arXiv:1003.4713 · Advances in Mathematics

Second-order corrections to mean field evolution of weakly interacting Bosons. II

Manoussos G. Grillakis, Matei Machedon, Dionisios Margetis

Published Jul 29, 2011• 91 citations

arXiv HTML PDF

DOI: 10.1016/j.aim.2011.06.028

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arXiv:1003.4713 · Advances in Mathematics

Second-order corrections to mean field evolution of weakly interacting Bosons. II

Manoussos G. Grillakis, Matei Machedon, Dionisios Margetis

Published Jul 29, 2011• 91 citations

arXiv HTML PDF

DOI: 10.1016/j.aim.2011.06.028

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arXiv:0904.0158Communications in Mathematical PhysicsOct 1, 2009141 citations

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We consider the evolution of $N$ bosons, where $N$ is large, with two-body interactions of the form $N^{3\beta}v(N^{\beta}\mathbf{\cdot})$ , $0\leq\beta\leq 1$ . The parameter $\beta$ measures the strength of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [25, 26]. For $0\leq \beta < 1/2$ , we derive an error bound of the form $p(t)/N^\alpha$ , where $\alpha>0$ and $p(t)$ is a polynomial, which implies a specific rate of convergence as $N\rightarrow\infty$ .We consider the evolution of $N$ bosons, where $N$ is large, with two-body interactions of the form $N^{3\beta}v(N^{\beta}\mathbf{\cdot})$, $0\leq\beta\leq 1$. The parameter $\beta$ measures the strength of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [25, 26]. For $0\leq \beta < 1/2$, we derive an error bound of the form $p(t)/N^\alpha$, where $\alpha>0$ and $p(t)$ is a polynomial, which implies a specific rate of convergence as $N\rightarrow\infty$.

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K. Nomura

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We consider the evolution of

N

bosons, where

N

is large, with two-body interactions of the form

N^{3\beta}v(N^{\beta}\mathbf{\cdot})

0\leq\beta\leq 1

. The parameter

\beta

measures the strength of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [25, 26]. For

0\leq \beta < 1/2

, we derive an error bound of the form

p(t)/N^\alpha

, where

\alpha>0

and

p(t)

is a polynomial, which implies a specific rate of convergence as

N\rightarrow\infty