On the Rate of Explosion for Infinite Energy Solutions of the Spatially Homogeneous Boltzmann Equation

Let μ ₀ be a probability measure on ℝ³ representing an initial velocity distribution for the spatially homogeneous Boltzmann equation for pseudo Maxwellian molecules. As long as the initial energy is finite, the solution μ _t will tend to a Maxwellian limit. We show here that if , then instead, all of the mass “explodes to infinity” at a rate governed by the tail behavior of μ ₀. Specifically, for L0, define

Molecular dynamics simulation of nanocolloidal amorphous silica particles: Part II

Explicit molecular dynamics simulations were applied to a pair of amorphous silica nanoparticles with diameter of 3.2 nm immersed in a background electrolyte. Mean forces acting between the pair of silica nanoparticles were extracted at four different background electrolyte concentrations. The dependence of the interparticle potential of mean force on the separation and the silicon to sodium ratio, as well as on the background electrolyte concentration, are demonstrated. The pH was indirectly accounted for via the ratio of silicon to sodium used in the simulations. The nature of the interaction of the counterions with charged silica surface sites (deprotonated silanols) was also investigated. The effect of the sodium double layer on the water ordering was investigated for three Si : Na(+) ratios. The number of water molecules trapped inside the nanoparticles was investigated as the Si : Na(+) ratio was varied. Differences in this number between the two nanoparticles in the simulations are attributed to differences in the calculated electric dipole moment. The implications of the form of the potentials for aggregation are also discussed.     

On the cases of equality in Bobkov's inequality and Gaussian rearrangement

We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the Faber-Krahn inequality. Received June 21, 1999 / Accepted June 12, 2000 / Published online November 9, 2000     

A remainder term for Hölder’s inequality for matrices and quantum entropy inequalities

A labelled oriented graph (LOG) group is a group given by a presentation constructed in a certain way from a labelled oriented graph: examples include Wirtinger presentations of knot groups. We show how to obtain generators for the Schur Multiplier \(H_2(G)\) of a LOG group from the underlying LOG, and by exhibiting the n-string braid group \(B_n\) as a LOG group, we compute \(H_2(B_n)\).     

On the Optimal Choice of Coefficients in a Truncated Wild Sum and Approximate Solutions for the Kac Equation

We study an approximate solution of the Boltzmann problem for Kac's caricature of a Maxwellian gas by using a truncated and modified expansion of Wild type. We choose the coefficients in the Wild sum approximation using a criterion based on exactly reproducing the behavior of the leading modes.     

Molecular dynamics simulation of nanocolloidal amorphous silica particles: Part I

Explicit molecular dynamics simulations were applied to a pair of amorphous silica nanoparticles in aqueous solution, with diameter of 4.4 nm and with four different background electrolyte concentrations, to extract the mean force acting between the two silica nanoparticles. Dependences of the interparticle forces on the separation and the background electrolyte concentration were demonstrated. The nature of the interaction of the counterions with charged silica surface sites (deprotonated silanols) was investigated. A "patchy" double layer of adsorbed sodium counterions was observed. Dependences of the interparticle potential of mean force on the separation and the background electrolyte concentration were demonstrated. Direct evidence of the solvation forces is presented in terms of changes of the water ordering at the surfaces of the isolated and double nanoparticles. The nature of the interaction of the counterions with charged silica surface sites (deprotonated silanols) was investigated in terms of quantifying the effects of the number of water molecules separately inside each pair of nanoparticles by defining an impermeability measure. A direct correlation was found between the impermeability (related to the silica surface "hairiness") and the disruption of water ordering. Differences in the impermeability between the two nanoparticles are attributed to differences in the calculated electric dipole moment.     

Inner-shell ionization in near-central collisions of 6.7 MeV/amu4He ions with mid-Z atoms

We report a self consistent tight binding calculation of the cohesion energy of small MgO clusters, up to 12 atoms. We discuss the size variation of the charges, the bond lengths, the electronic gaps. For the first time, we predict the existence of a strong dimerization of the bond length in the small clusters. We underline the consequences of the competition between covalent and electrostatic effects in these ionocovalent systems.     

On the relation between rates of relaxation and convergence of wild sums for solutions of the Kac equation

In the case of Maxwellian molecules, the Wild summation formula gives an expression for the solution of the spatially homogeneous Boltzmann equation in terms of its initial data F as a sum . Here, is an average over n-fold iterated Wild convolutions of F. If M denotes the Maxwellian equilibrium corresponding to F, then it is of interest to determine bounds on the rate at which tends to zero. In the case of the Kac model, we prove that for every ε>0, if F has moments of every order and finite Fisher information, there is a constant C so that for all n, where Λ is the least negative eigenvalue for the linearized collision operator. We show that Λ is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of f(·,t) to M. A key role in the analysis is played by a decomposition of into a smooth part and a small part. This depends in an essential way on a probabilistic construction of McKean. It allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regularity of the initial data.     

Displacement Convexity and Minimal Fronts at Phase Boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables.We use this to show that, in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals and, in the multi-component setting, jointly displacement convex functionals.