On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns $L^1$- 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 $L^1$-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 $L^1$-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 $L^1$-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 $S=k\log W$, 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 $1/n$ replaced by $2/n$. 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 $u_1,u_2 : (0,\infty) \times \R^d \to (0,\infty)$ are sufficiently well-behaved solutions to certain heat inequalities on $\R^d$ then the function $u: (0,\infty) \times \R^d \to (0,\infty)$ given by $u^{1/p}=u_1^{1/p_1} * u_2^{1/p_2}$ also satisfies a heat inequality of a similar type provided $\tfrac{1}{p_1} + \tfrac{1}{p_2} = 1 + \tfrac{1}{p}$. On iterating, this result leads to an analogous statement concerning $n$-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp $n$-fold Young convolution inequality and its reverse form.Comment: 12 page

Hypercontractivity on the qq-Araki-Woods algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-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 $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ 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