Fluctuations and Phase Transition Dynamics

Kibble and Zurek have provided a unifying causal picture for the appearance ofclassical defects like cosmic strings or vortices at the onset of phase transitionsin relativistic QFT and condensed matter systems, respectively. In condensedmatter the predictions are partially supported by agreement with experiments insuperfluid helium. We provide an alternative picture for the initial appearanceof defects that supports the experimental evidence. When the original predictionsfail, this is understood, in part, as a consequence of thermal fluctuations (noise),which play a comparable role in both condensed matter and QFT.     

Fluctuations for Kawasaki Dynamics

In this paper Kawasaki dynamics are considered. Lower bounds are obtained for the variance of the occupation time of a site in any dimension and for temperature above critical temperature. These lower bounds are expressed in terms of the density correlation function and hence relate the fluctuations to some phase transition quantities. At critical temperature, under a reasonable assumption of the static structure function, lower bounds for the variance of the occupation time are obtained. These lower bounds are consistent with the supposed value of the critical exponent. This paper also examines the same problem for Glauber dynamics and shows that the phase transition may not be of importance for the behavior of fluctuations.     

Spectral Analysis of Universal Conductance Fluctuations

Journal of Experimental and Theoretical Physics - Universal conductance fluctuations are usually observed in the form of aperiodic oscillations in the magnetoresistance of thin wires under...     

Dynamics and Universality of Unimodal Mappings with Infinite Criticality

We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.Partially supported by NSF grant DMS-0245358.     

Nonlinear Dynamics and Fluctuations of Dissipative Toda Chains

The dynamics of a ring of masses including dissipative forces (passive or active friction) and Toda interactions between the masses is investigated. The characteristic attractor structure and the influence of noise by coupling to a heat bath are studied. The system may be driven from the thermodynamic equilibrium to far from equilibrium states by including negative friction. We show, that over-critical pumping with free energy may lead to a partition of the phase space into attractor regions corresponding to several types of collective motions including uniform rotations, one- and multiple soliton-like excitations and relative oscillations. The distribution functions in the phase space and the correlation functions of the forces and the spectra of nonlinear excitations are calculated. We show that a finite-size Toda ring with weak thermal coupling develops at intermediate temperatures a broadband colored noise spectrum with an 1/f tail at low frequencies.     

Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class

This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the regions, when the size of the regions tend to infinity, together with the cuts. It leads to a new kernel, which is expected to have universality properties.     

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ _{β ,p}  > 0, such that for all exponential time scales, exp(γ N), with γ < γ _{β ,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β ² < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.     

On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles

 We consider real random symmetric N × N matrices H of the band-type form with characteristic length b. The matrix entries are independent Gaussian random variables and have the variance proportional to , where u(t) vanishes at infinity. We study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace . We examine on the local scale and show that its asymptotic behavior is determined by the rate of decay of u(t). In particular, if u(t) decays exponentially, then . This expression is universal in the sense that the particular form of u determines the value of C > 0 only. Our results agree with those detected in both numerical and theoretical physics studies of spectra of band random matrices. Received: 8 April 2000 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Present address: Département de Mathématiques, Université de Versailles Saint-Quentin, 78035 Versailles, France.     

Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N-body Schrödinger equation and the orthogonal projection onto the solution of the Hartree equation.     

透过率起伏光谱分析法测粒技术

提出的透过率起伏光谱分析法是一种新的颗粒测量方法。采用一细小光束照射匀速流动的颗粒系统,通过采集透射光起伏信号,经统计处理得到透过率的平均值与起伏谱。通过求解逆问题,从透过率的起伏谱中得到颗粒粒径分布信息,再结合透过率的平均值得到颗粒的体积分数信息。给出了关于单层颗粒透过率的平均值与起伏谱的理论表达式,并推广到三维单分散和多分散的颗粒系统。对粒径在32~425μm内的稀薄颗粒系进行了部分实验测试和模拟计算,结果表明该方法可同时对颗粒粒径分布和体积分数进行有效测量。     

Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous–Diaconis–Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working. Received: 4 October 2001 / Accepted: 12 March 2002     

On the Spectral Distribution of Equilibrium Radiation and Zero-Point Fluctuations in the Coulomb System

The consideration of equilibrium radiation in plasma-like media shows that the spectral energy distribution of such radiation differs from that of Planck equilibrium radiation. Based on the previously derived relation for the spectral energy density of equilibrium radiation in the system of charged particles, accounting for finite damping in a medium with spatial dispersion, the limiting case of infinitesimal damping dispersion is considered. It was shown that zero-point vacuum fluctuations being a component of the total spectral energy distribution in the medium should be renormalized when using certain models for the transverse plasma permittivity. In this case, renormalized zero-point vacuum fluctuations become dependent on plasma parameters. The possibility of the manifestation of this effect is discussed.     

Quantum Level Structures and Nonlinear Classical Dynamics

The classical structure underlying the quantum mechanics of molecular vibrations is illustrated by reference to changes in vibrational energy distributions induced by increased anharmonic coupling as the energy increases. Specific applications to Fermi resonance models and to the level structures arising from cylindrically symmetrical saddle points are used to illustrate the relevance of classical bifurcation analysis, catastrophe theory, and quantum monodromy.     

Quantum Dynamics in Noisy Backgrounds: from Sampling to Dissipation and Fluctuations

We investigate the dynamics of a quantum system coupled linearly to Gaussian white noise using functional methods. By performing the integration over the noisy field in the evolution operator, we get an equivalent non-Hermitian Hamiltonian, which evolves the quantum state with a dissipative dynamics. We also show that if the integration over the noisy field is done for the time evolution of the density matrix, a gain contribution from the fluctuations can be accessed in addition to the loss one from the non-hermitian Hamiltonian dynamics. We illustrate our study by computing analytically the effective non-Hermitian Hamiltonian, which we found to be the complex frequency harmonic oscillator, with a known evolution operator. It leads to space and time localisation, a common feature of noisy quantum systems in general applications.     

Spectral Properties of Effective Dynamics from Conditional Expectations

The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.