Convergence and multiplicity of normal modes of differentiated projected Liouville equation

Abstract

The Liouville equation is a traditional way of decribing non-equilibrium behavior of a statistical system. The use of projection techniques allows an exact equation for the momentum distribution function for N particles
∂ϕ_N/∂t = − iP₀Lϕ_N − iP₀L_iG(t)ϕ_N(0) − λ²P₀L_i² ∫₀^t G(t−s)ϕ_N(s) ds.
It was discovered that a further time derivative gives a second order equation that simplifies very nicely when projected over all but one momentum coordinate. Under a "weak coupling" approximation in the sense that the integral term was neglected, calculations regarding the time behavior of a system can be easily made. A further differentiation yielding a third order equation can, upon use of a weak coupling ansatz, lead to suggestions about turbulent behavior. However, the problem related to the validity of the introduction of extra time derivatives in exact physical equations remains to be addressed, so is the validity of the weak coupling ansatz.