A Universal Model: The Vlasov Equation

Abstract

Halfway between the N body model and the usual hydrodynamic, one the Vlasov equation (supplemented by the Poisson‐Maxwell equations) describes different media going from nuclear matter to the expanding universe (via semiconductors, plasmas, and stellar dynamics problems and the introduction of a quantum counterpart, the so‐called Wigner equation). In the first part we discuss the conditions of validity and how the equation can be pushed at the particle level through the dressed test particle picture. In a second part we discuss how, via water bag and multi water bag models, at least for one‐dimensional problems, the Vlasov model can be interpreted as a multifluid hydrodynamics and how mathematical similarities between 2D incompressible and the 2D phase space fluid can be used in vortex stability problems. The next part presents self similar group techniques and their embedding in the rescaling method to obtain the asymptotic solutions of expanding one species plasma and self‐gravitating universe. The last part shows how numerical methods must concentrate on keeping useful information and the interest of using as new numerical variables the dynamical invariants (energy, angular momentum, magnetic moment, etc.). This need to include analytical results into our numerical schemes for a better understanding of the Vlasov model will be our conclusion.

Keywords:

• Vlasov equation
• plasma physics
• stellar dynamics
• computational plasma physics
• Wigner equation
• water bag model
• self similarity
• dressed test particles
• phase space fluid

Notes

¹See also the classical textbook by Chandrasekhar (1942), for a historical review on the Vlasov equation the reader is also refered to a paper by M. Hénon (1982).

²For more details on the expansion of the universe, see Peebles (1980).

³See, for instance, phase space plots of Figure 4 in Huot et al. (2003).