Considered is the Wigner equation which accounts for the nondissipative part of the transport via the coherent free term and the Wigner potential and for dissipation processes via the Boltzmann collision operator . In the Monte Carlo simulation all three dimensions of the momentum space can be considered.

In contrast the finite difference methods consider the momentum space only as one-dimensional. Dissipation is only included in a relaxation time approximation [SHMS98].

For one-dimensional devices the equation reads:

The Boltzmann collision operator is defined by the scattering rate

is the probability density per unit time for scattering from state to state . is a cumulative quantity which accounts for different scattering sources such as phonons and impurities. The total out-scattering rate is defined by the integral over all after-scattering states as

We stress that the Boltzmann scattering operator is a superoperator and cannot be written as an ordinary commutator. Technically this is the way in which proper quantum mechanics - which is time-reversible - is extended. As the Wigner formalism is naturally a superoperator formalism this is more easily achieved in the Wigner picture [Roy91]. The superoperator formalism was favored by Prigogine [GP79] as a framework for time-irreversible quantum mechanics.

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