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

Abstract

The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in a lot of scenarios like, for example, driven soft matter or nuclear magnetic resonance. In this article, we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.

26 pages

References in corpus (20)

• The Magnus expansion and some of its applications
• Tuned, driven, and active soft matter
• Memory Effects In Nonequilibrium Quantum Impurity Models
• From Reversible Quantum Microdynamics to Irreversible Quantum Transport
• Mode-Coupling Theory for Active Brownian Particles
• On the "generalized Generalized Langevin Equation"
• Tomographically reconstructed master equations for any open quantum dynamics
• Derivation of Hydrodynamics from the Hamiltonian description of particle systems
• Non equilibrium mode coupling theory for supercooled liquids and glasses
• Bulk Viscosity and Relaxation Time of Causal Dissipative Relativistic Fluid Dynamics
• Mapping between dissipative and Hamiltonian systems
• On the dynamics of reaction coordinates in classical, time-dependent, many-body processes
• Extended dynamical density functional theory for colloidal mixtures with temperature gradients
• Coupling a nano-particle with isothermal fluctuating hydrodynamics: Coarse-graining from microscopic to mesoscopic dynamics
• Derivation of Transport Equations using the Time-Dependent Projection Operator Method
• Application of the projection operator formalism to non-Hamiltonian dynamics
• Microscopic approach to entropy production
• A New Expansion of the Heisenberg Equation of Motion with Projection Operator
• Duality and Conditional Expectations in the Nakajima-Mori-Zwanzig Formulation
• Zwanzig-Mori projection operators and EEG dynamics: deriving a simple equation of motion

Cited by in corpus (25)

• Classical dynamical density functional theory: from fundamentals to applications
• Effects of social distancing and isolation on epidemic spreading: a dynamical density functional theory model
• Coarse-Grained Modelling Out of Equilibrium
• Projection operators in statistical mechanics: a pedagogical approach
• Jerky active matter: a phase field crystal model with translational and orientational memory
• Perspective: New directions in dynamical density functional theory
• Field-theoretical approach to open quantum systems and the Lindblad equation
• Generalized Langevin dynamics simulation with non-stationary memory kernels: How to make noise
• Nonequilibrium Statistical Mechanics and Optimal Prediction of Partially-Observed Complex Systems
• Mori-Zwanzig formalism for general relativity: a new approach to the averaging problem
• The Five Problems of Irreversibility
• Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism
• The non-equilibrium solvent response force: What happens if you push a Brownian particle
• Modeling of biomolecular machines in non-equilibrium steady states
• Comments on the validity of the non-stationary Generalized Langevin Equation as a coarse-grained evolution equation for microscopic stochastic dynamics
• On the generalized Langevin equation and the Mori projection operator technique
• Decoupling of External and Internal Dynamics in Driven Two-level Systems
• Memory-dependent noise-induced resonance and diffusion in non-markovian systems
• Nonlinear Langevin functionals for a driven probe
• Coarse-grained Second Order Response Theory
• Modelling Interfacial Dynamics Using Hydrodynamic Density Functional Theory: Dynamic Contact Angles and the Role of Local Viscosity
• Nonequilibrium statistics of barrier crossings with competing pathways
• The concept of minimal dissipation and the identification of work in autonomous systems: A view from classical statistical physics
• Relativistic Langevin Equation derived from a particle-bath Lagrangian
• Thermodynamics of an Empty Box