The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of particles, by a one particle mean-field equation if the following general principles are satisfied: $\bullet$ The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles). $\bullet$ The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part. $\bullet$ There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of particles, by a one particle mean-field equation if the following general principles are satisfied: $\bullet$ The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles). $\bullet$ The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part. $\bullet$ There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.

arXiv:1809.01450Sep 5, 20182 citations

Dynamical Mean Field Solution of the Bose-Hubbard Model

Peter Anders, Emanuel Gull, Lode Pollet +2 more

We present the effective action and self-consistency equations for the bosonic dynamical mean field approximation to the bosonic Hubbard model and show that it provides remarkably accurate phase diagrams and correlation functions. To solve the bosonic dynamical mean field equations, we use a continuous-time Monte Carlo method for bosonic impurity models based on a diagrammatic expansion in the hybridization and condensate coupling. This method is readily generalized to bosonic mixtures, spinful bosons, and Bose-Fermi mixtures.We present the effective action and self-consistency equations for the bosonic dynamical mean field approximation to the bosonic Hubbard model and show that it provides remarkably accurate phase diagrams and correlation functions. To solve the bosonic dynamical mean field equations, we use a continuous-time Monte Carlo method for bosonic impurity models based on a diagrammatic expansion in the hybridization and condensate coupling. This method is readily generalized to bosonic mixtures, spinful bosons, and Bose-Fermi mixtures.

arXiv:1004.0510Physical Review LettersAug 24, 201089 citations

Local enhancement of the mean-field approximation for bosons

Marius Lemm, Simone Rademacher, Zhang, Jingxuan

We study the quantum many-body dynamics of a Bose-Einstein condensate (BEC) on the lattice in the mean-field regime. We derive a local enhancement of the mean-field approximation: At positive distance $ρ>0$ from the initial BEC, the mean-field approximation error at time $t\leq ρ/v$ is bounded as $ρ^{-n}$ , for arbitrarily large $n\geq 1$ . This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.We study the quantum many-body dynamics of a Bose-Einstein condensate (BEC) on the lattice in the mean-field regime. We derive a local enhancement of the mean-field approximation: At positive distance $ρ>0$ from the initial BEC, the mean-field approximation error at time $t\leq ρ/v$ is bounded as $ρ^{-n}$, for arbitrarily large $n\geq 1$. This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.

arXiv:2412.13868Dec 18, 20240 citations

\bullet

The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles).

\bullet

The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part.

\bullet

There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.

ρ>0

from the initial BEC, the mean-field approximation error at time

t\leq ρ/v

is bounded as

ρ^{-n}

, for arbitrarily large

n\geq 1

. This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.