Approximate solutions of the Liouville equation I. A truncation scheme for distribution functions

✍ Scribed by Eugene P Gross

• Publisher: Elsevier Science
• Year: 1972
• Tongue: English
• Weight: 690 KB
• Volume: 69
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(72)90004-8

✦ Synopsis

The Liouville equation is studied in the domain of validity of linear response theory. The approach is to assume a functional form for the N-body distribution which fits classes of initial distributions exactly. This determines all time-dependent, reduced distribution functions. In the first approximation, the deviation from the exact equilibrium function is assumed to be one-body additive in phase space for all times. The one-body function is fixed by the first equation of the timedependent hierarchy. This leads to Zwanzig's modification of the linearized Vlasov equation. In the second approximation, we include a two-body additive function and use the tirst two equations of the hierarchy. With the help of exact identities satisfied by the equilibrium correlations one obtains kinetic equations for the singlet and pair distributions in which the potential has been eliminated and only equilibrium correlation functions appear. In the second approximation, the second and fourth frequency moments of the inelastic scattering function are exact. The resulting truncation scheme is meaningful in any order for both strong short-range forces and for plasmas.