A Laval nozzle with its typical convergent and then divergent shape has the interesting thermodynamic properties of accelerating expanding gas above supersonic velocities and of cooling the gas in the process. This work gives a theoretical approach to the thermody- namics in a macroscopic Laval nozzle and studies the expansion of the gas in a microscopic Laval nozzle by means of non-equilibrium molecular dynamics simulations. A scheme for simulating such a non-equilibrium system is introduced for the nozzle with grand canon- ical Monte-Carlo exchange of particles to provide a stationary flow. The simulation is realized for a mono-atomic gas simulated with a Lennard-Jones-potential and for a nozzle with perfectly smooth walls. We investigate the thermodynamic state variables pressure, density, and temperature as well as the Knudsen number, Mach number and velocity of the gas for nozzles of different size. For the temperature we will have a closer look how well macroscopic assumptions are still fulfilled during the expansion in the nozzle. This investigation will also motivate a closer look at the velocity distribution inside the nozzle as well as a investigation of the velocity auto-correlation and pair-correlation function. Another focus of this work is on the speed of sound inside the nozzle and with this the position of the sonic horizon and if there is even a well defined sonic horizon on a micro- scopic scale. This work tries to answer this by studying density fluctuation correlation in the nozzle which provide information about the dispersion and attenuation of small per- turbations of the density which are naturally occurring in molecular dynamic simulations.