Fast electrostatic solvers for kinetic Monte Carlo simulations

William Saunders, James Grant, Eike Müller, Ian Thompson

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Kinetic Monte Carlo (KMC) is an important computational tool in theoretical physics and chemistry. In contrast to standard Monte Carlo, KMC permits the description of time dependent dynamical processes and is not restricted to systems in equilibrium. Compared to Molecular Dynamics, it allows simulations over significantly longer timescales. Recently KMC has been applied successfully in modelling of novel energy materials such as Lithium-ion batteries and organic/perovskite solar cells. Motivated by this, we consider general solid state systems which contain free, interacting particles which can hop between localised sites in the material. The KMC transition rates for those hops depend on the change in total potential energy of the system. For charged particles this requires the frequent calculation of electrostatic interactions, which is usually the bottleneck of the simulation. To avoid this issue and obtain results in reasonable times, many studies replace the long-range potential by a phenomenological short range approximation. This, however, leads to systematic errors and unphysical results. On the other hand standard electrostatic solvers such as Ewald summation or fast Poisson solvers are highly inefficient in the KMC setup or introduce uncontrollable systematic errors at high resolution. In this paper we describe how the Fast Multipole Method by Greengard and Rokhlin can be adapted to overcome this issue by dramatically reducing computational costs. We exploit the fact that each update in the transition rate calculation corresponds to a single particle move and changes the configuration only by a small amount. This allows us to construct an algorithm which scales linearly in the number of charges for each KMC step, something which had not been deemed to be possible before. We demonstrate the performance and parallel scalability of the method by implementing it in a performance portable software library, which was recently developed in our group. We describe the high-level Python interface of the code which makes it easy to adapt to specific use cases.

Original languageEnglish
Article number109379
Number of pages28
JournalJournal of Computational Physics
Early online date4 Mar 2020
Publication statusPublished - 1 Jun 2020

Bibliographical note

Funding Information:
This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath and the Isambard UK National Tier-2 HPC Service ( ). Isambard is operated by GW4 and the UK Met Office, and funded by EPSRC ( EP/P020224/1 ). This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service ( ), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/P020259/1 ), and DiRAC funding from the Science and Technology Facilities Council ( ). This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreements No 646176 and No 824158 . William Saunders was funded by an EPSRC studentship during his PhD. We would like to thank Bérenger Bramas and Olivier Coulaud for help with running ScalFMM on our cluster.

Publisher Copyright:
© 2020 Elsevier Inc.


  • Domain specific language
  • Electrostatics
  • Fast Multipole Method
  • Kinetic Monte Carlo
  • Parallel computing

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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