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The collisionless Boltzmann equation, also called free-streaming, is aconvenient approximation. It is rather simple to implement numerically, and andit is effective at reducing the irregularities of rough initial conditions. Itcan be obtained as a small $\hbar$ limit from a free scalar quantum field.Namely, by neglecting the $\hbar^2$ term in the dynamical evolution of theWigner distribution, the quantum precursor of the distribution function.  In this work it is presented the general form for the exact solutions of theWigner distribution, for a scalar field undergoing a $(1+1)$-dimensionalexpansion. Namely, with the symmetry constraint of rotation and translationinvariance in the transverse plane. It is very different from the (on-shell)free-streaming of classical particles. It is shown how to recognize theclassical, $\hbar\to 0$, limit from the general form. The numerical analysisfor a specific example shows very large quantum corrections to the classicalfree-streaming.

Quantum free-streaming: out of equilibrium expansion for the free scalar fields