Mathematical and Theoretical Physics Seminar (MTPS)

Eva Hackmann (Center of Applied Space Technology and Microgravity (ZARM) at Bremen University)

Equilibrium Configurations of Fluids around Black Holes

Abstract: The most luminous energy sources in our universe are active galactic nuclei, which are driven by the accretion of matter onto a supermassive black hole of millions to billions of solar masses. The accreting matter is mainly orbiting the black hole in form of an accretion disk. As the gravitational field close to the black hole is very strong, accretion disks may serve as a system to test our current understanding of the theory of gravity.
We will first review some basic aspects of General Relativity, that is currently the best available theory of gravity, and then discuss the construction of fluid configurations in strong gravitational fields. In general, these configurations are described by a set of partial differential equations derived from energy conservation and Maxwell equations. The fluid configurations can serve as a simplified model to understand some of the basic processes in accretion disks, and to study the interplay of electromagnetic and gravitational forces in these processes.

David Mitrouskas (IST Austria)

Effective Dynamics of Tracer Particles in a Dense Fermi Gas

Abstract: We discuss the dynamics of few tracer particles in a d-dimensional box coupled to N fermions via a suitable pair interaction. After taking the large volume limit at positive Fermi momentum we consider the regime of high density, that is, large Fermi momentum. Assuming that the fermions are initially in the ground state of the kinetic energy, we show that a single tracer particle effectively decouples from the fermions and evolves like a free particle. We explain that this is based on a separation of scales between the tracer particle and the fast electrons at the Fermi surface. In the rest of the talk we show how the picture changes for more than one tracer particle and/or if one starts with an excited state of the fermions.

Jan Olaf Mirko Härter (Niels Bohr Institute, University of Copenhagen; Jacobs University; Leibniz-Zentrum für Marine Tropenforschung)

Criticality and bistability in the tropical convective cloud field

Abstract: Whereas Rayleigh-Bénard convection is a classical problem in fluid dynamics and complex systems, atmospheric moist convection harbors a number of additional phenomena, such as large-scale segregation into cloudy and cloud-free subregions (known as convective self-aggregation) or the emergence of mesoscale convective systems - self-organized clusters of convective cells. Treating the population of convective cells as a form of interacting particle system, we discuss several simple toy models, which highlight some mechanisms involved in tropical cloud organization. We make contact to results obtained from idealized fluid dynamics simulations of the tropical atmosphere and highlight, where discrepancies between simulations and observations exist and could be bridged.

Ian Jauslin (Princeton University)

An effective equation to study Bose gases at all densities

Abstract: I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas. The interactions induce many-body correlations in the system, which makes it very difficult to study, be it analytically or numerically. A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to a one-particle problem, which incorporates the many-body correlations. The effective equation I will discuss is arrived at by making a very different set of approximations, and, like Bogolubov theory, ultimately reduces to a one-particle problem. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the many-body Bose gas at both low and at high densities. I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. That is, the analytical and numerical evidence suggest that this effective equation can capture many-body correlations in a one-particle picture beyond what Bogolubov can accomplish. Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas (about which there still are many important open questions). In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations.

Manish Jung Thapa (ETH Zürich)

Path Integral Methods in Quantum Rate Theories (link to recording)

Abstract: I present a new quantum transition-state theory (QTST) for calculating chemical reaction rates in complex multidimensional systems. The method is able to accurately include nuclear quantum effects such as tunneling and delocalization. The method can be computed by path-integral sampling and is applicable to treat molecular reactions in solution. A constraint functional based on energy conservation is introduced which ensures that the dominant semiclassical paths are sampled by our rate formula. We find that this dominant semiclassical paths looks like instanton, an imaginary-time periodic orbit. We demonstrate the accuracy of our approach by comparison to exact approach in model chemical systems.

Vlad Vicol (NYU)

Shock formation for the 3d Euler equations

Abstract: In this talk, I will discuss the shock formation process for the 3d compressible Euler equations, in which sounds waves interact with entropy waves to produce vorticity. Smooth solutions form a generic stable shock with explicitly computable blowup time, location, and direction. Our method establishes the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of three distinct wave families.
This is based on joint work with T. Buckmaster and S. Shkoller.