The Solutions for the Boundary Layer Problem of Boltzmann Equation in a Half-Space

We study the half space boundary layer problem for Boltzmann equation with cut-off potentials in all the cases −3<γ≤1, while the boundary condition is imposed on the incoming particles of Dirichlet type, and the solution is assumed to approach to a global Maxwellian at the far field. The same as for cut-off hard sphere model, there is an implicit solvability condition on the boundary data which gives the co-dimensions of the boundary data in terms of positive characteristic speeds.     

Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

The dynamics of an electron gas in a constant ion background can be decribed by the Vlasov-Poisson-Boltzmann system at the kinetic level, or by the compressible Euler-Poisson system at the fluid level. We prove that any solution of the Vlasov-Poisson-Boltzmann system near a smooth local Maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the Euler-Poisson system, as the mean free path ε goes to zero. We use a recent L ² − L ^∞ framework in the Boltzmann theory to control the higher order remainder in the Hilbert expansion uniformly in ε and globally in time.     

Low-Temperature Dynamics of the Curie-Weiss Model: Periodic Orbits,Multiple Histories,and Loss of Gibbsianness

We consider the Curie-Weiss model at initial temperature 0<β ⁻¹≤∞ in vanishing external field evolving under a Glauber spin-flip dynamics with temperature 0<β′⁻¹≤∞. We study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity (bad points). We provide a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending earlier work for the case of independent spin-flip dynamics.     

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?^* _ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?^* _ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables. Received: 28 April 2000 / Accepted: 6 December 2000     

On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions

The paper considers macroscopic behavior of a Fermi–Dirac particle system. We prove the L ¹-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box with the collision kernel b(cos θ)|ρ−ρ _*|^γ, which corresponds to very soft potentials: −5 < γ ≤ −3 with a weak angular cutoff: ∫₀ ^π b(cos θ)sin ³θ dθ < ∞. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff–Young inequality, the L ^∞-bounds of the solutions, and a specific property of the value-range of the exponent γ. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy.     

Zero-Temperature Dynamics of ±J Spin Glasses¶and Related Models

We study zero-temperature, stochastic Ising models σ ^t on Z ^d with (disordered) nearest-neighbor couplings independently chosen from a distribution μ on R and an initial spin configuration chosen uniformly at random. Given d, call μ type ℐ (resp., type ℱ) if, for every x in Z ^d , σ _x ^t flips infinitely (resp., only finitely) many times as t→∞ (with probability one) – or else mixed type ℳ. Models of type ℒ and ℳ exhibit a zero-temperature version of “local non-equilibration”. For d=1, all types occur and the type of any μ is easy to determine. The main result of this paper is a proof that for d=2, ±J models (where μ=αδ _J +(1-α)δ_{- J} ) are type ℳ, unlike homogeneous models (type ℐ) or continuous (finite mean) μ's (type ℳ). We also prove that all other noncontinuous disordered systems are type ℳ for any d≥ 2. The ±J proof is noteworthy in that it is much less “local” than the other (simpler) proof. Homogeneous and ±J models for d≥ 3 remain an open problem. Received: 3 November 1999 / Accepted: 10 April 2000     

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

 We consider the quasi-geostrophic equation with the dissipation term, κ (-Δ)^α θ, In the case , Constantin-Cordoba-Wu [6] proved the global existence of strong solution in H ¹ and H ² under the assumption of small L ^∞ -norm of initial data. In this paper, we prove the global existence in the scale invariant Besov space, B ^2−2α _2,1 , for initial data small in the B ^2−2α _2,1 norm. We also prove a global stability result in B ¹ _2,1 . Received: 24 April 2002 / Accepted: 29 July 2002 Published online: 10 December 2002 Communicated by P. Constantin     

Nuclear Spin-Lattice Relaxation of 82BrFe

The field dependence of the nuclear spin-lattice relaxation (SLR) of cold implanted ⁸²Br (T ≤ 25 mK) in α-Fe single crystals was investigated with nuclear magnetic resonance of oriented nuclei (NMR/ON) at low temperatures as experimental technique. The SLR at the lattice sites with the hyperfine fields found by earlier NMR/ON experiments was measured as a function of the applied external magnetic field B _ext parallel to the three principle axes [100], [110] and [111] of the iron single crystal. The data were evaluated with the full relaxation formalism in the single impurity limit and for comparison also with the often employed model of a single exponential function with an effective relaxation time T ₁′. With a phenomenological model the high field values of the relaxation rates r _{∞, [100]′} = 6.6(2) · 10⁻¹⁵ T²sK⁻¹, r _{∞, [110]} = 5.4(2) · 10⁻¹⁵ T²sK⁻¹ and r _{∞, [111]} = 5.2(1) · 10⁻¹⁵ T²sK⁻¹ were obtained.     

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

L<Superscript>1</Superscript> Stability of Spatially Periodic Solutions in Relativistic Gas Dynamics

This paper proves the well posedness of spatially periodic solutions of the relativistic isentropic gas dynamics equations. The pressure is given by a γ-law with initial data of large amplitude, provided γ − 1 is sufficiently small. As a byproduct of our techniques, we obtain the same results for the classical case. At the limit c → + ∞, the solutions of the relativistic system converge to the solutions of the classical one, the convergence rate being 1/c ². We also construct the semigroup of solutions of the Cauchy problem for initial data with bounded total variation, which can be large, as long as γ − 1 is small.     

Stability Theorems for Chiral Bag Boundary Conditions

We study asymptotic expansions of the smeared L ²-traces Fe^{−t P^2} and FPe^{−tP^2}, where P is an operator of Dirac type and F is an auxiliary smooth endomorphism. We impose chiral bag boundary conditions depending on an angle θ. Studying the θ-dependence of the above trace invariants, θ-independent pieces are identified. The associated stability theorems allow one to show the regularity of the eta function for the problem and to determine the most important heat kernel coefficient on a four dimensional manifold. Mathematics Subject Classification (2000). 58J50     

Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Semilinear PDEs on Self-Similar Fractals

A Laplacian may be defined on self-similar fractal domains in terms of a suitable self-similar Dirichlet form, enabling discussion of elliptic PDEs on such domains. In this context it is shown that that semilinear equations such as Δu+u ^p = 0, with zero Dirichlet boundary conditions, have non-trivial non-negative solutions if 0<ν≤ 2 and p>1, or if ν >2 and 1<p< (ν+ 2)/(ν− 2), where ν is the “intrinsic dimension” or “spectral dimension” of the system. Thus the intrinsic dimension takes the r\^{o}le of the Euclidean dimension in the classical case in determining critical exponents of semilinear problems. Received: 11 December 1998 / Accepted: 22 March 1999     

Asymptotics of Determinants of Bessel Operators

 For aL ^∞ (ℝ₊)∩L ¹ (ℝ₊) the truncated Bessel operator B _τ (a) is the integral operator acting on L ² [0,τ] with the kernel

Infinite Operator-Sum Representation of Density Operator for a Dissipative Cavity with Kerr Medium Derived by Virtue of Entangled State Representation

By using the thermo entangled state representation we solve the master equation for a dissipative cavity with Kerr medium to obtain density operators’ infinite operator-sum representation ρ(t)=∑ _m,n,l=0^∞ M _m,n,l ρ ₀ℳ _m,n,l ^† . It is noticeable that M _m,n,l is not Hermite conjugate to ℳ _m,n,l ^† , nevertheless the normalization ∑ _m,n,l=0^∞ℳ _n,m,l ^† M _m,n,l =1 still holds, i.e., they are trace-preserving in a general sense. This example may stimulate further studying if general superoperator theory needs modification.     

KdV-soliton dynamics in a random field

We consider the interaction between soliton and a spatially uniform external random field within the framework of the forced Korteweg-de Vries equation. In the general case, the averaged soliton field is transformed to a Gaussian pulse whose amplitude falls off with time as t^−α, while its width increases as t^α, where the parameter α is characterized by the statistical properties of the external force. We obtain an analytical solution for α = 2, which corresponds to the limiting case of an infinitely long correlation time (τ₀ → ∞). The obtained solution is compared with the well-known Wadati solution for the case of a delta-correlated external force (τ₀ → 0) where the soliton is transformed to a Gaussian pulse with amplitude falling off at a lower rate α = 3/2. The numerical solutions of the forced Korteweg-de Vries equation, which demonstrate an increase in the parameter α from 3/2 to 2 with increasing correlation time, are given for the intermediate case corresponding to 0 < τ₀ < ∞. It is shown that the amplitude of the averaged soliton in a periodic random field falls off as t⁻¹ for the long times t. In this case, two pulses propagating in different directions are formed. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 7, pp. 599–606, July 2006.     

Current Reservoirs in the Simple Exclusion Process

We consider the symmetric simple exclusion process in the interval [−N,N] with additional birth and death processes respectively on (N−K,N], K>0, and [−N,−N+K). The exclusion is speeded up by a factor N ², births and deaths by a factor N. Assuming propagation of chaos (a property proved in a companion paper, De Masi et al., ) we prove convergence in the limit N→∞ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non-linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.     

Nonlinear Stability of Boundary Layers of the Boltzmann Equation,I. The case <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript><−1

This is a continuation of the paper [15] on nonlinear boundary layers of the Boltzmann equation where the existence is established and shown to be strongly dependent on the Mach number M of the Maxwellian state at far field. In this paper, when M <–1, we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers.     

Dielectronic recombination of lithium-like gold: Towards QED tests

Dielectronic recombination (DR) and radiative recombination (RR) of lithium-like gold in the energy range of 0 to 225 eV have been studied at the Experimental Storage Ring (ESR) of the GSI in Darmstadt. Main objective of the measurements is the precise determination of the 2s_1/2−2p_1/2 energy splitting as an additional QED test. This novel method, developed at the ESR [1], is based on the extrapolation of a multitude of measured resonances Au⁷⁵⁺ (1s²2p_1/2 nl_j) up to the series limit (n = ∞). Furthermore experimental data for the Au⁷⁵⁺ (1s²2p_3/26l_j) resonance manifold are presented. This revised version was published online in July 2006 with corrections to the Cover Date.