The Vlasov Poisson System with Steady Spatial Asymptotics

Abstract

A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as |x|→∞ is considered. Hence the total positive charge and the total negative charge are infinite. Smooth solutions with appropriate asymptotic behavior are shown to exist locally in time. Conditions for continuation of these solutions are also established.     

Sanov results for Glauber spin-glass dynamics

Summary. In this paper we prove a Sanov result, i.e. a Large Deviation Principle (LDP) for the distribution of the empirical measure, for the annealed Glauber dynamics of the Sherrington-Kirkpatrick spin-glass. Without restrictions on time or temperature we prove a full LDP for the asymmetric dynamics and the crucial upper large deviations bound for the symmetric dynamics. In the symmetric model a new order-parameter arises corresponding to the response function in [SoZi83]. In the asymmetric case we show that the corresponding rate function has a unique minimum, given as the solution of a self-consistent equation. The key argument used in the proofs is a general result for mixing of what is known as Large Deviation Systems (LDS) with measures obeying an independent LDP. Received: 18 May 1995 / In revised form: 14 March 1996     

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Summary For Gibbsian systems of particles inR ^d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.     

Gauge symmetries and percolation in ±J Ising spin glasses

We consider the ±J spin glass on a finite graph G=(V,E), with i.i.d. couplings. Our approach considers the Z ₂ local gauge invariance of the system. We see the gauge group as a graph theoretic linear code ? over GF(2). The gauge is fixed by choosing a convenient linear supplement of ?. Assuming some relation between the disorder parameter p and the inverse temperature of the thermal bath β _pb , we study percolation in the random interaction random cluster model, and extend the results to dilute spin glasses. Received: 5 May 1997 / Revised version: 9 April 1998     

On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Abstract

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formula

Q_k f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k² ? 1)f (y)

for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Q_k f (x, y) = 0, after it, we show that if Q_k f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].     

Application of extended kinetic theory to particle transport with inelastic scattering

In the frame of extended kinetic theory, the linear Boltzmann equation for test particles in an absorbing and inelastically scattering background leads to a partial-integral-difference equation which is studied in the proper mathematical setting. As an application, penetration of a beam of particles in a plane slab is considered in steady state conditions, and the relevant problem is solved by a rigorous algorithm. Accurate results for particle and energy distributions, and for transmission, reflection, and absorption coefficients are provided and briefly discussed.

---

Riassunto Viene studiata, nell'ambito della teoria cinetica estesa, 1'equazione di Boltzmann lineare per il trasporto di particelle in un mezzo assorbente e scatterante anelasticamente. L'equazione integro-differenziale alle differenze è applicata al problema stazionario della penetrazione di un fascio di particelle in una lastra piana. Vengono presentati risultati numerici rigorosi per le distribuzioni di particelle ed energia, e per i coefficient: di trasmissione, riflessione ed assorbimento.

     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.     

Hydrodynamical limit for spatially heterogeneous simple exclusion processes

Summary.   We prove hydrodynamical limit for spatially heterogeneous, asymmetric simple exclusion processes on Z ^d . The jump rate of particles depends on the macroscopic position x through some nonnegative, smooth velocity profile α(x). Hydrodynamics are described by the entropy solution to a spatially heterogeneous conservation law of the form

How to Choose the Right Dean for Your University: Remembering the Five P's of Deanship

Abstract

In this column, as in many other print and electronic forums, information technology (IT) is discussed at length in terms of applications for students, faculty, staff, and administrators; in terms of hardware, software, and infrastructure; and in terms of strategic planning, budgeting, and fund-raising. There is only one person on every campus who must constantly assess and balance all these factors and constituencies: the president. However, a presidential perspective—the view from 30,000 feet—does not appear often in discussions about IT in higher education. This column shares my recent conversation on the subject with Carol A. Cartwright, president of Kent State University, a CEO to whom I am privileged to have unique access.     

Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity

Summary. In this paper we consider the numerical solutions of the nonlinear time-dependent Ginzburg-Landau model which describes the phase transitions taking place in superconducting films. We propose a semi-implicit finite element scheme which is based on a linear finite element approximation of the order parameter and a mixed finite element discretization for the equation involving the magnetic potential A. The error estimates of the scheme are derived. Received September 5, 1994 / Revised version received April 23, 1995     

Analysis of a Quantum Subband Model for the Transport of Partially Confined Charged Particles

This paper is devoted to the analysis of a quantum subband model, which is presented as an alternative to the standard 3D Schr?dinger-Poisson system for modeling the transport of electrons strongly confined along one direction. This subband model is composed of quasistatic 1D Schr?dinger equations in the direction of the confinement, coupled to 2D time-dependent Schr?dinger equations describing the transport in the non-confined directions. Selfconsistent electrostatic interactions are also taken into account via the Poisson equation. This system is studied in the framework of the strong partial confinement asymptotics introduced in the article “Adiabatic approximation of the Schr?dinger-Poisson system with a partial confinement”, by Ben Abdallah, Méhats and Pinaud (SIAM J. Math. Anal. 36 (2005), 986–1013).     

Estimates on path delocalization for copolymers at selective interfaces

Starting from the simple symmetric random walk {S_n}_n, we introduce a new process whose path measure is weighted by a factor exp with α,h≥0, {W_n}_n a typical realization of an IID process and N a positive integer. We are looking for results in the large N limit. This factor favors S_n>0 if W_n+h>0 and S_n<0 if W_n+h<0. The process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating two selective solvents. It has been shown [6] that this model undergoes a (de)localization transition: more precisely there exists a continuous increasing function λ↦h_c(λ) such that if h<h_c(λ) then the model is localized while it is delocalized if h≥h_c(λ). However, localization and delocalization were not given in terms of path properties, but in a free energy sense. Later on it has been shown that free energy localization does indeed correspond to a (strong) form of path localization [3]. On the other hand, only weak results on the delocalized regime have been known so far. We present a method, based on concentration bounds on suitably restricted partition functions, that yields much stronger results on the path behavior in the interior of the delocalized region, that is for h>h_c(λ). In particular we prove that, in a suitable sense, one cannot expect more than O( log N) visits of the walk to the lower half plane. The previously known bound was o(N). Stronger O(1)–type results are obtained deep inside the delocalized region. The same approach is also helpful for a different type of question: we prove in fact that the limit as α tends to zero of h_c(λ)/λ exists and it is independent of the law of ω₁, at least when the random variable ω₁ is bounded or it is Gaussian. This is achieved by interpolating between this class of variables and the particular case of ω₁ taking values ±1 with probability 1/2, treated in [6].     

An asymptotic formula for the t-core partition function and a conjecture of Stanton

For a positive integer t, a partition is said to be a t-core if each of the hook numbers from its Ferrers-Young diagram is not a multiple of t. In 1996, Granville and Ono proved the t-core partition conjecture, that at(n), the number of t-core partitions of n, is positive for every nonnegative integer n as long as t?4. As part of their proof, they showed that if p?5 is prime, the generating function for ap(n) is essentially a multiple of an explicit Eisenstein Series together with a cusp form. This representation of the generating function leads to an asymptotic formula for ap(n) involving L-functions and divisor functions. In 1999, Stanton conjectured that for t?4 and n?t+1, at(n)?at+1(n). Here we prove a weaker form of this conjecture, that for t?4 and n sufficiently large, at(n)?at+1(n). Along the way, we obtain an asymptotic formula for at(n) which, in the cases where t is coprime to 6, is a generalization of the formula which follows from the work of Granville and Ono when t=p?5 is prime.     

Infinite paths with bounded or recurrent partial sums

We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or −1, with probability 1 −p. We ask whether an infinite path π exists, with the property that the partial sums of the ±1s along π are uniformly bounded, and whether there exists an infinite path π' with the property that the partial sums along π' are equal to zero infinitely often. The answers to these question depend on the type of path one allows, the value of p and the uniform bound specified. We show that phase transitions occur for these phenomena. Moreover, we make a surprising connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each vertex at most finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson process in the plane. For the recurrence problem, we also show that the probability of finding such a path is monotone in p, for p≥?. Received: 10 January 2000 / Revised version: 14 August 2000 / Published online: 9 March 2001     

The diffusive phase of a model of self-interacting walks

Summary We consider simple random walk onZ ^d perturbed by a factor exp[T ^–P J _T], whereT is the length of the walk and . Forp=1 and dimensionsd2, we prove that this walk behaves diffusively for all – < <₀, with ₀ > 0. Ford>2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real (positive or negative). Ford>2 the scaling limit is Brownian motion, but ford2 it is the Edwards model (with the wrong sign of the coupling when >0) which governs the limiting behaviour; the latter arises since for ,T ^–p J _T is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.     

Temperely-Lieb Algebras and the Four-Color Theorem

The Temperley–Lieb algebra T_n with parameter 2 is the associative algebra over Q generated by 1,e₀,e₁, . . .,e_n, where the generators satisfy the relations if |i–j|=1 and e_ie_j=e_je_i if |i–j|2. We use the Four Color Theorem to give a necessary and sufficient condition for certain elements of T_n to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.* Partially supported by NSF under Grant DMS-9802859 and by NSA under grant MDA904-97-1-0015. Partially supported by NSF under Grant DMS-9623031 and by NSA under Grant MDA904-98-1-0517.     

Feynman graph representation of the perturbation series for general functional measures

A representation of the perturbation series of a general functional measure is given in terms of generalized Feynman graphs and rules. The graphical calculus is applied to certain functional measures of Lévy type. A graphical notion of Wick ordering is introduced and is compared with orthogonal decompositions of the Wiener-Itô-Segal type. It is also shown that the linked cluster theorem for Feynman graphs extends to generalized Feynman graphs. We perturbatively prove existence of the thermodynamic limit for the free energy density and the moment functions. The results are applied to the gas of charged microscopic or mesoscopic particles—neutral in average—in d=2 dimensions generating a static field φ with quadratic energy density giving rise to a pair interaction. The pressure function for this system is calculated up to fourth order. We also discuss the subtraction of logarithmically divergent self-energy terms for a gas of only one particle type by a local counterterm of first order.     

Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions

Summary We show a strong type of conditionally mixing property for the Gibbs states ofd-dimensional Ising model when the temperature is above the critical one. By using this property, we show that there is always coexistence of infinite (+ *)-and (–*)-clusters when is smaller than _c andh=0 in two dimensions. It is also possible to show that this coexistence region extends to some non-zero external field case, i.e., for every < _c, there exists someh _c()>0 such that |h|<h _c() implies the coexistence of infinite (*)-clusters with respect to the Gibbs state for (,h).work supported in part by Grant in Aid for Cooperative research no. 03302010, Grant in Aid for Scientific Research no. 03640056 and ISM Cooperative research program (91-ISM,CRP-3)To the memory of Professor Haruo Totoki     

A pattern theorem for lattice clusters

We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration (pattern) of sites and bonds can occur in large clusters, then for some constantc>0, it occurs at leastcn times in most clusters of sizen. An analogous theorem for self-avoiding walks was proven in 1963 by Kesten [9]. We use the pattern theorem to prove the convergence of lim _n a _n+1 /a _n , wherea _n is the number of clusters of sizen, up to translation. The results also apply to weighted sums, and in particular, we can takea _n to be the probability that the percolation cluster containing the origin consists of exactlyn sites. Another consequence is strict inequality of connective constants for sublattices and for certain subclasses of clusters.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.The author was visiting the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, while writing this paper.     

Hands-on tutorial for parallel computing with R

Due to the increasing availability of powerful hardware resources, parallel computing is becoming an important issue, as a noticeable speedup may be achieved. The statistical programming language R allows for parallel computing on computer clusters as well as multicore systems through several packages. This tutorial gives a short, practical overview of four, in view of the authors, important packages for parallel computing in R, namely multicore, snow, snowfall and nws. First, the general principle of parallelizing simple tasks is briefly illustrated based on a statistical cross-validation example. Afterwards, the usage of each of the introduced packages is being demonstrated on the example. Furthermore, we address some specific features of the packages and provide guidance for selecting an adequate package for the computing environment at hand.