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\title{Monte Carlo Branch Method for Simulation of Nonlinear
 Electron Quantum Kinetics in one-band Semiconductor\thanks{
  Supported by the Ministry of Education, Science and Technology
   of Bulgaria under Grant \# I 501/95.}}

\author{ T. Gurov$^1$, M. Nedjalkov$^1$, I. Dimov$^1$ }
\begin{document}
\maketitle

\footnotetext[1]{  Bulgarian Academy of Sciences,
 Central Laboratory for Parallel
    Processing, Department of High Performance Computing and
    Parallel Algorithms,
    Acad. G. Bonchev St., bl. 25 A, 1113  Sofia,
    Bulgaria, e-mail: gurov@iscbg.acad.bg,
 mixi@parallel.acad.bg, dimov@amigo.acad.bg, \,
Web site: http://www.acad.bg/BulRTD/math/dimov2.html}


\begin{abstract}

In this paper Monte Carlo method  for solving a
quantum-kinetic equation describing  an ultrafast semiconductor
carrier transport is proposed and studied.
This equation contains polynomial non-linearity which allows
 to use the link between non-stationary iterative processes and the
 branched Markov chains.

The presented Monte Carlo algorithm uses an appropriate simulation of
 branching stochastic processes.
It is proved that the mathematical expectation of the introduced
random variable is equal to a linear functional of the solution.

 A variety of numerical experiments are performed.  Particularly
 some nonlinear solutions are compared with the corresponding
 linear ones. It is shown that, depending on the initial condition,
 the nonlinearity can have an important effect on the evolution of
the solution.
The work demonstrates that the proposed stochastic
 method can be used for studying of a wide class  problems in this
 field.


\end{abstract}

{\it keywords:}
 Monte Carlo method, quantum-kinetic equation, random variable,
 branching stochastic processes


\end{document}