Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from
1Weapons and Materials Research Directorate, U.S. Army DEVCOM Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005, USA.
Physical Review. E
|September 16, 2021
Summary
We developed a generalized Langevin equation (GLE) for open systems with time-dependent potentials. This framework accurately describes system dynamics, including memory effects and time-dependent friction for externally forced systems.
Area of Science:
- Statistical Mechanics
- Classical Mechanics
- Non-equilibrium Thermodynamics
Background:
- Classical Hamiltonians with time-dependent potentials describe open systems.
- The Mori-Zwanzig projection operator method is used for reduced descriptions of Hamiltonian systems.
Purpose of the Study:
- To present a generalized Langevin equation (GLE) governing the exact time evolution of phase-space observables in finite open systems with time-dependent potentials.
- To demonstrate that GLE dynamics in such systems are determined by conservative, dissipative memory, and projected force fields.
- To formulate particle-based, coarse-grained (CG) GLE dynamics and extend dissipative particle dynamics to open systems.
Main Methods:
- Utilizing the Mori-Zwanzig projection operator method within a Heisenberg picture.
- Employing time-independent Zwanzig projection operators for a reduced description of Hamiltonian systems.
- Deriving canonical and generalized canonical GLEs using Zwanzig operators based on probability densities.
- Transitioning to Jacobi coordinates for particle-based, coarse-grained (CG) GLE dynamics.
- Applying a Markovian approximation to the canonical CG GLE for extending dissipative particle dynamics.
Main Results:
- The derived GLE dynamics are governed by conservative, dissipative memory, and projected force fields.
- Memory functions exhibit a relationship with the projected force, similar to equilibrium fluctuation-dissipation relations.
- The memory kernel generally depends on momentum gradients of the irrelevant subsystem's entropy.
- Canonical CG GLE for relevant momenta generalizes known equations of motion for closed systems.
- Explicitly time-dependent frictions are introduced, reflecting changes in dissipation rates due to time-dependent bath coupling.
Conclusions:
- The developed GLE formalism provides a general and viable framework for microscopically informed coarse-grained treatments.
- This framework is applicable to externally forced systems far from equilibrium.
- The inclusion of time-dependent frictions offers a novel approach to modeling dissipation in open systems.
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