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Chapter 3
Signed-Particle Method

This chapter discusses the mathematical foundation of the signed-particle method used to solve the Wigner-Boltzmann equation using Monte Carlo techniques. The integral formulation of the WBE is introduced. The integral form is developed into a Neumann series, which is evaluated using Monte Carlo integration. Finally, the basic building blocks and architecture of the algorithm implemented in the Wigner Ensemble Monte Carlo (WEMC) simulator – as part of the open source project ViennaWD [102] – are presented.

 3.1 Background
 3.2 Outline
 3.3 Integral formulation
  3.3.1 Fredholm integral form
  3.3.2 Adjoint equation
 3.4 Neumann series
 3.5 Monte Carlo integration
 3.6 Computational task
  3.6.1 Wigner potential as a scattering mechanism
 3.7 Algorithm
  3.7.1 Structure
  3.7.2 Discretization
  3.7.3 Particle initialization
  3.7.4 Free-flight
  3.7.5 Scattering
  3.7.6 Annihilation process
  3.7.7 Implementation

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