The dynamics of a quantum system with a large number $N$ of identical bosonic particles interacting by means of weak two-body potentials can be simplified by using mean-field equations in which all interactions to any one body have been replaced with an average or effective interaction in the mean-field limit $N \rightarrow \infty$ . In order to show these mean-field equations are accurate, one needs to show convergence of the quantum $N$ -body dynamics to these equations in the mean-field limit. Previous results on convergence in the mean field limit have been derived for certain initial conditions in the case of one species of bosonic particles, but no results have yet been shown for multi-species. In this thesis, we look at a quantum bosonic system with two species of particles. For this system, we derive a formula for the rate of convergence in the mean-field limit in the case of an initial coherent state, and we also show convergence in the mean-field limit for the case of an initial factorized state. The analysis for two species can then be extended to multiple species.The dynamics of a quantum system with a large number $N$ of identical bosonic particles interacting by means of weak two-body potentials can be simplified by using mean-field equations in which all interactions to any one body have been replaced with an average or effective interaction in the mean-field limit $N \rightarrow \infty$. In order to show these mean-field equations are accurate, one needs to show convergence of the quantum $N$-body dynamics to these equations in the mean-field limit. Previous results on convergence in the mean field limit have been derived for certain initial conditions in the case of one species of bosonic particles, but no results have yet been shown for multi-species. In this thesis, we look at a quantum bosonic system with two species of particles. For this system, we derive a formula for the rate of convergence in the mean-field limit in the case of an initial coherent state, and we also show convergence in the mean-field limit for the case of an initial factorized state. The analysis for two species can then be extended to multiple species.

Aug 1, 20141 citation

Interacting boson systems

Xiao-Gang Wen

Oxford University Press eBooksSep 6, 20070 citations

Interacting boson systems

WORLD SCIENTIFIC eBooksApr 1, 20050 citations

Higher order effective interactions and effective bosonized model for 2-N particle states

Fabio L. Braghin

In this work, an effective fermion model with particular higher order interactions given by: , for finite N, is investigated by means of the auxiliary field method by taking into account 2n-particle states as proposed in Ref. [1]. The bosonized model was found to exhibit an approximate symmetry when expanded for weak field fluctuations around the mean field solutions. In the present work, the role of the mean field solutions for the corresponding auxiliary fields is investigated. With the integration of fermions, the resulting determinant is expanded in a polynomial boson model in the weak field approximation and the approximate symmetry found for this series does not appear if the boson mean fields are considered to be zero.In this work, an effective fermion model with particular higher order interactions given by: , for finite N, is investigated by means of the auxiliary field method by taking into account 2n-particle states as proposed in Ref. [1]. The bosonized model was found to exhibit an approximate symmetry when expanded for weak field fluctuations around the mean field solutions. In the present work, the role of the mean field solutions for the corresponding auxiliary fields is investigated. With the integration of fermions, the resulting determinant is expanded in a polynomial boson model in the weak field approximation and the approximate symmetry found for this series does not appear if the boson mean fields are considered to be zero.

arXiv:1511.04365EPJ Web of ConferencesJan 1, 20160 citations

Higher order effective interactions and effective bosonized model for 2-N particle states

Fábio L. Braghin

\nIn this work, an effective fermion model with particular higher order interactions given by: ${I_{II}} = \\mathop \\sum \\limits_n^N g{2^n}{({\\bar \\psi _a}{\\psi _a})^{{2^n}}}$ , for finite N, is investigated by means of the auxiliary field method by taking into account 2n-particle states as proposed in Ref. [1]. The bosonized model was found to exhibit an approximate symmetry when expanded for weak field fluctuations around the mean field solutions. In the present work, the role of the mean field solutions for the corresponding auxiliary fields is investigated. With the integration of fermions, the resulting determinant is expanded in a polynomial boson model in the weak field approximation and the approximate symmetry found for this series does not appear if the boson mean fields are considered to be zero.\n\nIn this work, an effective fermion model with particular higher order interactions given by: ${I_{II}} = \\mathop \\sum \\limits_n^N g{2^n}{({\\bar \\psi _a}{\\psi _a})^{{2^n}}}$, for finite N, is investigated by means of the auxiliary field method by taking into account 2n-particle states as proposed in Ref. [1]. The bosonized model was found to exhibit an approximate symmetry when expanded for weak field fluctuations around the mean field solutions. In the present work, the role of the mean field solutions for the corresponding auxiliary fields is investigated. With the integration of fermions, the resulting determinant is expanded in a polynomial boson model in the weak field approximation and the approximate symmetry found for this series does not appear if the boson mean fields are considered to be zero.\n

Mar 25, 20160 citations

Interacting boson systems

WORLD SCIENTIFIC eBooksMay 1, 20080 citations

The dynamics of a quantum system with a large number

N

of identical bosonic particles interacting by means of weak two-body potentials can be simplified by using mean-field equations in which all interactions to any one body have been replaced with an average or effective interaction in the mean-field limit

N \rightarrow \infty

. In order to show these mean-field equations are accurate, one needs to show convergence of the quantum

N

-body dynamics to these equations in the mean-field limit. Previous results on convergence in the mean field limit have been derived for certain initial conditions in the case of one species of bosonic particles, but no results have yet been shown for multi-species. In this thesis, we look at a quantum bosonic system with two species of particles. For this system, we derive a formula for the rate of convergence in the mean-field limit in the case of an initial coherent state, and we also show convergence in the mean-field limit for the case of an initial factorized state. The analysis for two species can then be extended to multiple species.

\nIn this work, an effective fermion model with particular higher order interactions given by:

{I_{II}} = \\mathop \\sum \\limits_n^N g{2^n}{({\\bar \\psi _a}{\\psi _a})^{{2^n}}}

, for finite N, is investigated by means of the auxiliary field method by taking into account 2n-particle states as proposed in Ref. [1]. The bosonized model was found to exhibit an approximate symmetry when expanded for weak field fluctuations around the mean field solutions. In the present work, the role of the mean field solutions for the corresponding auxiliary fields is investigated. With the integration of fermions, the resulting determinant is expanded in a polynomial boson model in the weak field approximation and the approximate symmetry found for this series does not appear if the boson mean fields are considered to be zero.\n