Quantum surface diffusion in Bohmian Mechanics

TL;DR

This paper investigates surface diffusion of small adsorbates using Bohmian mechanics, extending classical and quantum models to include a stochastic trajectory approach and identifying distinct diffusion regimes.

Contribution

It introduces a Bohmian framework for surface diffusion analysis, combining classical stochastic trajectories with quantum effects via the Schr"odinger-Langevin equation.

Findings

Identifies ballistic, Brownian, and intermediate diffusion regimes.

Shows weak anomalous diffusion in the Bohmian framework under certain conditions.

Derives velocity autocorrelation functions consistent with classical results.

Abstract

Surface diffusion of small adsorbates is analyzed in terms of the so-called intermediate scattering function and dynamic structure factor, observables in experiments using the well-known quasielastic Helium atom scattering and Helium spin echo techniques. The linear theory used is an extension of the neutron scattering due to van Hove and considers the time evolution of the position of the adsorbates in the surface. This approach allows us to use a stochastic trajectory description following the classical, quantum and Bohmian frameworks. Three regimes of motion are clearly identified in the diffusion process: ballistic, Brownian and intermediate which are well characterized, for the first two regimes, through the mean square displacements and Einstein relation for the diffusion constant. The Langevin formalism is used by considering Ohmic friction, moderate surface temperatures and small coverages. In the Bohmian framework, the starting point is the so-called Schr\"odinger-Langevin equation which is a nonlinear, logarithmic differential equation. By assuming a Gaussian function for the probability density, the corresponding quantum stochastic trajectories are given by a dressing scheme consisting of a classical stochastic trajectory of the center of the Gaussian wave packet, issued from solving the Langevin equation (particle property), plus the time evolution of its width governed by the damped Pinney differential equation (wave property). The velocity autocorrelation function is the same as the classical one when the initial spread rate is assumed to be zero. If not, in the diffusion regime, the Brownian-Bohmian motion shows a weak anomalous diffusion.