Path Integral Methods in Atomistic Modelling: An Introduction

Abstract

This book provides an introduction to path integral methods and their application to modeling atomistic processes. The book covers both the foundational theory and recently developed simulation techniques. The text provides a self-contained resource and was originally developed for the CECAM schools on Path Integral Methods.

Figures (38)

• Figure 1: Evolution of a configuration in phase space and of the corresponding volume element along a molecular dynamics simulation.
• Figure 2: Energy conservation for a simulation of liquid water at room temperature, using a velocity Verlet integrator with three different values of the time step.
• Figure 3: A collection of sample paths for a random process. Also shown is how the one-time probability density $\mathcal{P}\left(\mathbf{x},t\right)$ can be constructed as the distribution of the points of all the sample paths at a given time.
• Figure 4: Autocorrelation time for different observables for a harmonic oscillator of frequency $\omega$, as a function of the Langevin friction $\gamma$. Both the friction and the autocorrelation times are expressed in terms of the intrinsic time scale of the oscillator.
• Figure 5: Discretized possible paths. A particle at point $q_0$ at time $t=0$ tries all possible ways to arrive at $q$ at time $t=P\delta t$. Each path is weighted by the respective action as given by the path-integral time evolution amplitudes $K(q, q_0, t)$.