^{1}, Nuria Plattner

^{1,2}, David L. Freeman

^{3}, Yufei Liu

^{4}and Paul Dupuis

^{4}

### Abstract

In the present paper we identify a rigorous property of a number of tempering-based Monte Carlo sampling methods, including parallel tempering as well as partial and infinite swapping. Based on this property we develop a variety of performance measures for such rare-event sampling methods that are broadly applicable, informative, and straightforward to implement. We illustrate the use of these performance measures with a series of applications involving the equilibrium properties of simple Lennard-Jones clusters, applications for which the performance levels of partial and infinite swapping approaches are found to be higher than those of conventional parallel tempering.

The authors gratefully acknowledge grant support of this research through the (U.S.) Department of Energy (DOE) Multiscale and Optimization for Complex Systems Program No. DE-SC0002413. N.P. wishes to thank the Swiss National Science Foundation for postdoctoral support and J.D.D. wishes to acknowledge support through the DOE departmental Program No. DE-00015561. P.D. wishes to acknowledge support from the Army Research Office (W911NF-12-1-0222) and P.D. gratefully acknowledges support from the National Science Foundation (DMS-1008331). The authors also wish to thank Dr. James Gubernatis, Los Alamos National Laboratory, for helpful discussions concerning the present work.

I. INTRODUCTION

II. BACKGROUND AND AN OBSERVATION

III. ILLUSTRATIVE APPLICATIONS

IV. DISCUSSION AND SUMMARY

### Key Topics

- Entropy
- 13.0
- Monte Carlo methods
- 6.0
- Heat capacity
- 5.0
- Numerical modeling
- 3.0
- Statistical properties
- 3.0

## Figures

Occupation traces for three-temperature PT (top row) and PINS (bottom row) simulations for Ar_{13}. T_{1} = 30 K and T_{3} = 40 K for all simulations, while T_{2} = 35 K for simulations in left column and 39 K for those in right column.

Occupation traces for three-temperature PT (top row) and PINS (bottom row) simulations for Ar_{13}. T_{1} = 30 K and T_{3} = 40 K for all simulations, while T_{2} = 35 K for simulations in left column and 39 K for those in right column.

Plots of the average number of moves required for a round-trip transit of the computational ensemble, ⟨n_{rt}⟩ as function of T_{2}, for extended versions of the three-temperature Ar_{13} simulations of the type in Fig. 1 .

Plots of the average number of moves required for a round-trip transit of the computational ensemble, ⟨n_{rt}⟩ as function of T_{2}, for extended versions of the three-temperature Ar_{13} simulations of the type in Fig. 1 .

Approach of S_{f}(n_{move}) (c.f., Eq. (2.3) ) to its uniform limiting value for the three-temperature PINS and PT simulations of Ar_{13} used in Table I and described in the text. T_{1} = 30 K, T_{3} = 40 K, T_{2} = 35 K or 39 K.

Plot of ln(S_{max} − S_{f}) for the three-temperature Ar_{13} results of Fig. 3 .

A portion of occupation traces for 66-temperature Ar_{38} PINS (black) and PT (red) simulations discussed in the text. The vertical axis denotes the temperature index (1–66) as a function of the number of moves in the simulation.

A portion of occupation traces for 66-temperature Ar_{38} PINS (black) and PT (red) simulations discussed in the text. The vertical axis denotes the temperature index (1–66) as a function of the number of moves in the simulation.

A histogram of the PINS occupation trace shown in Fig. 6 showing the number of times the various temperature indices are visited, M(n), as a function of n.

A histogram of the PINS occupation trace shown in Fig. 6 showing the number of times the various temperature indices are visited, M(n), as a function of n.

A plot of S_{f}(n_{move}) obtained for Ar_{13} using PINS and PT methods for the various 24-temperature ensembles described in the text. The apparent “break” in the PINS-24a results occurs at an S_{f} value of roughly 2.5, a value that corresponds to an active number of temperatures (N_{a}) of ∼12.

A plot of S_{f}(n_{move}) obtained for Ar_{13} using PINS and PT methods for the various 24-temperature ensembles described in the text. The apparent “break” in the PINS-24a results occurs at an S_{f} value of roughly 2.5, a value that corresponds to an active number of temperatures (N_{a}) of ∼12.

Plot of ln(S_{max} − S_{f}) for results in Fig. 8 .

C(s) for the PINS Ar_{13} results obtained using the three, 24-temperature ensembles described in the text. PT results for ensemble-c are shown for comparison (dashed line near top of plot).

C(s) for the PINS Ar_{13} results obtained using the three, 24-temperature ensembles described in the text. PT results for ensemble-c are shown for comparison (dashed line near top of plot).

Brief portions of occupation traces for PINS (top panel) and PT (bottom panel) simulations for Ar_{13} obtained using 24-temperature ensemble-c (see text for details).

Brief portions of occupation traces for PINS (top panel) and PT (bottom panel) simulations for Ar_{13} obtained using 24-temperature ensemble-c (see text for details).

Plots of the occupation entropy, S_{f}(nmove), for 66-temperature PINS and PT simulations of the Ar_{38} system. The two PINS results correspond to simulations that are initiated in the global minimum geometry (black curve) or lowest-lying icosahedral minimum (red curve). The limiting S_{f} value of ln(66) is shown for reference.

Plots of the occupation entropy, S_{f}(nmove), for 66-temperature PINS and PT simulations of the Ar_{38} system. The two PINS results correspond to simulations that are initiated in the global minimum geometry (black curve) or lowest-lying icosahedral minimum (red curve). The limiting S_{f} value of ln(66) is shown for reference.

Plot of ln(S_{max} − S_{f}) for results in Fig. 12 .

A plot of ⟨Q_{4}(T)⟩ for the Ar_{38} cluster obtained by the PINS simulations described in the text. Results in black (red) are obtained using a simulation initialized using the fcc global minimum (icosahedral local minimum) structure. For clarity and as an aid in comparing the two simulations ⟨Q_{4}(T)⟩ only values for every other (every fourth) temperature are shown for the fcc (icosahedral) results.

A plot of ⟨Q_{4}(T)⟩ for the Ar_{38} cluster obtained by the PINS simulations described in the text. Results in black (red) are obtained using a simulation initialized using the fcc global minimum (icosahedral local minimum) structure. For clarity and as an aid in comparing the two simulations ⟨Q_{4}(T)⟩ only values for every other (every fourth) temperature are shown for the fcc (icosahedral) results.

Q_{4} values for an extended portion of the global minimum initiated occupation trace of Fig. 6 .

Q_{4} values for an extended portion of the global minimum initiated occupation trace of Fig. 6 .

A history of the number of configurations (out of 66) in the two PINS Ar_{38} simulations described in the text for which Q_{4} ≤ 0.09.

A history of the number of configurations (out of 66) in the two PINS Ar_{38} simulations described in the text for which Q_{4} ≤ 0.09.

Block averages of Q_{4} for Ar_{38} for T_{1} = 10 K, T_{2} = 14.9350 K for the PINS simulations described in the text.

Block averages of Q_{4} for Ar_{38} for T_{1} = 10 K, T_{2} = 14.9350 K for the PINS simulations described in the text.

Shown are the Q_{4} values for T = 22.727 K for a short, post warm up portion of the icosahedral minimum initiated Ar_{38} PINS simulation. Compare with Fig. 11 of Ref. ^{ 29 } .

Shown are the Q_{4} values for T = 22.727 K for a short, post warm up portion of the icosahedral minimum initiated Ar_{38} PINS simulation. Compare with Fig. 11 of Ref. ^{ 29 } .

## Tables

Observed fractions of total moves (f_{n}) spent at each of the ensemble temperatures by two three-temperature Ar_{13} parallel tempering (PT) and partial swapping (PINS) simulations. In both ensembles, T_{1} = 30 K and T_{3} = 40 K, while in one ensemble T_{2} = 35 K and in the other T_{2} = 39 K.

Observed fractions of total moves (f_{n}) spent at each of the ensemble temperatures by two three-temperature Ar_{13} parallel tempering (PT) and partial swapping (PINS) simulations. In both ensembles, T_{1} = 30 K and T_{3} = 40 K, while in one ensemble T_{2} = 35 K and in the other T_{2} = 39 K.

The temperatures used in the PINS Ar_{38} computational ensemble.

The temperatures used in the PINS Ar_{38} computational ensemble.

Average of S_{ρ} values, ⟨S_{ρ}⟩ , for Ar_{38} PINS simulation obtained using the computational ensemble shown in Table II . For reference, the maximum values for S_{ ρ} correspond to ln(3!) = 1.792 and ln(6!) = 6.579.

Average of S_{ρ} values, ⟨S_{ρ}⟩ , for Ar_{38} PINS simulation obtained using the computational ensemble shown in Table II . For reference, the maximum values for S_{ ρ} correspond to ln(3!) = 1.792 and ln(6!) = 6.579.

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