Dynamics of colloidal particles with capillary interactions

We investigate the dynamics of colloids at a fluid interface driven by attractive capillary interactions. At submillimeter length scales, the capillary attraction is formally analogous to two-dimensional gravity. In particular it is a non-integrable interaction and it can be actually relevant for collective phenomena in spite of its weakness at the level of the pair potential. We introduce a mean-field model for the dynamical evolution of the particle number density at the interface. For generic values of the physical parameters the homogeneous distribution is found to be unstable against large-scale clustering driven by the capillary attraction. We also show that for the instability to be observable, the appropriate values for the relevant parameters (colloid radius, surface charge, external electric field, etc.) are experimentally well accessible. Our analysis contributes to current studies of the structure and dynamics of systems governed by long-ranged interactions and points towards their experimental realizations via colloidal suspensions.Comment: Matches version accepted for publication. New refs. added, misprints corrected in figs.6,8,9,1

The geometry of r-adaptive meshes generated using optimal transport methods

The principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods, and a metric tensor M and mesh metric are useful theoretical tools for understanding a methods level of mesh alignment, or anisotropy. We consider a mesh redistribution method based on the Monge-Ampere equation, which combines equidistribution of a given scalar density function with optimal transport. It does not involve explicit use of a metric tensor M, although such a tensor must exist for the method, and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh. For model problems with a linear feature and with a radially symmetric feature, we derive the exact form of the metric M, which involves expressions for its eigenvalues and eigenvectors. The eigenvectors are shown to be orthogonal and tangential to the feature, and the ratio of the eigenvalues (corresponding to the level of anisotropy) is shown to depend, both locally and globally, on the value of the density function and the amount of curvature. We thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function. Numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features, including for a non-trivial time dependant nonlinear PDE which evolves narrow and curved reaction fronts

Quantum dynamical semigroups for diffusion models with Hartree interaction

We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global-in-time, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup. The mathematical difficulties stem from combining an unbounded Lindblad generator with the Hartree nonlinearity.Comment: 30 pages; Introduction changed, title changed, easier and shorter proofs due to new energy norm. to appear in Comm. Math. Phy

Classical and Quantum Mechanical Models of Many-Particle Systems

The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and “to a lesser extent” numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping