Variational methods for time-dependent classical many-particle systems

Abstract

A variational method for the classical Liouville equation is introduced that facilitates the development of theories for non-equilibrium classical systems. The method is based on the introduction of a complex-valued auxiliary quantity Ψ that is related to the classical position-momentum probability density ρ via ρ = Ψ*Ψ. A functional of Ψ is developed whose extrema imply that ρ satisfies the Liouville equation. Multiscale methods are used to develop trial functions to be optimized by the variational principle. The present variational principle with multiscale trial functions can capture both the microscopic and the coarse-grained descriptions, thereby yielding theories that account for the two way exchange of information across multiple scales in space and time. Equations of the Smoluchowski form for the coarse-grained state probability density are obtained. Constraints on the initial state of the N-particle probability density for which the aforementioned equation is closed and conserves probability are presented. The methodology has applicability to a wide range of systems including macromolecular assemblies, ionic liquids, and nanoparticles.

A multiscale variational approach to the kinetics of viscous classical liquids: The coarse-grained mean field approximation

“…The CGMF ansatz is based on the notion that over long times and distances a migrating particle interacts with many others, i.e., experiences a mean field; in contrast, the Vlasov equation follows from the variational method but there the ansatz is directed at the microscopic level. 4 With this, the CGMF trial function takes the form…”

“…4 With this, an equation for υ is determined by finding the extrema of C with respect to υ * ( R, τ ) using a rationale similar to that for a Dirac action of quantum mechanics. Discussion of variational principles cast in terms of complex-valued functions was presented earlier, 40 and in the context of the classical Liouville equation as well.…”

“…Discussion of variational principles cast in terms of complex-valued functions was presented earlier, 40 and in the context of the classical Liouville equation as well. 4 The derivation proceeds according to the following steps. (1) Write C as the functional of the single-particle auxiliary function υ via the CGMF ansatz (16).…”

“…The question then emerges as to whether this CGMF concept might allow for the derivation of a closed kinetic equation for classical many-particle systems. Here, multiscale perturbation theory, 3 a variational method for the LE, 4 and a CGMF ansatz 24 are used to develop a kinetic equation for the single-particle density.…”

“…4 The resulting equation for the CG number density ϕ is not guaranteed to be conserving. Equation (A30) can be made closed and mass-conserving in ϕ by constraining o 1 that is embedded into Z (23), as in Eq.…”