Bosonic Many-particle Systems at Positive Temperature

The motion of atoms and molecules is governed by the laws of quantum mechanics, which distinguish between two fundamental types of particles: bosons and fermions. Fermions obey the Pauli exclusion principle, meaning that no two fermions can occupy the same quantum state. Bosons, by contrast, exhibit a tendency to accumulate in the same state. This phenomenon gives rise to the Bose–Einstein condensation phase transition: below a critical temperature, a macroscopic fraction of all particles occupies a single quantum state, causing this part of the system to behave collectively as one giant quantum object. Systems in the condensed phase exhibit remarkable properties, such as superfluidity, that is, flow without resistance. Developing a rigorous mathematical understanding of Bose–Einstein condensation starting from the fundamental laws of quantum mechanics is a challenging and important problem. Over the past two decades, there has been substantial progress in the study of Bose gases at zero temperature. In contrast, the mathematical theory of Bose gases at finite temperature remains far less developed. The goal of this project is to develop new analytic, functional-analytic, probabilistic, and quantum-information-theoretic techniques to address this gap. The project also includes a broad educational component. Through research seminars and brownbag lunches, the investigator promotes mathematical physics, analysis, and quantum theory to a broad audience of students. In addition, the project provides valuable training opportunities for undergraduate and graduate students through direct involvement in research-level mathematics. The focus of the project is the mathematical analysis of bosonic many-particle systems at positive temperatures. In this regime, a macroscopic number of particles occupies excited states outside the condensate, and mathematical techniques that have proved successful at zero temperature no longer apply. This is natural, since collective properties at positive temperature depend on a much larger portion of the spectrum of the Hamiltonian describing the system, and analyzing high-energy excitations is typically far more complicated than studying the low-energy sector. Among the questions addressed are a proof of superfluidity via spontaneous symmetry breaking for weakly interacting Bose gases and the derivation of accurate approximations for the Gibbs state of the dilute Bose gas in the Gross–Pitaevskii limit. Such results contribute to a more complete mathematical description of these fundamental models in the large-particle-number limit. To make progress on these problems, the investigator aims to develop new correlation inequalities for Gibbs state in order to better understand and quantify their interaction-induced correlation structure. Another goal of the project is to uncover new connections between the analysis of Bose gases, probability theory, and quantum information theory. The project may also lead to new predictions for experiments with ultracold quantum gases. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2555747 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Andreas Deuchert | Institution: Virginia Polytechnic Institute and State University, BLACKSBURG, VA | Award Amount: $200,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2555747 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2555747.html