75 research outputs found
On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
The paper concerns - convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials (-4\le
\gm<0), with and without angular cutoff. We prove the time-averaged
-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than 2+|\gm|. For the
usual -convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with -1\le \gm<0, there
are algebraic upper and lower bounds on the rate of -convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.Comment: This version contains a strengthened theorem 3, on rate of
convergence, considerably relaxing the hypotheses on the initial data, and
introducing a new method for avoiding use of poitwise lower bounds in
applications of entropy production to convergence problem
Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality
In noncommutative probability theory independence can be based on free
products instead of tensor products. This yields a highly noncommutative
theory: free probability . Here we show that the classical Shannon's entropy
power inequality has a counterpart for the free analogue of entropy .
The free entropy (introduced recently by the second named author),
consistently with Boltzmann's formula , was defined via volumes of
matricial microstates. Proving the free entropy power inequality naturally
becomes a geometric question.
Restricting the Minkowski sum of two sets means to specify the set of pairs
of points which will be added. The relevant inequality, which holds when the
set of "addable" points is sufficiently large, differs from the Brunn-Minkowski
inequality by having the exponent replaced by . Its proof uses the
rearrangement inequality of Brascamp-Lieb-L\"uttinger
Phase Transition in a Vlasov-Boltzmann Binary Mixture
There are not many kinetic models where it is possible to prove bifurcation
phenomena for any value of the Knudsen number. Here we consider a binary
mixture over a line with collisions and long range repulsive interaction
between different species. It undergoes a segregation phase transition at
sufficiently low temperature. The spatially homogeneous Maxwellian equilibrium
corresponding to the mixed phase, minimizing the free energy at high
temperature, changes into a maximizer when the temperature goes below a
critical value, while non homogeneous minimizers, corresponding to coexisting
segregated phases, arise. We prove that they are dynamically stable with
respect to the Vlasov-Boltzmann evolution, while the homogeneous equilibrium
becomes dynamically unstable
Heat Kernel Bounds for the Laplacian on Metric Graphs of Polygonal Tilings
We obtain an upper heat kernel bound for the Laplacian on metric graphs
arising as one skeletons of certain polygonal tilings of the plane, which
reflects the one dimensional as well as the two dimensional nature of these
graphs.Comment: 8 page
Closure properties of solutions to heat inequalities
We prove that if are
sufficiently well-behaved solutions to certain heat inequalities on then
the function given by
also satisfies a heat inequality of a
similar type provided . On
iterating, this result leads to an analogous statement concerning -fold
convolutions. As a corollary, we give a direct heat-flow proof of the sharp
-fold Young convolution inequality and its reverse form.Comment: 12 page
Hypercontractivity on the -Araki-Woods algebras
Extending a work of Carlen and Lieb, Biane has obtained the optimal
hypercontractivity of the -Ornstein-Uhlenbeck semigroup on the
-deformation of the free group algebra. In this note, we look for an
extension of this result to the type III situation, that is for the
-Araki-Woods algebras. We show that hypercontractivity from to
can occur if and only if the generator of the deformation is bounded.Comment: 17 page
Froth-like minimizers of a non local free energy functional with competing interactions
We investigate the ground and low energy states of a one dimensional non
local free energy functional describing at a mean field level a spin system
with both ferromagnetic and antiferromagnetic interactions. In particular, the
antiferromagnetic interaction is assumed to have a range much larger than the
ferromagnetic one. The competition between these two effects is expected to
lead to the spontaneous emergence of a regular alternation of long intervals on
which the spin profile is magnetized either up or down, with an oscillation
scale intermediate between the range of the ferromagnetic and that of the
antiferromagnetic interaction. In this sense, the optimal or quasi-optimal
profiles are "froth-like": if seen on the scale of the antiferromagnetic
potential they look neutral, but if seen at the microscope they actually
consist of big bubbles of two different phases alternating among each other. In
this paper we prove the validity of this picture, we compute the oscillation
scale of the quasi-optimal profiles and we quantify their distance in norm from
a reference periodic profile. The proof consists of two main steps: we first
coarse grain the system on a scale intermediate between the range of the
ferromagnetic potential and the expected optimal oscillation scale; in this way
we reduce the original functional to an effective "sharp interface" one. Next,
we study the latter by reflection positivity methods, which require as a key
ingredient the exact locality of the short range term. Our proof has the
conceptual interest of combining coarse graining with reflection positivity
methods, an idea that is presumably useful in much more general contexts than
the one studied here.Comment: 38 pages, 2 figure
Ground state at high density
Weak limits as the density tends to infinity of classical ground states of
integrable pair potentials are shown to minimize the mean-field energy
functional. By studying the latter we derive global properties of high-density
ground state configurations in bounded domains and in infinite space. Our main
result is a theorem stating that for interactions having a strictly positive
Fourier transform the distribution of particles tends to be uniform as the
density increases, while high-density ground states show some pattern if the
Fourier transform is partially negative. The latter confirms the conclusion of
earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and
Likos et al. (2007). Other results include the proof that there is no Bravais
lattice among high-density ground states of interactions whose Fourier
transform has a negative part and the potential diverges or has a cusp at zero.
We also show that in the ground state configurations of the penetrable sphere
model particles are superposed on the sites of a close-packed lattice.Comment: Note adde
Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration
Optimal hypercontractivity bounds for the fermion oscillator semigroup are
obtained. These are the fermion analogs of the optimal hypercontractivity
bounds for the boson oscillator semigroup obtained by Nelson. In the process,
several results of independent interest in the theory of non-commutative
integration are established. {}.Comment: 18 p., princeton/ecel/7-12-9
Continuity and Stability of Partial Entropic Sums
Extensions of Fannes' inequality with partial sums of the Tsallis entropy are
obtained for both the classical and quantum cases. The definition of kth
partial sum under the prescribed order of terms is given. Basic properties of
introduced entropic measures and some applications are discussed. The derived
estimates provide a complete characterization of the continuity and stability
properties in the refined scale. The results are also reformulated in terms of
Uhlmann's partial fidelities.Comment: 9 pages, no figures. Some explanatory and technical improvements are
made. The bibliography is extended. Detected errors and typos are correcte
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