# A Novel Hybrid Monte Carlo Algorithm for Sampling Path Space

## Abstract

**:**

## 1. Introduction

## 2. Time Evolution of the Hamiltonian

#### 2.1. Splitting: ABA

#### 2.2. “Energy” Error

## 3. Numerical Experiments

#### 3.1. Equilibrium Distribution

#### 3.2. Parameter Tuning

#### 3.2.1. Path Length

#### 3.2.2. Deterministic Integration Time

#### 3.2.3. OU Bridge Parameter ${A}_{ou}^{(x,\phantom{\rule{0.166667em}{0ex}}y)}$

#### 3.2.4. MD Time Step Size (h)

## 4. Path Sampling

## 5. Continuous-Time Limit

## 6. Discussion

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Onsager–Machlup Functionals

## Appendix B. Constructing Ornstein–Uhlenbeck Bridges

## Appendix C. BAB Splitting: Numerical Integration

## Appendix D. BAB Splitting: Energy Error

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**Figure 1.**Contour plot of the two-dimensional potential. The horizontal axis corresponds to the x value; the vertical axis, the y axis. The value of the white contour is two and of the dashed contour is $0.05$. The solid black contour enclosed by the dashed contour has a value of $0.001$. The potential at the saddle point is approximately unity. The potential at the local maximum is ≈1.6.

**Figure 2.**The equilibrium (Boltzmann) distribution for a temperature $\u03f5=0.05$ plotted as a “density” plot. The black areas denote the higher probability areas. The narrow channel connects the left and right probability basins. The lack of an energy barrier is seen by the lack of variation in the shading along the channel. The angle $\mathsf{\Theta}$ is pictured here.

**Figure 3.**The function $\overline{P}\left(\mathsf{\Theta}\right)$ plotted as a function of the angle $\mathsf{\Theta}$. See Equation (23) for its definition. The value of $\delta =\pi /40$ was used. Notice that the single-peak structure on the left differs from the twin peaks on the right. This structure is caused by the geometric factor, which arises when the radius line slices the distribution function.

**Figure 4.**The results of 10,000 forward integrations of Brownian dynamics with temperature $\u03f5=0.05$. All had the same initial conditions (in the left basin). The plot gives the fraction of paths with positive values of x as a function of time. As the dotted lines indicate, after $T=125$, over $35\%$ of the integrations have ending points in the right well. As the integration time exceeds 1000, the fraction approaches the equilibrium distribution.

**Figure 5.**The correlation functions ${d}^{\left(x\right)}\left(\tau \right)$ and ${d}^{\left(y\right)}\left(\tau \right)$ for two runs. For the black curves, the parameters were ${A}_{OU}^{x}={A}_{OU}^{y}=1$; for the gray curves, ${A}_{OU}^{x}={A}_{OU}^{y}=0$. The functions ${d}^{\left(x\right)}\left(\tau \right)$ are almost identical for the two runs: the lower black curve lies on top of the lower gray curve, hiding it.

**Figure 6.**The energy error plotted as a function of MD integration time. The black curves designate the error for the case ${A}_{ou}^{\left(x\right)}={A}_{ou}^{\left(y\right)}=\phantom{\rule{0.166667em}{0ex}}1$; the gray curves for ${A}_{ou}^{\left(x\right)}={A}_{ou}^{\left(y\right)}=\phantom{\rule{0.166667em}{0ex}}0$. The time step parameter for the former case is $h=6.667\times {10}^{-4}$ and for the latter $h=6.667\times {10}^{-5}$. In the latter case, the smaller h meant that an order of magnitude more computing resources were required to generated the gray curve, as compared to the black one.

**Figure 7.**The correlation functions ${d}^{\left(x\right)}\left(\tau \right)$ and ${A}_{ou}^{\left(y\right)}$ for various runs. For all the runs, ${A}_{ou}^{\left(x\right)}=1$. From the top, the curves for ${d}^{\left(y\right)}\left(\tau \right)$ have the values of ${A}_{ou}^{\left(y\right)}$ of $1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}8$, and 16, respectively. The curves for ${d}^{\left(x\right)}\left(\tau \right)$ all lie on top of one other; only one is plotted.

**Figure 8.**The effective-energy error plotted as a function of the MD integration time ($\tau $). The three curves correspond to runs with values ${A}_{ou}^{\left(x\right)}={A}_{ou}^{\left(y\right)}=1$. The gray curve gives the error for the run with $h=0.002$; the black curve, $h=0.001$; and the white curve, $h=0.00067$.

**Figure 9.**The Metropolis–Hastings acceptance rate for two runs. For the curve on the left, the parameters were ${A}_{OU}^{x}={A}_{OU}^{y}=0$ and for the curve on the right, ${A}_{OU}^{x}=1$ and ${A}_{OU}^{y}=8$. For both runs, the number of molecular dynamics steps were chosen to be $(1+\eta )\pi /\left(4\phantom{\rule{0.166667em}{0ex}}h\right)$ with $\eta $ being a uniformly distributed random number in the unit interval.

**Figure 10.**The one-dimensional representation of a typical path. The angle $\mathsf{\Theta}$ plotted as a function of time. See Figure 2 for the definition of $\mathsf{\Theta}$. At $t=0$, the particle starts out in the left basin, makes its way through the narrow channel at $t\approx 50$, and ends in the right basin.

**Figure 11.**Results for a calculation with ${A}_{OU}^{x}=1$ and ${A}_{OU}^{y}=8$. The bottom (black) curve is the fraction of the path with $\left|x\right|<1/2$; the middle (gray) curve is the fraction of the path with $x<-1/2$; and the top (black) curve is the fraction of the path with $x>1/2$.

**Figure 12.**Results for a calculation with ${A}_{OU}^{x}=1$ and ${A}_{OU}^{y}=4$. The bottom (black) curve is the fraction of the path with $\left|x\right|<1/2$; the middle (gray) curve is the fraction of the path with $x<-1/2$; and the top (black) curve is the fraction of the path with $x>1/2$.

**Figure 13.**Results for a calculation with ${A}_{OU}^{x}=1$ and ${A}_{OU}^{y}=4$. The solid black curve is the histogram of the $\overline{P}\left(\mathsf{\Theta}\right)$ for the 200,000 Metropolis steps pictured above in Figure 12. The dashed line represents the histogram that corresponds to the equilibrium distribution.

**Figure 14.**Results for a calculation with ${A}_{OU}^{x}={A}_{OU}^{y}=1$. The black curve is the fraction of the path with $x>1/2$; the light gray curve is the fraction of the path with $x<-1/2$; and the bottom (dark gray) curve is the fraction of the path with $\left|x\right|<1/2$.

**Figure 15.**Results for a calculation with ${A}_{OU}^{x}=1$ and ${A}_{OU}^{y}=4$ using Equation (A11) as the effective Hamiltonian. The black curve labeled “Center” is the fraction of the path with $\left|x\right|<1/2$; the solid gray curve is the fraction of the path with $x>1/2$; and the dashed gray curve is the fraction of the path with $x<-1/2$.

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Pinski, F.J. A Novel Hybrid Monte Carlo Algorithm for Sampling Path Space. *Entropy* **2021**, *23*, 499.
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Pinski FJ. A Novel Hybrid Monte Carlo Algorithm for Sampling Path Space. *Entropy*. 2021; 23(5):499.
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Pinski, Francis J. 2021. "A Novel Hybrid Monte Carlo Algorithm for Sampling Path Space" *Entropy* 23, no. 5: 499.
https://doi.org/10.3390/e23050499