Mathematical Physics

The Mathematical Physics group is primarily focused on Quantum fluids with problems originating from a range of superfluid systems.

Quantum fluids are distinct from classical fluids in that both quantum mechanics and quantum statistics are required in order to explain their properties. One of the most profound macroscopic manifestations of such quantum effects is the phenomena of superfluidity in which a fluid can flow without any dissipation and can rotate by creating quantized vortices. Members of the School of Mathematics who conduct research in this area are Davide Proment and Hayder Salman. We have close international links with a number of theoretical and experimental groups to model a range of phenomena arising in ultracold atomic Bose-Einstein condensates, vortex dynamics in liquid He-4, and nonequilibrium phenomena associated with systems driven far from equilibrium. Details of some of our past and present projects are included below.

A Bose-Einstein condensate is a state of matter realisable at temperatures close to absolute zero. Under such conditions, quantum effects give rise to nonclassical phenomena on a macroscopic scale. Predicted by the physicists Bose and Einstein in the 1920s, they were first realised experimentally in 1995 using ultracold alkali atoms. These gases permit a remarkable degree of control of the systems' parameters and are realized under very controlled conditions making them ideal systems to test fundamental physical models of superfluids. We are interested in a number of fundamental questions concerning the modelling of such systems in two and three spatial dimensions. These include:

Berezinskii-Kosterlitz-Thouless (BKT) phase transition in 2D

The phenomena is Bose-Einstein condensation in two-dimensional systems is a delicate problem since (thermal or quantum) fluctuations are sufficiently strong to destroy the long-range order that is necessary for a condensate to exist. In fact, it is now understood that the phase transition in two-dimensions is associated with the unbinding of vortex-antivortex pairs which ultimately gives rise to a gas of free quantized vortices. We have been developing measures to accurately characterize and explain the phase transition in such low-dimensional superfluid systems.

Finite temperature effects

In practice, Bose-Einstein condensates are always realized at finite temperature. The remnant thermal cloud of finite temperature excitations can interact with the Bose-Einstein condensate to dramatically influence its dynamics. One of the questions we have been focused on is how to model such finite temperature effects. We have been developing a classical fields model of a Bose gas that permits a non-perturbative treatment of the non-equilibrium phenomena associated with these finite temperature systems. Below we illustrate the occupation number as a function of wavenumber for a two-component system but one that has an imbalance in the particle number of each component. This leads to condensation of one component with thermally populated states elsewhere.

Numerical methods

The need to resolve a wide range of phenomena in atomic condensates including quantized vortices, phonon excitations, thermal excitations, and out of equilibrium phenomena, as would occur in turbulent systems requires very accurate high order and computationally efficient numerical schemes that correctly describe the underlying energy levels associated with the excitations in our system. For this purpose, we have developed high order spectral methods in periodic domains using FFT based methods, and in systems with harmonic potentials using generalized-Laguerre basis functions. Examples of results obtained with our numerical schemes for a range of physical scenarios are included below. These demonstrate the formation of a vortex lattice in rotating 2D condensate and a Kelvin-Helmholtz instability arising at the interface of a two-component system.

In superfluids, vortices have quantized circulation and correspond topological defects at which the density of a superfluid vanishes. Quantized vortices turn out to be of crucial importance in understanding the dynamics of superfluids. For example, in the turbulent state at extremely low temperatures, such vortices are responsible for a Kolmogorov like cascade in superfluids . At very small scales, the quantized nature of the vorticity modifies the dynamics from the quasi-classical Kolmogorov picture. In this case the motion of individual vortices needs to be modelled. We are currently interested in modeling the dynamics and excitations of vortices in a range of problems. These include

Lighthill theory of sound radiation for superfluid vortices

We have developed a theory of the radiation of sound by moving superfluid vortices by extending analogous results for classical compressible and geophysical vortices. This so-called Lighthill theory of radiation reveals that superfluid vortices continually radiate sound as they propagate leading to a decay of energy associated with the vortical component of the flow.

Kelvin waves, solitons and breathers

Vortices can admit different types of excitations propagating along the length of the vortex. The most commonly studies forms of such excitations are Kelvin waves, low amplitude excitations that can support a so-called Kelvin wave cascade on vortices. We have also been considering solitons and breathers propagating on vortices that are associated with nonlinear large amplitude excitations. Our recent work on breathers on such vortices reveals a striking similarity with similar structures arising in other contexts in nature and promises to enhance our understanding of how energy is dissipated in superfluid turbulence.

Vortex knots

Knots can be easily thought of as closed wires that cannot be deformed into rings without cutting and tying them up again. The simplest example of a knot is the trefoil which is the easiest type of knot that one can create by knotting a wire and tying it. Recently, experimental groups have realized a knotted vortex structure in a classical fluid. We are currently studying of how knotted vortices propagate and decay in superfluids. This problem raises a number of mathematically challenging questions and brings together elements of geometry and topology to uncover the basic dynamics of such knotted vortices. Such an understanding of nontrivially knotted topological defects can have an impact on other disciplines such as classical fluid dynamics and magnetohydrodynamics.

Many systems in nature are in a state that is driven out of equilibrium. In superfluids non-equilibrium states can also play an important role in understanding phase transitions providing an ideal testing ground for some of the important theories of condensed matter systems. In general the non-equilibrium phenomena in superfluids are either characterised by a system of random waves interacting nonlinearly under particular resonant conditions, or are dominated by superfluid vortices in 2D or a tangle of vortices in 3D. While possessing some similarities, these two regimes are quite different and require contrastingly different approaches. We have relied on a range of mathematical approaches to study such phenomena, including statistical mechanics, wave turbulence theory and hydrodynamic turbulence. This theoretical work is commonly complemented by state of the art numerical computations to uncover the underlying physical processes that dictate the emergent phenomena in such systems. Problems we are currently interested in include:

Superfluid turbulence

A common problem that is encountered in superfluids which blends together these different scenarios in a single setting is superfluid turbulence. Below we provide a demonstration of a Bose-Einstein condensate that is in a turbulent state. The vortex tangle forms following a rapid quench under the so-called Kibble-Zurek scenario. Throughout the evolution the system passes from wave turbulence dominated by four wave resonances to strong turbulence with vortices, followed by wave-turbulence with three-wave resonances.

Kolmogorov-Zhakarov spectra

Wave turbulence is a mathematical theory that may provide exact results when nonlinear interactions are weak in the system. Other than predicting thermodynamic equilibrium spectra, other steady state out-of-equilibrium distributions may be analytically obtained when considering forcing and dissipation acting at very different scales. Similar to the celebrated Kolmogorov cascade of energy in classical fluid, such spectra describe a constant flux of conserved quantities (e.g. mass, energy) over a range of scales and are referred to as Kolmogorov-Zakharov spectra. Under certain assumptions, Bose-Einstein condensates possess weakly nonlinear interactions and out-of-equilibrium Kolmogorov-Zakharov spectra may appear.