Renormalization of Hierarchically Interacting Isotropic Diffusions

We study a renormalization transformation arising in an infinite system of interacting diffusions. The components of the system are labeled by the N-dimensional hierarchical lattice (N≥2) and take values in the closure of a compact convex set $\bar D \subset \mathbb{R}^d (d \geqslant 1)$ . Each component starts at some θ ∈ D and is subject to two motions: (1) an isotropic diffusion according to a local diffusion rate g: $\bar D \to [0,\infty ]$ chosen from an appropriate class; (2) a linear drift toward an average of the surrounding components weighted according to their hierarchical distance. In the local mean-field limit N→∞, block averages of diffusions within a hierarchical distance k, on an appropriate time scale, are expected to perform a diffusion with local diffusion rate F ^(k) g, where $F^{(k)} g = (F_{c_k } \circ ... \circ F_{c_1 } )$ g is the kth iterate of renormalization transformations F _c (c>0) applied to g. Here the c _k measure the strength of the interaction at hierarchical distance k. We identify F _c and study its orbit (F ^(k) g) _k≥0. We show that there exists a “fixed shape” g* such that lim _k→∞ σ_k F ^(k) g = g* for all g, where the σ _k are normalizing constants. In terms of the infinite system, this property means that there is complete universal behavior on large space-time scales. Our results extend earlier work for d = 1 and $\bar D = [0,1]$ , resp. [0, ∞). The renormalization transformation F _c is defined in terms of the ergodic measure of a d-dimensional diffusion. In d = 1 this diffusion allows a Yamada–Watanabe-type coupling, its ergodic measure is reversible, and the renormalization transformation F _c is given by an explicit formula. All this breaks down in d≥2, which complicates the analysis considerably and forces us to new methods. Part of our results depend on a certain martingale problem being well-posed.     

Spectrum Localization of Infinite Matrices

The paper deals with linear operators in a separable Hilbert space represented by infinite matrices with compact off diagonal parts. Bounds for the spectrum are established. In particular, new estimates for the spectral radius are proposed. These results are new even in the finite-dimensional case. Also applications to integral, differential and integro-differential operators are discussed.     

On the Motion of a Charged Particle Interacting with an Infinitely Extended System

 We study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. In the framework of fully Hamiltonian models, we discuss conditions on the particle/medium interaction which are necessary for the particle to reach a finite limit velocity. We first consider the case when the charged particle is confined in an unbounded tube of ℝ³. The electric field E is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system. The background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic (equilibrium or non-equilibrium) state. We prove that, for large E and bounded interactions between the charged particle and the background, the velocity v(t) of the charged particle does not reach a finite limit velocity, but it increases to infinite as: |v(t)−Et|≤C ₀ (1+t), where C ₀ is a constant independent of E. As a corollary we obtain that, if the initial conditions of the background system are distributed according to any Gibbs state, then the average velocity of the charged particle diverges as time goes to infinite. This result is obtained for E large enough in comparison with the mean energy of the Gibbs state. We next study the one-dimensional case, in which the estimates can be improved. We finally discuss, at an heuristic level, the existence of a finite limit velocity for unbounded interactions, and give some suggestions about the case of small electric fields. Received: 7 March 2002 / Accepted: 23 September 2002 Published online: 8 January 2003 RID="*" ID="*" Work partially supported by the GNFM-INDAM and the Italian Ministry of the University. Communicated by J.L. Lebowitz     

Diffusive Hydrodynamic Limits for Systems of Interacting Diffusions with Finite Range Random Interaction

We consider a system of interacting diffusive particles with finite range random interaction. The variables can be interpreted as charges at sites indexed by a periodic multidimensional lattice. The equilibrium states of the system are canonical Gibbs measures with finite range random interaction. Under the diffusive scaling of lattice spacing and time, we derive a deterministic nonlinear diffusion equation for the time evolution of the macroscopic charge density. This limit is almost sure with respect to the random environment. Received: 3 October 1996 / Accepted: 13 February 1997     

Weyl Semimetallic Phase in an Interacting Lattice System

By using Wilsonian renormalization group methods we rigorously establish the existence of a Weyl semimetallic phase in an interacting three dimensional fermionic lattice system, by showing that the zero temperature Schwinger functions are asymptotically close to the ones of massless Dirac fermions. This is done via an expansion which is convergent in a region of parameters, which includes the quantum critical point discriminating between the semimetallic and the insulating phase.     

On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The location of the essential spectrum of H is described. The existence of infinitely many eigenvalues (resp. the finiteness of eigenvalues) below the bottom τ_ess(H) of the essential spectrum of H is proved for the case where the associated Friedrichs model has a threshold energy resonance (resp. a threshold eigenvalue). For the number N(z) of eigenvalues of H lying below z < τ_ess(H) the following asymptotics is found

Ornstein-Uhlenbeck Limit for the Velocity Process of an N-Particle System Interacting Stochastically

An N-particle system with stochastic interactions is considered. Interactions are driven by a Brownian noise term and total energy conservation is imposed. The evolution of the system, in velocity space, is a diffusion on a (3N?1)-dimensional sphere with radius fixed by the total energy. In the N→∞ limit, a finite number of velocity components are shown to evolve independently and according to an Ornstein-Uhlenbeck process.     

Kerr介质中双原子与相干态光场Raman过程的腔场谱

为探讨Kerr介质和原子间的耦合作用对两个等同二能级原子与双模光场Raman相互作用过程腔场谱的影响,通过求解本征方程导出了双模相干态光场处于任意强度时腔场谱的计算公式,给出了数值计算结果.发现在两模初始场均为弱场时,Kerr介质和原子间的耦合作用对谱结构的影响较小;在双模初始场均为强场时,有Kerr介质加入后两模的中心主峰迅速下降,但随Kerr效应的增强,产生出越来越多的边峰,且在两模主峰的两侧产生出拱形的梳状边峰群,并伴随着峰位的向右偏移.     

On Diffraction of a Point Sound Source by an Infinite Wedge

Acoustical Physics - We consider a two-dimensional problem of diffraction of a harmonic sound wave emitted by a point sound source located near the sharp angle of an infinite wedge asymmetrically...     

Low-Frequency Spectrum of the Gyrotropic Modes of a Finite Chain of Interacting Ferromagnetic Disks

Technical Physics - The problem of the frequency spectrum of collective gyrotropic modes in a finite linear chain of magnetostatically interacting ferromagnetic disks has been considered. It has...     

QED中相互作用场的量子化

提出在QED中利用自由场的量子化条件和场方程,可以全面解决独立和非独立的相互作用场中￠(x),Aμ(x)的量子化问题.证明了:(1)当外规范场(x)≠0时,￠(x)与Aμ(x)相互独立,它与常规的场量子化方法一致.(2)当(x)=0时,￠(x)与Aμ(x)不再相互独立,常规的量子化方法不再适用,而本文提出的方法继续有效.并以1+1维QED作为实例.     

Spectrum of a Problem of Elasticity Theory in the Union of Several Infinite Layers

The essential spectrum of the Dirichlet problem for the system of Lamé equations in a three-dimensional domain formed by three mutually perpendicular elastic layers occupies the ray [Λ_?,+∞). The lower bound Λ_? > 0 is the least eigenvalue (its existence is established) of the problem of elasticity theory in an infinite two-dimensional cross-shaped waveguide. It is proved that the discrete spectrum of the spatial problem is nonempty. Other configurations of layers and the scalar problem of the junction of quantum waveguides are also considered.     

Entropy Dynamics in the System of Interacting Qubits

The classical second law of thermodynamics demands that an isolated system evolves with a nondiminishing entropy. This holds as well in quantum mechanics if the evolution of the energy-isolated system can be described by a unital quantum channel. At the same time, the entropy of a system evolving via a nonunital channel can, in principle, decrease. Here, we analyze the behavior of entropy in the context of the H-theorem. As exemplary phenomena, we discuss the action of a Maxwell demon (MD) operating a qubit and the processes of heating and cooling in a two-qubit system. Further we discuss how small initial correlations between a quantum system and a reservoir affect the entropy increase during the quantum-system evolution.     

On the Generalized Thermoelasticity Problem for an Infinite Fibre-Reinforced Thick Plate under Initial Stress

In this paper, the generalized thermoelasticity problem for an infinite fiber-reinforced transversely-isotropic thick plate subjected to initial stress is solved. The lower surface of the plate rests on a rigid foundation and temperature while the upper surface is thermally insulated with prescribed surface loading. The normal mode analysis is used to obtain the analytical expressions for the displacements, stresses and temperature distributions. The problem has been solved analytically using the generalized thermoelasticity theory of dual-phase-lags. Effect of phase-lags, reinforcement and initial stress on the field quantities is shown graphically. The results due to the coupled thermoelasticity theory, Lord and Shulman's theory, and Green and Naghdi's theory have been derived as limiting cases. The graphs illustrated that the initial stress, the reinforcement and phase-lags have great effects on the distributions of the field quantities.     

非等同两原子与光场相互作用系统的腔场谱

研究了两个二能级原子与单模腔场具有不同耦合常量系统的腔场谱,讨论了量子化光场分别处于不同的光子数态时腔场谱结构随相对耦合常量R变化的新特性。发现随着R由0到1的增加,腔场谱各对应峰峰位相对腔场原共振频率ω0对称偏移;真空场（n=0）的峰高在0〈R〈0．3内变化较快,其拉比峰个数按2→6→4规律变化;弱场（n=1）峰高在0〈R〈0．5内变化较快,其峰数量按2→6→12→9→7规律变化;强场（n=8）峰高在0〈R〈0．1内改变迅速,其峰数量按2→8→11→5规律变化,谱结构显得更加复杂;进一步的计算表明,当n〉〉8时。其峰的数量按3→5→3规律变化。同时发现,R=0和R=1时,峰的数量相对较少。各峰峰高之和由最小单调增至最大,这在物理上反映了原子的协作效应。