Monte Carlo generation of localized particle trajectories

We introduce modifications to Monte Carlo simulations of the Feynman path integral that improve sampling of localized interactions. The algorithms generate trajectories in simple background potentials designed to concentrate them around the interaction region, reminiscent of importance sampling. This improves statistical sampling of the system and overcomes a long-time undersampling problem caused by the spatial diffusion inherent in Brownian motion. We prove the validity of our approach using previous analytic work on the distribution of values of the Wilson line over path integral trajectories and illustrate the improvements on some simple quantum mechanical systems.

Figure 1

An illustrative WMC estimation of the kernel, $K$, for the harmonic oscillator using $N_L=25000$ trajectories and $N_p=5000$ points. Undersampling is manifest in the late time deviation from linearity as predicted by (10).

Figure 2

The distribution $\mathcal{P}\left(v\right|0,0,40)$ for a harmonic oscillator ($m=1, \omega =1$) sampled with $N_L={10}^6$ trajectories. The histogram of sampled values of $v\left[x\right]$ is shown against the analytic result (blue solid line), with good agreement.

Figure 3

WMC estimate of the kernel $K(0,0;T)$ for the harmonic oscillator ($\omega =1, N_L=25000, N_p=5000, \mathrm{\Omega }=0.75$) using free particle (fp) paths and trajectories generated in a quadratic background potential (qbp), compensated via (20).

Figure 4

WMC estimate of the kernel $K(0,0;T)$ for the linear potential ($k=0.5, N_L=25000, N_p=5000, \kappa =0.48$) using free particle (fp) trajectories and trajectories generated in a linear background potential (lbp), compensated via (21).

Figure 5

WMC estimate of $K(0,0;T)$ for the harmonic oscillator ($\omega =1, N_L=25000, N_p=5000, \mathrm{\Omega }=0.75$) using fp trajectories and a qbp, with the potential subtraction (32).

Figure 6

WMC estimate of $K(0,0;T)$ for the linear potential ($k=0.5, N_L=25000, N_p=5000, \kappa =0.48$) with fp trajectories and a lbp, compensated with potential subtraction (33).

Figure 7

WMC estimate of the kernel $K(0,0;T)$ for the Pöschl-Teller potential ($\lambda =1, \alpha =1, N_L=25000, N_p=5000, \mathrm{\Omega }=0.75$) using fp trajectories and qbp with potential subtraction.

Figure 8

WMC estimate of the kernel $K(0,0;T)$ for a cubic memory kernel ($\mu =0.025, N_L=25000, N_p=5000, \kappa =0.1$) using fp and lbp trajectories (potential subtraction).