Periodic Reporting for period 1 - MathBEC (Mathematics of Bose-Einstein Condensation)
Okres sprawozdawczy: 2023-08-01 do 2026-01-31
This is a project in the mathematics of dilute atomic gases. A main focus is the mathematics around Bose Einstein Condensation (BEC). Since the 1990s it has been possible to produce BEC in laboratories around the world, and we aim to improve the mathematical tools for analyzing such large systems of interacting particles.
That a large system of particles exhibits BEC is a very detailed property of the system and as such very difficult to establish mathematically. A more accessible quantity is the energy of the system. Therefore, most progress in the area has been on energy asymptotics for large (dilute) systems of bosons. Furthermore, most of our proofs of BEC at different length scales proceed via a precise control of the energy at those length scales. Therefore, a key part of the project is to understand the energy of different systems of bosons in different spatial dimensions and at different length scales.
The situation where the particles in the gas are fermions instead of bosons is also very important and an intense area of research, but is or course not directly relevant for Bose Einstein Condensation. However, there is a large degree of mathematical knowledge/technology transfer between the two areas of research.
That a large system of particles exhibits BEC is a very detailed property of the system and as such very difficult to establish mathematically. A more accessible quantity is the energy of the system. Therefore, most progress in the area has been on energy asymptotics for large (dilute) systems of bosons. Furthermore, most of our proofs of BEC at different length scales proceed via a precise control of the energy at those length scales. Therefore, a key part of the project is to understand the energy of different systems of bosons in different spatial dimensions and at different length scales.
The situation where the particles in the gas are fermions instead of bosons is also very important and an intense area of research, but is or course not directly relevant for Bose Einstein Condensation. However, there is a large degree of mathematical knowledge/technology transfer between the two areas of research.
We have carried out research on the energy of dilute bose gases in 2 and 3 dimensions. Also, the case of fermi gases for different scalings has been considered.
The main achievements are:
- Proof of a 2-term asymptotic expansion of the ground state energy density for the dilute 2-dimensional bose gas.
- Proof of a 3-term asymptotic expansion of the free energy for the dilute 3-dimensional bose gas interacting through singular potentials and at very low temperatures.
The main achievements are:
- Proof of a 2-term asymptotic expansion of the ground state energy density for the dilute 2-dimensional bose gas.
- Proof of a 3-term asymptotic expansion of the free energy for the dilute 3-dimensional bose gas interacting through singular potentials and at very low temperatures.
Both main achievements mentioned above are clearly beyond the state of the art.
The 2-dimensional bose gas displays logarithmic length scales related to the logarithmic fundamental solution of the 2D Laplace operator. This causes very fundamental differences between the 2D and 3D systems, making the 2D bose gas much more singular than the 3D counterpart. To handle these difficulties - even including the truly singular hard sphere interactions - for both upper and lower bounds is a true step forward.
For the 3D gas at positive temperature, we adapt a recent approach using the convenient Neumann bracketing technique in the localization step. In the case of singular interactions we succeed in using a convexity argument to realize that the free energy does not require a detailed understanding of the excitation spectrum of the gas. Only a somewhat averaged information on the spectrum is needed.
The 2-dimensional bose gas displays logarithmic length scales related to the logarithmic fundamental solution of the 2D Laplace operator. This causes very fundamental differences between the 2D and 3D systems, making the 2D bose gas much more singular than the 3D counterpart. To handle these difficulties - even including the truly singular hard sphere interactions - for both upper and lower bounds is a true step forward.
For the 3D gas at positive temperature, we adapt a recent approach using the convenient Neumann bracketing technique in the localization step. In the case of singular interactions we succeed in using a convexity argument to realize that the free energy does not require a detailed understanding of the excitation spectrum of the gas. Only a somewhat averaged information on the spectrum is needed.