Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics

Abstract

In these lecture notes we discuss recent progress in the rigorous derivation of effective evolution equations for the description of the dynamics of quantum mechanical many-body systems.

56 pages, 2 figures. Lecture Notes for the mini-course held at the 2008 CMI Summer School "Evolution Equations" at ETH Zurich

References in corpus (5)

• Proof of Bose-Einstein Condensation for Dilute Trapped Gases
• Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems
• Rigorous Derivation of the Gross-Pitaevskii Equation
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy

Cited by in corpus (17)

• Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $\mathbb{R}$
• On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy
• Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies
• Mean-field quantum dynamics for a mixture of Bose-Einstein condensates
• Randomization and the Gross-Pitaevskii hierarchy
• Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations
• Local existence of solutions to Randomized Gross-Pitaevskii hierarchies
• Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
• The NLS limit for bosons in a quantum waveguide
• Bosons in a double well: two-mode approximation and fluctuations
• Effective non-linear spinor dynamics in a spin-1 Bose-Einstein condensate
• Reduced fluctuations for bosons in a double well
• Limites de champ moyen bosoniques {à} temp{é}rature positive
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate
• Quantum Kinetic Equations and Evolution of Many-Particle Systems
• The law of large numbers for quantum stochastic filtering and control of many particle systems