figure 5
(a) The reference energy landscape of the Wolfe-Quapp potential [Eq. (61)]. (b) The energy landscapes estimated from sampling using histogramming projected on the y-direction. Colored solid lines are from simulations with birth-death events using different kernel widths $σ$ . We note that the lines for $σ = 0.55$ and $σ = 0.75$ are hardly distinguishable because they basically are on top of each other. The dashed line is from a Langevin simulation without birth-death events but the same number of particles. For clarity, the dashed line is omitted in the inset. The black line is the reference energy landscape calculated from the potential. (c) Kullback-Leibler divergences from the estimated probability distribution to the equilibrium distribution for simulations with different kernel widths $σ$ . For comparison, the dashed horizontal line is from a Langevin simulation without the birth-death algorithm. (d) Fraction of particles in the state with $y > 0$ as a function of simulation time. The black horizontal line is the reference equilibrium value. The different lines represent the same simulations as in (b).
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