Converging positive temperature equilibrium states in many-body bosonic quantum systems

The many-body problem is about finding the properties of (often microscopic) systems that are made up of a large number of interacting particles. The vast complexity and many degrees of freedom of the many-body problem renders them, except in the simplest cases, numerically insoluble.

A new article published in the Journal of Mathematical Physics takes a step toward understanding how simplified equations that could be solved numerically can replace the often impenetrable many-body problem. Specifically, the work deals with Gibbs measures, which specify the state of a given system in a probabilistic manner and can provide insight into simple bosonic systems. The authors show they can obtain Gibbs measures, based on 1-D defocusing nonlinear Schrödinger functionals with subharmonic trapping, as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons.

The authors’ new achievement was to relate the positive temperature equilibrium states of the two main theories of many-body bosonic quantum systems. The first principles theory is based on the many-body linear Schrödinger Hamiltonian, while the effective mean-field theory is based on the one-body nonlinear Schrödinger equation. While previous works found that the zero-temperature states converge, and the time-evolution of zero-temperature states are related, a proof of converging positive temperature states had yet to be found.

The authors worked with a limit where the particle number in the states of interest was not uniformly bounded, yet renormalization methods were not necessary to make sense of the problem. For future work, they are exploring the more difficult case where renormalization is needed.

Source: “Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits,” by Mathieu Lewin, Phan Thánh Nam, and Nicolas Rougerie, Journal of Mathematical Physics (2018). The article can be accessed at https://doi.org/10.1063/1.5026963 .

(Image credited to Immanuel Bloch, Max Planck Institute of Quantum Optics.)