A Central Limit Theorem in Many-Body Quantum Dynamics (original) (raw)
Related papers
Letters in Mathematical Physics, 2008
The mean-field limit for the dynamics of bosons with random two-body interactions and in the presence of a random external potential is rigorously studied, both for the Hartree dynamics and the Gross Pitaevskii dynamics. First, it is shown that, for interactions and potentials that are almost surely bounded, the many-body quantum evolution can be replaced in the mean-field limit by a single particle nonlinear evolution that is described by the Hartree equation. This is an Egorov-type theorem for many-body quantum systems with random interactions. The analysis is then extended to derive the Gross Pitaevskii equation with random interactions.
A Large Deviation Principle in Many-Body Quantum Dynamics
Annales Henri Poincaré
We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.
On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
Communications in Mathematical Physics, 2009
In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg-picture dynamics of "observables", thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a "semi-classical"
Bogoliubov correction to the mean-field dynamics of interacting bosons
Advances in Theoretical and Mathematical Physics, 2017
We consider the dynamics of a large quantum system of N identical bosons in 3D interacting via a two-body potential of the form N 3β−1 w(N β (x − y)). For fixed 0 ≤ β < 1/3 and large N , we obtain a norm approximation to the many-body evolution in the N-particle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.
Stochastic quantum dynamics beyond mean field
The European Physical Journal A, 2014
Mean-field approaches where a complex fermionic many-body problem is replaced by an ensemble of independent particles in a self-consistent mean-field can describe many static and dynamical aspects. It generally provides a rather good approximation for the average properties of one-body degrees of freedom. However, the mean-field approximation generally fails to produce quantum fluctuations of collective motion. To overcome this difficulty, noise can be added to the mean-field theory leading to a stochastic description of the many-body problem. In the present work, we summarize recent progress in this field and discuss approaches where fluctuations have been added either to the initial time, like in the Stochastic Mean-Field theory or continuously in time as in the Stochastic Time-Dependent Hartree-Fock. In some cases, the initial problem can even be re-formulated exactly by introducing Quantum Monte-Carlo methods in real-time. The possibility to describe superfluid systems is also invoked. Successes and shortcomings of the different beyond mean-field theories are discussed and illustrated.
1996
We present numerical evidence that in a system of interacting bosons there exists a correspondence between the spectral properties of the exact quantum Hamiltonian and the dynamical chaos of the associated mean field evolution. This correspondence, analogous to the usual quantum-classical correspondence, is related to the formal parallel between the second quantization of the mean field, which generates the exact dynamics of the quantum N-body system, and the first quantization of classical canonical coordinates. The limit of infinite density and the thermodynamic limit are then briefly discussed.
Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
We discuss the Hartree equation arising in the mean-field limit of large systems of bosons and explain its importance within the class of nonlinear Schroedinger equations. Of special interest to us is the Hartree equation with focusing nonlinearity (attractive two-body interactions). Rigorous results for the Hartree equation are presented concerning: 1) its derivation from the quantum theory of large systems of bosons, 2) existence and stability of Hartree solitons, and 3) its point-particle (Newtonian) limit. Some open problems are described.
Mean-Field Approximation of Quantum Systems and Classical Limit
Mathematical Models and Methods in Applied Sciences, 2003
We prove that, for a smooth two-body potential, the quantum mean-field approximation to the nonlinear Schrödinger equation of the Hartree type is stable at the classical limit h → 0, yielding the classical Vlasov equation.
On the Stochastic Limit of Quantum Field Theory
2012
The weak coupling limit for a quantum system, with discrete energy spectrum, coupled to a Bose reservoir with the most general linear interaction is considered: under this limit we have a quantum noise processes substituting for the field. We obtain a limiting evolution unitary on the system and noise space which, when reduced to the system's degrees of freedom, provide the master and Langevin equations that are postulated on heuristic grounds by physicists. In addition we give a concrete application of our results by deriving the evolution of an atomic system interacting with the electrodynamic field without recourse to either rotating wave or dipole approximations.