^{1,a)}, Daan Frenkel

^{1}and Pieter Rein ten Wolde

^{1,b)}

### Abstract

We analyze the efficiency of several simulation methods which we have recently proposed for calculating rate constants for rare events in stochastic dynamical systems in or out of equilibrium. We derive analytical expressions for the computational cost of using these methods and for the statistical error in the final estimate of the rate constant for a given computational cost. These expressions can be used to determine which method to use for a given problem, to optimize the choice of parameters, and to evaluate the significance of the results obtained. We apply the expressions to the two-dimensional nonequilibrium rare event problem proposed by Maier and Stein [Phys. Rev. E48, 931 (1993)]. For this problem, our analysis gives accurate quantitative predictions for the computational efficiency of the three methods.

The authors thank Axel Arnold for his careful reading of the manuscript. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). One of the authors (R.J.A.) was supported by the European Union Marie Curie program.

I. INTRODUCTION

II. BACKGROUND: FFS-TYPE METHODS

A. Forward flux sampling

B. The branched growth method

C. The Rosenbluth method

III. COMPUTATIONAL EFFICIENCY

A. Computational cost

1. Expressions for the cost

2. Illustration

B. Statistical error

1. Expressions for the variance

2. Illustration

3. Landscape variance

C. Efficiency

IV. THE MAIER-STEIN SYSTEM

A. Measuring the parameters

B. Testing the expressions

V. DISCUSSION

### Key Topics

- Stochastic processes
- 7.0
- Polymers
- 5.0
- Reaction rate constants
- 5.0
- Free energy
- 3.0
- Interface dynamics
- 3.0

## Figures

Schematic illustration of the definition of regions and and the interfaces (Here, ). Three transition paths are shown.

Schematic illustration of the definition of regions and and the interfaces (Here, ). Three transition paths are shown.

Cost for evenly spaced interfaces , , , , and . (a) as a function of for . (b) as a function of for .

Cost for evenly spaced interfaces , , , , and . (a) as a function of for . (b) as a function of for .

Relative variance , for , , and . The circles show the function . (a) as a function of for . (b) as a function of for .

Relative variance , for , , and . The circles show the function . (a) as a function of for . (b) as a function of for .

Relative variance in , as predicted by Eqs. (50)–(52), for the model problem of Figs. 2 and 3 with and . The upper curves in each group correspond to , the middle curves to , and the lower curves to . (a) as a function of , keeping . (b) as a function of , keeping .

Relative variance in , as predicted by Eqs. (50)–(52), for the model problem of Figs. 2 and 3 with and . The upper curves in each group correspond to , the middle curves to , and the lower curves to . (a) as a function of , keeping . (b) as a function of , keeping .

Efficiency , calculated using Eq. (4), for the simple model of Figs. 2–4. For each method, results are plotted with (lower curves) and (upper curves). (a) as a function of for . (b) as a function of for .

Efficiency , calculated using Eq. (4), for the simple model of Figs. 2–4. For each method, results are plotted with (lower curves) and (upper curves). (a) as a function of for . (b) as a function of for .

Typical trajectory for a brute-force simulation of the Maier-Stein system, with , , and .

Typical trajectory for a brute-force simulation of the Maier-Stein system, with , , and .

Costs of trial runs between interfaces for the Maier-Stein system. The average lengths, in simulation steps, of “successful” trials (to ) are shown as filled circles. For these trials, and . The average lengths of “unsuccessful” trials (to ) are shown as open circles. For these trials, . The solid lines show the linear approximation [Eq. (6)], with .

Costs of trial runs between interfaces for the Maier-Stein system. The average lengths, in simulation steps, of “successful” trials (to ) are shown as filled circles. For these trials, and . The average lengths of “unsuccessful” trials (to ) are shown as open circles. For these trials, . The solid lines show the linear approximation [Eq. (6)], with .

Predicted and measured values of for the Maier-Stein problem, as described in Sec. IV. The lines show the theoretical predictions for the FFS (solid line), BG (dotted line), and RB (dashed line) methods. The symbols show the simulation results. Circles: FFS method, squares: BG method, and triangles: RB method (with Metropolis acceptance/rejection). Simulation results were obtained with 400 blocks of starting points for FFS and 2000 starting points per block for BG and RB. (a) as a function of for . (b) as a function of for for evenly spaced interfaces.

Predicted and measured values of for the Maier-Stein problem, as described in Sec. IV. The lines show the theoretical predictions for the FFS (solid line), BG (dotted line), and RB (dashed line) methods. The symbols show the simulation results. Circles: FFS method, squares: BG method, and triangles: RB method (with Metropolis acceptance/rejection). Simulation results were obtained with 400 blocks of starting points for FFS and 2000 starting points per block for BG and RB. (a) as a function of for . (b) as a function of for for evenly spaced interfaces.

Predicted and measured values of for the Maier-Stein problem. The lines show the theoretical predictions for the FFS (solid line), BG (dotted line), and RB (dashed line) methods. The symbols show the simulation results. Circles: FFS method, squares: BG method, and triangles: RB method (with Metropolis acceptance/rejection). Simulation results were obtained with 400 blocks of starting points for FFS and 2000 starting points per block for BG and RB. Interfaces were evenly spaced between and . (a) as a function of for . (b) as a function of for . In (b), the landscape contribution is not included in the theoretical calculation.

Predicted and measured values of for the Maier-Stein problem. The lines show the theoretical predictions for the FFS (solid line), BG (dotted line), and RB (dashed line) methods. The symbols show the simulation results. Circles: FFS method, squares: BG method, and triangles: RB method (with Metropolis acceptance/rejection). Simulation results were obtained with 400 blocks of starting points for FFS and 2000 starting points per block for BG and RB. Interfaces were evenly spaced between and . (a) as a function of for . (b) as a function of for . In (b), the landscape contribution is not included in the theoretical calculation.

Predicted and measured values of for the Maier-Stein problem for the FFS method. Solid line: Eq. (50) (with the landscape variance), dotted line: Eq. (23) (no landscape variance), and circles: simulation results.

Predicted and measured values of for the Maier-Stein problem for the FFS method. Solid line: Eq. (50) (with the landscape variance), dotted line: Eq. (23) (no landscape variance), and circles: simulation results.

Predicted and measured efficiency for the Maier-Stein system. The lines show the theoretical predictions for the FFS (solid line), BG (dotted line), and RB (dashed line) methods. The symbols show the simulation results. Circles: FFS method, squares: BG method, and triangles: RB method (with Metropolis acceptance/rejection). Simulation results were obtained with 400 blocks. For FFS, each block had starting points, and for BG and RB each blocks had 2000 starting points. Interfaces were evenly spaced. (a) vs for . (b) vs for . No landscape contribution to is included in panel b.

Predicted and measured efficiency for the Maier-Stein system. The lines show the theoretical predictions for the FFS (solid line), BG (dotted line), and RB (dashed line) methods. The symbols show the simulation results. Circles: FFS method, squares: BG method, and triangles: RB method (with Metropolis acceptance/rejection). Simulation results were obtained with 400 blocks. For FFS, each block had starting points, and for BG and RB each blocks had 2000 starting points. Interfaces were evenly spaced. (a) vs for . (b) vs for . No landscape contribution to is included in panel b.

(a) “Simulated” and predicted acceptance probabilities for interfaces for the “simulated simulation” described in the text, for . (b) Simulated and predicted values of for for the Maier-Stein problem of Sec. 4, for . In both plots, solid lines represent predicted values for , dotted lines, , and dashed lines, . Symbols represent simulation results: circles: , squares: , and triangles: .

(a) “Simulated” and predicted acceptance probabilities for interfaces for the “simulated simulation” described in the text, for . (b) Simulated and predicted values of for for the Maier-Stein problem of Sec. 4, for . In both plots, solid lines represent predicted values for , dotted lines, , and dashed lines, . Symbols represent simulation results: circles: , squares: , and triangles: .

(solid line) and (dashed line), as functions of , calculated using FFS as described in Appendix D, for the Maier-Stein problem of Sec. IV with 10 000 points at the first interface .

(solid line) and (dashed line), as functions of , calculated using FFS as described in Appendix D, for the Maier-Stein problem of Sec. IV with 10 000 points at the first interface .

## Tables

Positions of the interfaces and measured values of and for the Maier-Stein problem.

Positions of the interfaces and measured values of and for the Maier-Stein problem.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content