A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

We revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is

$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) = \left({\rm Tr}\left[\left( \, {\sum\limits_{j=1}^m A_j^p } \, \right) ^{q/p} \right] \right)^{1/q} $

for m positive definite operators A _j . In our earlier paper, we only considered the case q = 1 and proved the concavity of Φ _p,1 for 0 < p ≤ 1 and the convexity for p = 2. We conjectured the convexity of Φ _p,1 for 1 < p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q ≥ 1 and still retain the convexity. Among other things this leads to a definition of an L ^q (L ^p ) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces – which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

     

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

Dielectrophoretic alignment of metal and metal oxide nanowires and nanotubes: a universal set of parameters for bridging prepatterned microelectrodes

Nanowires and nanotubes were synthesized from metals and metal oxides using templated cathodic electrodeposition. With templated electrodeposition, small structures are electrodeposited using a template that is the inverse of the final desired shape. Dielectrophoresis was used for the alignment of the as-formed nanowires and nanotubes between prepatterned electrodes. For reproducible nanowire alignment, a universal set of dielectrophoresis parameters to align any arbitrary nanowire material was determined. The parameters include peak-to-peak potential and frequency, thickness of the silicon oxide layer, grounding of the silicon substrate, and nature of the solvent medium used. It involves applying a field with a frequency >10(5) Hz, an insulating silicon oxide layer with a thickness of 2.5 μm or more, grounding of the underlying silicon substrate, and the use of a solvent medium with a low dielectric constant. In our experiments, we obtained good results by using a peak-to-peak potential of 2.1 V at a frequency of 1.2 × 10(5) Hz. Furthermore, an indirect alignment technique is proposed that prevents short circuiting of nanowires after contacting both electrodes. After alignment, a considerably lower resistivity was found for ZnO nanowires made by templated electrodeposition (2.2-3.4 × 10(-3) Ωm) compared to ZnO nanorods synthesized by electrodeposition (10 Ωm) or molecular beam epitaxy (MBE) (500 Ωm).     

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L ^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder’s inequality, which we believe to be new.     

Spectral gap for the Kac model with hard sphere collisions

We prove the analog of the Kac conjecture for hard sphere collisions, giving a computable, close estimate on the spectral gap that is independent of the number of particles. Previous work has focused on the case in which the collision rates are independent of the particle velocities, the case of so-called Maxwellian molecules. The new methods introduced here allow us to deal with collision rates that are not bounded from below. We also obtain information on the structure of the gap eigenfunction.     

Propagation of Chaos for a Thermostated Kinetic Model

We consider a system of N point particles moving on a d-dimensional torus $\mathbb{T}^{d}$ . Each particle is subject to a uniform field E and random speed conserving collisions $\mathbf{v}_{i}\to\mathbf{v}_{i}'$ with $|\mathbf{v}_{i}|=|\mathbf{v}_{i}'|$ . This model is a variant of the Drude-Lorentz model of electrical conduction (Ashcroft and Mermin in Solid state physics. Brooks Cole, Pacific Grove, 1983). In order to avoid heating by the external field, the particles also interact with a Gaussian thermostat which keeps the total kinetic energy of the system constant. The thermostat induces a mean-field type of interaction between the particles. Here we prove that, starting from a product measure, in the limit N→∞, the one particle velocity distribution f(q,v,t) satisfies a self consistent Vlasov-Boltzmann equation, for all finite time t. This is a consequence of “propagation of chaos”, which we also prove for this model.     

Fast and Slow Convergence to Equilibrium for Maxwellian Molecules via Wild Sums

We consider the spatially homogeneous Boltzmann equation for Maxwellian molecules and general finite energy initial data: positive Borel measures with finite moments up to order 2. We show that the coefficients in the Wild sum converge strongly to the equilibrium, and quantitatively estimate the rate. We show that this depends on the initial data F essentially only through on the behavior near r=0 of the function J _F (r)=_|v|>1/r |v|² dF(v). These estimates on the terms in the Wild sum yield a quantitative estimate, in the strongest physical norm, on the rate at which the solution converges to equilibrium, as well as a global stability estimate. We show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds. These are the first examples of solutions of the Boltzmann equation that converge to equilibrium more slowly than exponentially.