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Probabilistic analysis of the Single Particle Wigner Monte-Carlo method

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  • Dimov, I.
  • Savov, M.

Abstract

In this work we formulate the Wigner equation as an operator equation in a suitable L2 space. This allows us to rigorously express its solution and functionals thereof in terms of von Neumann series and in an already established fashion to represent each term of this series as the contribution of a single signed particle. Then by applying classical Berry–Esseen bounds and majorizing the times of sign change by the particle by Gamma random variables we are able to estimate the theoretical error of the so-called Single Particle Wigner Monte-Carlo method (SPWMC) for the Wigner equation in terms of the supremum of the potential of the Wigner equation, say γ∗. We have shown that if τ>0 is the time-length of the simulation then one needs to consider 27γ∗τ terms of the von Neumann series to ensure the theoretical stability of the SPWMC. Since γ∗ may be of the order of 1015 and for the Monte-Carlo scheme one would need a multiple of 27γ∗τ iterations to estimate each term in the series, this explains the numerical unstability which SPWMC has been observed to exhibit in some numerical simulations.

Suggested Citation

  • Dimov, I. & Savov, M., 2020. "Probabilistic analysis of the Single Particle Wigner Monte-Carlo method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 173(C), pages 32-50.
    • Handle: RePEc:eee:matcom:v:173:y:2020:i:c:p:32-50
    DOI: 10.1016/j.matcom.2020.01.008
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    References listed on IDEAS

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    1. Sellier Jean Michel & Nedjalkov Mihail & Dimov Ivan & Selberherr Siegfried, 2014. "A benchmark study of the Wigner Monte Carlo method," Monte Carlo Methods and Applications, De Gruyter, vol. 20(1), pages 43-51, March.
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    1. Sellier, J.M. & Nedjalkov, M. & Dimov, I. & Selberherr, S., 2015. "A comparison of approaches for the solution of the Wigner equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 107(C), pages 108-119.

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