Variational methods for time-dependent classical many-particle systems
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.
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Volume (Year): 392 (2013)
Issue (Month): 4 ()
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- Pankavich, S. & Shreif, Z. & Ortoleva, P., 2008. "Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(16), pages 4053-4069.
- Shea, Joan-Emma & Oppenheim, Irwin, 1997. "Fokker-Planck equation and non-linear hydrodynamic equations of a system of several Brownian particles in a non-equilibrium bath," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 247(1), pages 417-443.
- Shea, Joan-Emma & Oppenheim, Irwin, 1998. "Fokker–Planck and non-linear hydrodynamic equations of an inelastic system of several Brownian particles in a non-equilibrium bath," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 250(1), pages 265-294.
- Shreif, Z. & Ortoleva, P., 2009. "Multiscale derivation of an augmented Smoluchowski equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(5), pages 593-600.
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