Derivation of meanfield equations and nextorder corrections for bosons and fermions
Derivation of meanfield equations and nextorder corrections for bosons and fermions
This thesis is about the derivation of effective mean field equations and their nextorder corrections starting from nonrelativistic manybody quantum theory. Mean field equations provide an approximate ansatz for the description of interacting manyparticle systems. In this ansatz the interaction between the particles is replaced by a selfconsistent external potential leading to an effective onebody description of the manyparticle system. Nextorder 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. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a manybody Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its nextorder corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three selfcontained chapters.
In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective oneparticle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of BoseEinstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single oneparticle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the BoseEinstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the Nparticle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the Nparticle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space.
Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding $l$ particles out of their total number N not in the condensate is exponentially small in the number $l$. 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 lowlying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energymomentum 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. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. 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 nexttoleading order, one has to consider an additional correction to the mean field energy which is due to socalled recollision processes.
Mean field equations, Bose gas, Fermi gas
03. Mar 2017
2017
English
Universitätsbibliothek der LudwigMaximiliansUniversität München
Mitrouskas, David
(2017):
Derivation of meanfield equations and nextorder corrections for bosons and fermions.
Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics

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Abstract
This thesis is about the derivation of effective mean field equations and their nextorder corrections starting from nonrelativistic manybody quantum theory. Mean field equations provide an approximate ansatz for the description of interacting manyparticle systems. In this ansatz the interaction between the particles is replaced by a selfconsistent external potential leading to an effective onebody description of the manyparticle system. Nextorder 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. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a manybody Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its nextorder corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three selfcontained chapters.
In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective oneparticle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of BoseEinstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single oneparticle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the BoseEinstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the Nparticle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the Nparticle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space.
Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding $l$ particles out of their total number N not in the condensate is exponentially small in the number $l$. 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 lowlying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energymomentum 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. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. 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 nexttoleading order, one has to consider an additional correction to the mean field energy which is due to socalled recollision processes.