The problem of a two-component, collisionless plasma expansion into vacuum is investigated from the viewpoint of the Vlasov–Poisson model. The set of equations is treated both analytically (through the rescaling transformations) and numerically, using a one-dimensional Eulerian code. In planar geometry, the rescaling allows to conjecture the existence of a self-similar expansion over long times. Numerical results subsequently confirm the conjecture and show that the plasma becomes neutral over a smaller and smaller scale. A few thermodynamical properties are studied: the temperature is shown to decrease as t−2; the polytropic relation (d/dt)(pn−γ)=0 (with γ=3) is verified asymptotically via a semianalytical argument. Finally, the same problem is studied in a spherical one-dimensional geometry. The time-asymptotic solution is again self-similar. Numerical simulations show that a non-neutral, multiple-layer structure appears, which is proved to be stable over long times.

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The expansion of a quantum electron gas (non-relativistic, no spin) is investigated via the one-particle Schrödinger–Poisson model. Classically, the nonlinear term enhances the formation of a very regular asymptotic state. By means of rescaling methods, we conjecture that the quantum asymptotic solution is identical to the classical one. Subsequent numerical simulations confirm the above conjecture and define precisely the way in which the classical limit is approached.

For a special class of potentials, the dynamical evolution of any quantum wavepacket is entirely determined by the laws of classical mechanics. Here, the properties of this class are investigated both from the viewpoint of the Ehrenfest theorem (which provides the evolution of the average position and momentum), and the Wigner representation (which expresses quantum mechanics in a phase space formalism). Finally, these results are extended to the case of a charged particle in a uniform magnetic field.