# A Central Limit Theorem in Many-Body Quantum Dynamics

@article{Arous2011ACL, title={A Central Limit Theorem in Many-Body Quantum Dynamics}, author={G{\'e}rard Ben Arous and Kay L Kirkpatrick and Benjamin Schlein}, journal={Communications in Mathematical Physics}, year={2011}, volume={321}, pages={371-417} }

We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing…

## 51 Citations

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## References

SHOWING 1-10 OF 34 REFERENCES

### Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics

- Mathematics, Physics
- 2007

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the…

### Mean-Field Dynamics: Singular Potentials and Rate of Convergence

- Physics, Mathematics
- 2010

We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N−1. We choose the initial wave function to describe a condensate in which all particles…

### Quantum Dynamics with Mean Field Interactions: a New Approach

- Physics, Mathematics
- 2009

We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new…

### Mean field dynamics of boson stars

- Physics
- 2005

We consider a quantum mechanical system of N bosons with relativistic dispersion interacting through a mean field Coulomb potential (attractive or repulsive). We choose the initial wave function to…

### Derivation of the nonlinear Schr\"odinger equation from a many body Coulomb system

- Mathematics, Physics
- 2001

We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the…

### A quantum central limit theorem for non-equilibrium systems: exact local relaxation of correlated states

- Mathematics
- 2010

We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a…

### The classical limit for quantum mechanical correlation functions

- Physics
- 1974

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the…

### The classical field limit of scattering theory for non-relativistic many-boson systems. II

- Physics, Mathematics
- 1979

We study the classical field limit of non relativistic many-boson theories in space dimensionn≧3, extending the results of a previous paper to more singular interactions. We prove the expected…

### Rate of Convergence Towards Hartree Dynamics

- Mathematics, Physics
- 2011

We consider a system of N bosons interacting through a two-body potential with, possibly, Coulomb-type singularities. We show that the difference between the many-body Schrödinger evolution in the…