Entropy, Duality and Cross Diffusion

This paper is devoted to the use of the entropy and duality methods for the existence theory of reaction-cross diffusion systems consisting of two equations, in any dimension of space. Those systems appear in population dynamics when the diffusion rates of individuals of two species depend on the concentration of individuals of the same species (self-diffusion), or of the other species (cross diffusion)

Cross-diffusion systems with entropy structure

Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak solutions is recalled, and some open problems are presented

Strongly Correlated Quantum Fluids: Ultracold Quantum Gases, Quantum Chromodynamic Plasmas, and Holographic Duality

Strongly correlated quantum fluids are phases of matter that are intrinsically quantum mechanical, and that do not have a simple description in terms of weakly interacting quasi-particles. Two systems that have recently attracted a great deal of interest are the quark-gluon plasma, a plasma of strongly interacting quarks and gluons produced in relativistic heavy ion collisions, and ultracold atomic Fermi gases, very dilute clouds of atomic gases confined in optical or magnetic traps. These systems differ by more than 20 orders of magnitude in temperature, but they were shown to exhibit very similar hydrodynamic flow. In particular, both fluids exhibit a robustly low shear viscosity to entropy density ratio which is characteristic of quantum fluids described by holographic duality, a mapping from strongly correlated quantum field theories to weakly curved higher dimensional classical gravity. This review explores the connection between these fields, and it also serves as an introduction to the Focus Issue of New Journal of Physics on Strongly Correlated Quantum Fluids: from Ultracold Quantum Gases to QCD Plasmas. The presentation is made accessible to the general physics reader and includes discussions of the latest research developments in all three areas.Comment: 138 pages, 25 figures, review associated with New Journal of Physics special issue "Focus on Strongly Correlated Quantum Fluids: from Ultracold Quantum Gases to QCD Plasmas" (http://iopscience.iop.org/1367-2630/focus/Focus%20on%20Strongly%20Correlated%20Quantum%20Fluids%20-%20from%20Ultracold%20Quantum%20Gases%20to%20QCD%20Plasmas