Einstein-Helfand form for transport coefficients from coarse-grained descriptions.

Area of Science:

• Statistical Mechanics
• Theoretical Physics
• Computational Chemistry

Background:

• Coarse-graining complex systems requires understanding underlying detailed dynamics.
• Existing methods often use Fokker-Planck equations for coarse-grained dynamics.
• Generalizing Zwanzig projection operator theory is key for improved models.

Purpose of the Study:

• To develop a robust statistical mechanics framework for systems with pre-coarse-grained dynamics.
• To address the challenge of ensuring positive definiteness of the friction matrix in coarse-grained models.
• To investigate the implications of time reversal and detailed balance in the derived coarse-grained dynamics.

Main Methods:

• Generalizing Zwanzig's projection operator formalism.
• Deriving a friction matrix from a non-autocorrelation function.
• Reformulating the Green-Kubo transport matrix into the Einstein-Helfand form.

Main Results:

• A generalized Zwanzig theory applicable to systems with existing coarse-grained dynamics.
• A friction matrix expression not obviously positive definite.
• Demonstration that the Green-Kubo transport matrix can be expressed in a manifestly positive definite Einstein-Helfand form.

Conclusions:

• The developed method provides a rigorous approach to coarse-graining systems with pre-existing coarse-grained dynamics.
• Ensuring the positive definiteness of the friction matrix is achieved through the Einstein-Helfand formulation.
• The study clarifies the role of time reversal and detailed balance in the resulting coarse-grained dynamics.