Non-equilibrium Gross-Pitaevskii dynamics of boson lattice models

Motivated by recent experiments on trapped ultra-cold bosonic atoms in an optical lattice potential, we consider the non-equilibrium dynamic properties of such bosonic systems for a number of experimentally relevant situations. When the number of bosons per lattice site is large, there is a wide parameter regime where the effective boson interactions are strong, but the ground state remains a superfluid (and not a Mott insulator): we describe the conditions under which the dynamics in this regime can be described by a discrete Gross-Pitaevskii equation. We describe the evolution of the phase coherence after the system is initially prepared in a Mott insulating state, and then allowed to evolve after a sudden change in parameters places it in a regime with a superfluid ground state. We also consider initial conditions with a "pi phase" imprint on a superfluid ground state (i.e. the initial phases of neighboring wells differ by pi), and discuss the subsequent appearance of density wave order and "Schrodinger cat" states.Comment: 16 pages, 11 figures; (v2) added reference

High order asymptotic preserving and well-balanced schemes for the shallow water equations with source terms

In this study, we investigate the Shallow Water Equations incorporating source terms accounting for Manning friction and a non-flat bottom topology. Our primary focus is on developing and validating numerical schemes that serve a dual purpose: firstly, preserving all steady states within the model, and secondly, maintaining the late-time asymptotic behavior of solutions, which is governed by a diffusion equation and coincides with a long time and stiff friction limit. Our proposed approach draws inspiration from a penalization technique adopted in {\it{[Boscarino et. al, SIAM Journal on Scientific Computing, 2014]}}. By employing an additive implicit-explicit Runge-Kutta method, the scheme can ensure a correct asymptotic behavior for the limiting diffusion equation, without suffering from a parabolic-type time step restriction which often afflicts multiscale problems in the diffusive limit. Numerical experiments are performed to illustrate high order accuracy, asymptotic preserving, and asymptotically accurate properties of the designed schemes

Perturbative calculation of quasi-potential in non-equilibrium diffusions: a mean-field example

In stochastic systems with weak noise, the logarithm of the stationary distribution becomes proportional to a large deviation rate function called the quasi-potential. The quasi-potential, and its characterization through a variational problem, lies at the core of the Freidlin-Wentzell large deviations theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory%, ~\cite{bertini2014},which formally fits within Freidlin-Wentzell's framework with a weak noise proportional to $1/\sqrt{N}$, where $N$ is the number of particles. The quasi-potential then appears as a natural generalization of the equilibrium free energy to non-equilibrium particle systems. A key physical and practical issue is to actually compute quasi-potentials from their variational characterization for non-equilibrium systems for which detailed balance does not hold. We discuss how to perform such a computation perturbatively in an external parameter $\lambda$, starting from a known quasi-potential for $\lambda=0$. In a general setup, explicit iterative formulae for all terms of the power-series expansion of the quasi-potential are given for the first time. The key point is a proof of solvability conditions that assure the existence of the perturbation expansion to all orders. We apply the perturbative approach to diffusive particles interacting through a mean-field potential. For such systems, the variational characterization of the quasi-potential was proven by Dawson and Gartner%. ~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit results about the quasi-potential and about fluctuations of one-particle observables in a simple example of mean field diffusions: the Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively

Numerical Algorithms for finding Black Hole solutions of Einstein's Equations

Einstein's Theory of General Relativity has proven remarkably successful at modelling a wide range of gravitational phenomena. Amongst some of the novel features in this description is the existence of black holes; regions of space-time where gravity is so strong that light cannot escape. The properties of black holes have been extensively studied within General Relativity, culminating in the result that the few known space-times are the only allowed stationary black hole solutions in four dimensions. In the past half century, research has focused on how to unify the distinct theories of gravity and quantum mechanics. A common theme amongst several strong candidates is that space-time, the backdrop for gravity, is fundamentally higher dimensional. In these theories, the structure of black hole solutions is relatively unknown and expected to be much richer; finding such solutions is, however, a very hard task. In this thesis, we introduce new numerical methods to study higher dimensional black holes. The methods, based on refinements of existing work and the novel application of standard techniques, are then used to study a number of black hole space-times. Namely the structure of black holes on a Kaluza-Klein background, and rotating Kerr black holes. We demonstrate that these algorithms can be applied in a wide class of situations and yield good quality results with comparative ease. New results are presented in both cases studied. We examine the predicted merger between non-uniform black strings and localised black holes on a Kaluza-Klein background. We find evidence for a new type of non-uniform black string with one Euclidean negative mode and lower entropy than the uniform strings. We discover a window of localised black holes with one Euclidean negative mode but positive specific heat. We also look at the local structure of the merger point and find consistency with Kol's cone prediction

CFT dual of the AdS Dirichlet problem: Fluid/Gravity on cut-off surfaces

We study the gravitational Dirichlet problem in AdS spacetimes with a view to understanding the boundary CFT interpretation. We define the problem as bulk Einstein's equations with Dirichlet boundary conditions on fixed timelike cut-off hypersurface. Using the fluid/gravity correspondence, we argue that one can determine non-linear solutions to this problem in the long wavelength regime. On the boundary we find a conformal fluid with Dirichlet constitutive relations, viz., the fluid propagates on a `dynamical' background metric which depends on the local fluid velocities and temperature. This boundary fluid can be re-expressed as an emergent hypersurface fluid which is non-conformal but has the same value of the shear viscosity as the boundary fluid. The hypersurface dynamics arises as a collective effect, wherein effects of the background are transmuted into the fluid degrees of freedom. Furthermore, we demonstrate that this collective fluid is forced to be non-relativistic below a critical cut-off radius in AdS to avoid acausal sound propagation with respect to the hypersurface metric. We further go on to show how one can use this set-up to embed the recent constructions of flat spacetime duals to non-relativistic fluid dynamics into the AdS/CFT correspondence, arguing that a version of the membrane paradigm arises naturally when the boundary fluid lives on a background Galilean manifold.Comment: 71 pages, 2 figures. v2: Errors in bulk metrics dual to non-relativistic fluids (both on cut-off surface and on the boundary) have been corrected. New appendix with general results added. Fixed typos. 82 pages, 2 figure