Variational methods for time-dependent classical many-particle systems.

Area of Science:

• Statistical Mechanics
• Theoretical Physics
• Computational Chemistry

Background:

• The classical Liouville equation describes the time evolution of probability density in phase space for classical systems.
• Developing theories for non-equilibrium classical systems presents significant challenges, particularly in bridging microscopic and coarse-grained descriptions.
• Existing methods often struggle to capture the two-way exchange of information across multiple scales.

Purpose of the Study:

• Introduce a novel variational method for the classical Liouville equation.
• Facilitate the development of theories for non-equilibrium classical systems.
• Develop a framework that accounts for information exchange across multiple scales.

Main Methods:

• Introduced a complex-valued auxiliary quantity Ψ, related to the probability density ρ by ρ = Ψ*Ψ.
• Developed a functional of Ψ whose extrema satisfy the Liouville equation.
• Employed multiscale methods to construct trial functions for variational principle optimization.

Main Results:

• The variational principle with multiscale trial functions successfully captures both microscopic and coarse-grained descriptions.
• Derived Smoluchowski-form equations for the coarse-grained state probability density.
• Presented constraints for the initial N-particle probability density ensuring equation closure and probability conservation.

Conclusions:

• The developed variational method provides a robust framework for studying non-equilibrium classical systems.
• The methodology effectively integrates information across different scales, crucial for complex systems.
• Applicable to diverse systems like macromolecular assemblies, ionic liquids, and nanoparticles.