The Monte Carlo technique was developed in 1946 by Ulam and Von Neumann [103] to solve neutron transport problems in the scope of the development of the atomic bomb. Monte Carlo simulation was later also adopted for kinetic transport problems in semiconductors, around 1966, by Kurosawa [104] and others. Monte Carlo algorithms were initially developed with the idea to emulate the physical behaviour of particles rather than to solve the Boltzmann transport equation per se. However, as the supporting mathematical theory in statistical sampling developed, it emerged that the Monte Carlo technique can be used as a general mathematical tool with applications going far beyond solving kinetic transport equations.
Indeed, there is a well-established Monte Carlo theory to solve (especially higher-dimensional) integrals or large systems of linear equations and integral equations efficiently [105]. Monte Carlo algorithms are better suited, compared to deterministic methods, to solve problems with a low regularity (smoothness) [106] and can be much more memory efficient, if large systems/domains are considered.
One can distinguish between grid and grid-free Monte Carlo algorithms to solve an equation: The former approach entails a discretization of the equation, whereafter the resulting system of linear algebraic equations is solved using Monte Carlo techniques. In certain cases, e.g. large integration domains, grid Monte Carlo algorithms show a superior computational complexity compared to grid-free algorithms [107]. Grid-free Monte Carlo algorithms, on the other hand, consider the integral form of the equation – this approach is applied to solve the WBE in this chapter.
The Monte Carlo approach is by now very well established in the field of semiconductor transport simulations; the de facto standard reference literature on the topic [108, 109, 110] can be consulted for an in-depth treatment of the topic. Gradually Monte Carlo algorithms have moved beyond a direct emulation of particle behaviour and issues of statistical enhancement were approached by adding a statistical weight to particles [111], or using backward-in-time trajectories [112]. These techniques were quickly recognised to be special cases of solving the BTE expressed as an integral equation and applying numerical Monte Carlo integration techniques [113]. This generalized approach to devising Monte Carlo algorithms has been used to develop the novel signed-particle method to solve the semi-discrete Wigner-Boltzmann equation and will be discussed in the remainder of this chapter.