On the Poisson Structure of the Time-Dependent Mean-Field Equations for Systems of Bosons out of Equilibrium

TL;DR

This paper investigates the Poisson structure of time-dependent mean-field equations for bosonic systems out of equilibrium, deriving a Lie-Poisson bracket and analyzing their stability and Hamiltonian properties.

Contribution

It constructs the Lie-Poisson bracket for these equations and explores conditions under which they can be expressed in Hamiltonian form.

Findings

Derived the Lie-Poisson bracket for bosonic mean-field equations

Performed stability analysis of the equations

Identified cases with Hamiltonian structure

Abstract

We analyze the Poisson structure of the time-dependent mean-field equations for bosons and construct the Lie-Poisson bracket associated to these equations. The latter follow from the time-dependent variational principle of Balian and Veneroni when a gaussian Ansatz is chosen for the density operator. We perform a stability analysis of both the full and the linearized equations. We also search for the canonically conjugate variables. In certain cases, the evolution equations can indeed be cast in a Hamiltonian form.