Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

Area of Science:

• Statistical Mechanics
• Theoretical Physics

Background:

• The Mori-Zwanzig projection operator formalism is crucial for deriving mesoscopic and macroscopic theories from microscopic equations.
• Current limitations prevent its direct application to systems with time-dependent Hamiltonians, which are common in areas like driven soft matter and nuclear magnetic resonance.

Purpose of the Study:

• To generalize the Mori-Zwanzig formalism to accommodate time-dependent Hamiltonians.
• To extend the formalism's applicability to classical and quantum systems, near and far from thermodynamic equilibrium, and for time-dependent observables.

Main Methods:

• Derivation of a generalized Mori-Zwanzig formalism for time-dependent systems.
• Development of approximation techniques for equations of motion and correlation functions.
• Application to spin relaxation in a time-dependent external magnetic field.

Main Results:

• A novel Mori-Zwanzig formalism capable of treating time-dependent Hamiltonians has been derived.
• Approximation techniques have been developed to enhance practical usability.
• The Bloch equations for spin relaxation were derived from first principles, including microscopic expressions for relaxation times.

Conclusions:

• The generalized Mori-Zwanzig formalism successfully extends the theory to systems with time-dependent Hamiltonians.
• The developed approximations and demonstrated application to spin relaxation highlight the formalism's practical utility.
• This work opens new avenues for theoretical modeling in diverse fields of physics and chemistry.