One of the main goal of statistical mechanics is the derivation of effective evolution equations for the time evolution of large systems starting from a microscopic description of their dynamics. In this minicourse, I am going to review mathematically rigorous derivations of effective equations starting from first- principle quantum dynamics. The presentation will be self-contained.
The minicourse is divided in three parts.
Part 1: Derivation of the nonlinear Hartree equation for mean field systems.
n the first lecture I am going to illustrate a method, introduced in this context by Spohn in 1980, which can be used to obtain rigorous derivations of effective Hartree equations for interacting quantum mechanical systems in the so-called mean field limit. I am also going to present recent progress obtained in the application of Spohn's technique to systems with singular interaction potentials.
Part 2: The Gross-Pitaevskii equation for the dynamics of Bose Einstein condensates.
Bose Einstein condensates are states of many body quantum mechanical systems where almost all particles are in the same one- particle state. Extending the techniques used in the rigorous derivation of the non-linear Hartree equation, one can prove that, in a certain scaling limit (and in an appropriate sense), Bose- Einstein condensation is preserved by the time evolution, and that the condensate wave function evolves according to a certain cubic non-linear Schroedinger equation known as the Gross-Pitaevskii equation.
Part 3: Rate of convergence towards Hartree dynamics
In my last lecture I am going to present a different approach to obtain rigorous derivations of the nonlinear Hartree equation for the evolution of mean field systems. This approach, introduced by Hepp in 1973 to study the classical limit of quantum mechanics, makes use of techniques developed in quantum field theory (second quantization, Fock space representation, coherent states) and, compared with Spohn's method, it gives a better control of the fluctuations around the limiting Hartree dynamics. In particular, I will discuss recent results, obtained by extensions of Hepp's original technique, on the rate of convergence of the true quantum mechanical evolution towards the effective Hartree dynamics.
We will survey techniques developed for the treatment of nonlinear Schrodinger equations at the critical (or scaling) regularity. This incorporates the work of many authors over the last ten years or so.
The minicourse is divided into three parts.
Part 1: Refinements of Strichartz inequality.
To a large extent, recent progress has been made possible by developments in Harmonic Analysis. First we will describe linear and bilinear restriction theorems. These will be used to derive a concentration compactness principle for the linear propagator.
Part 2: Minimal blowup solutions.
We will discuss perturbation theory for the nonlinear equation. Together with the concentration compactness principle, this will lead us to a proof of the existence of minimal mass (or energy) blowup solutions. Such solutions have additional compactness properties, as we will describe.
Part 3: A monotone end.
Monotonicity formulae related to the virial identity and Morawetz inequality play a crucial role in deducing the non-existence of blowup solutions and hence, global well-posedness. These and how they are applied are the topic of the final part of this course.
Many of the results on wave maps seem highly technical and require deep results from harmonic analysis for a complete understanding. In these three lectures we present direct approaches to certain global aspects of the wave map problem, with powerful conclusions.
1.The Cauchy problem for wave maps
We present an elementary approach in configuration space for showing global existence and uniqueness for the Cauchy problem for wave maps from the (1+ m)-dimensional Minkowski space, m ≥ 4, to any complete Riemannian manifold with bounded curvature, provided the initial data are small in the critical norm.
2. Wave maps with symmetries I
Singularities of co-rotational wave maps from (1 + 2)-dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to non-constant co-rotational harmonic maps from S2 into N. In consequence, for non-compact target surfaces of revolution the Cauchy problem for wave maps is globally well-posed.
3. Wave maps with symmetries II
By using results from the previous lecture we show global existence of smooth solutions to the Cauchy problem for wave maps from the (1 + 2)-dimensional Minkowski space to an arbitrary smooth, compact Riemannian manifold without boundary, for arbitrary smooth, radially symmetric data.
This minicourse will place quantum N-body scattering in the context of propagation phenomena for partial differential equations, emphasizing the analogy with diffraction of waves from edges. The last lecture will explain the connection to scattering theory on symmetric spaces, such as SL(N,R)/SO(N,R).
Lecture 1: Set-up of N-body scattering, compactifications. Propagation of singularities for wave equation on manifolds with corners and in N-body scattering.
Lecture 2: Methods of proofs: pseudodifferential operator algebras, microlocal energy estimates.
Lecture 3: Set-up of symmetric spaces, reduction to N-body like problems, parametrix constructions, analytic continuation of the resolvent of the Laplacian.