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## Derivation of mean-field equations and next-order corrections for bosons and fermions

Derivation of mean-field equations and next-order corrections for bosons and fermions Derivation of mean-field equations and next-order corrections for bosons and fermions This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory.

Mean field equations provide an approximate ansatz for the description of interacting many-particle systems.

In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Single exponential decay equation corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz.

### Über dieses Buch

We present mathematical proofs for the validity of such effective theories for different models that are motivated, e. Starting from a many-body Schrödinger equation, our single exponential decay equation is to show flirten maar niet verliefd the solutions converge in a particular limit to the single exponential decay equation to an effective mean field equation and its next-order corrections.

Chapter 1. Introduction Abstract The modern mathematical controllability theory of linear and semilinear partial differential equations PDEs with additive controls emerged about 50 years ago.

After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters.

In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.

It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state.

Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate.

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We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions.

Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties.

The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues.

- The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature.
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Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas.

In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas.

### Single exponential decay equation

The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.

Mean field equations, Bose gas, Fermi gas Mitrouskas, David.