Random matrices, fermions, collective fields, and universality

We first relate the random matrix model to a Fokker-Planck Hamiltonian system, such that the correlation functions of the model are expressed as the vacuum expectation values of equal-time products of density operators. We then analyze the universality of the random matrix model by solving the Focker-Planck Hamiltonian system for large N. We use two equivalent methods to do this, namely the method of relating it to a system of interacting fermions in one space dimension and the method of collective fields for large N matrix quantum mechanics. The final result using both these methods is the same Hamiltonian system of chiral bosons on a circle, which manifestly exhibits the universality of the random matrix model.     

Random external fields

The various theoretical considerations for the effects of quenched random fields (RF) on second-order transitions as well as the experimental situation are briefly reviewed. Some of the physical realizations of the RF models are discussed, with an emphasis on solid-state first-order transitions in impure systems. The physical arguments for the RF effects in the bulk as well as on phase interfaces are discussed. In the latter case it is suggested that scattering experiments can probe the details of the interface fluctuations. The role of long relaxation times and metastability in Ising RF systems is emphasized.Research at Tel Aviv University partially supported by the U.S.-Israel Binational Science Foundation (BSF), Jerusalem, Israel.     

Random fields and quantum dynamics

We shall discuss quantum mechanical operators depending on the time or on the manifold in space. There is a similarity to the case of stochastic processes or random fields, where the innovation approach is one of the powerful tools to investigate their probabilistic structure. Having had some review of the innovation, similar attempt is made for some cases in quantum dynamics.     

Reconstruction, disordering and roughening of metal surfaces

Until recently, surface reconstruction and roughening have been studied as separate surface phenomena and have been described theoretically by means of separate models. This approach is reasonable when the deconstruction transition has a displacive character as occurs for example on Mo(001) and W(001) surfaces. In these cases only in-plane degrees of freedom are involved in the reconstruction process. Conversely, a unified approach to reconstruction and roughening seems more appropriate when the reconstructed phase is produced by the low-temperature ordering of a large concentration of point defects (vacancies, adatoms) or extended defects (steps). The reconstruction in these cases involves off-plane degrees of freedom. Since roughening derives from the proliferation of thermally excited steps, the interplay between steps and deconstructive defects may provide the connection between the two transitions. A notable example of these latter systems is provided by the (110) surface of noble and near-noble metals; the (110) surface of the heavier metals (Au, Pt) reconstructs at low temperature in the 2 x 1 missing-row structure. In the present article we review some statistical mechanics models able to display both deconstruction and roughening of the fcc(110) missing-row phase, chosen as a paradigmatic example of the interplay between the two transitions. The properties of the phase between deconstruction and roughening, produced by the different models are analysed, and compared with the results of molecular-dynamics simulations with continous potentials. The fingerprint of the different phases in scattering measurements are discussed and compared with the experimental data available.     

Random fields and quantum effects in proton glasses

The effects of quantum fluctuations on the proton glass phase in mixed hydrogen-bonded ferro-antiferroelectric systems are considered. The system is described in terms of the infinite-ranged Ising pseudospin glass model in a transverse tunneling field in the presence of random parallel fields. The stability limit of the high-temperature proton glass phase is determined within the thermofield dynamic approach, and the behavior or linear and nonlinear susceptibility is evaluated.     

Degeneration of turbulence in simple systems

The results of numerical studies of a simple dissipative system are presented including the appearance of the stochastic attractor under sufficiently strong dissipation. The degeneration of turbulent motion into a periodical one under a weak dissipation is emphasized and studied both qualitatively and quantitatively including the dependence on the number of degrees of freedom.     

Evolution of magnetic fields in freely decaying magnetohydrodynamic turbulence

We study the evolution of magnetic fields in freely decaying magnetohydrodynamic turbulence. By quasilinearizing the Navier-Stokes equation, we solve analytically the induction equation in the quasinormal approximation. We find that, if the magnetic field is not helical, the magnetic energy and correlation length evolve in time, respectively, as E(B) proportional to t(-2(1+p)/(3+p)) and xi(B) proportional to t(2/(3+p)), where p is the index of initial power-law spectrum. In the helical case, the magnetic helicity is an almost conserved quantity and forces the magnetic energy and correlation length to scale as E(B) proportional to (logt)(1/3)t(-2/3) and xi(B) proportional to (logt)(-1/3)t(2/3).     

Color changes in stochastic light fields propagating in non-Kolmogorov turbulence

The dependence of spectral shifts and switches in optical stochastic beams propagating through nonclassic turbulent medium on the slope of the power spectrum of fluctuations in the refractive index is revealed.     

Random fields in Ge-Si alloys produced by doping

A calculation is performed and estimates made of the binary correlation functions of the random field produced by the difference in the atomic pseudopotentials of solvent and dissolved materials. Also evaluated are elastic deformation fields in unordered Ge-Si solid replacement solutions with low Si content (up to 10 at. %).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 41–46, February, 1987.The author heartily thanks V. L. Bonch-Bruevich for his interest in the current study and many valuable discussions.     

Suppress Winfree turbulence by local forcing excitable systems

The occurrence of Winfree turbulence is currently regarded as one of the principal mechanisms underlying cardiac fibrillation. We develop a local stimulation method that suppresses Winfree turbulence in three-dimensional excitable media. We find that Winfree turbulence can be effectively suppressed by locally injecting periodic signals to only a very small subset (around some surface region) of total space sites. Our method for the first time demonstrates the effectiveness of local low-amplitude periodic excitations in suppressing turbulence in 3D excitable media and has fundamental improvements in efficiency, convenience, and turbulence suppression speed compared with previous strategies. Therefore, it has great potential for developing into a practical low-amplitude defibrillation approach.     

Different ways to turbulence in dissipative dynamical systems

The Lorenz model is studied in details for σ = 10, $\text{b =}\text{8}\text{3}$ and 145 < r < 170. Between r = 145 and r = 148.4 the Lore nz attractor disaggregates itself into a limit cycle through a cascade of bifurcation with successive undoubling of periods. At r = 166.07 this limit cycle looses its stability through “intermittency”, giving rise to a second aperiodic attractor. We give a semi-quantitative interpretation of these processes and discuss their relation with the different transitions to turbulence observed experimentally.     

Random fields on Riemannian manifolds: A constructive approach

We extend to Euclidean fields on a wide class of Riemannian manifolds two results which have proven to be crucial in the construction of interacting quantum fields in the flat case, namely local regularity properties of the free covariance in two dimensions and Osterwalder-Schrader positivity.Work supported in part by Ministero della Pubblica Istruzione     

Defect-mediated turbulence in systems with local deterministic chaos

Defect-mediated turbulence is shown to exist in media where the underlying local dynamics is deterministically chaotic. While many of the characteristics of defect-mediated turbulence, such as the exponential decay of correlations and a squared Poissonian distribution for the number of defects, are identical to those seen in oscillatory media, the fluctuations in the number of defects differ significantly. The power spectra suggest the existence of underlying correlations that lead to a different and nonuniversal scaling structure in chaotic media.     

Intermittent transition to turbulence in dissipative dynamical systems

We study some simple dissipative dynamical systems exhibiting a transition from a stable periodic behavior to a chaotic one. At that transition, the inverse coherence time grows continuously from zero due to the random occurrence of widely separated bursts in the time record.     

Random channel kinetics for reaction-diffusion systems

A random channel approach is developed for reaction-diffusion processes in disordered systems. Although the starting point of our research is the kinetic study of the decay and preservation of marine organic carbon, our approach can be used for describing other disordered kinetic catalytic processes with random pathways. We consider a generic catalytic mechanism with two species: (a) a catalyst, which is continuously produced by a variable number of independent sources randomly distributed in space; this catalyst diffuses from the sources and is degrading according to a first order kinetic law; the generation, the degradation and the diffusion of the catalyst balance each other out and a stationary concentration field is generated; (b) an active species, which decays according to a second order kinetic law; the decay rate is proportional to the product of the concentrations of the catalyst and the concentration of the active species. We show that the catalyst concentration field can be represented by the sum of a random number of Yukawa-like potentials. The average value of the survival function of the active species can be expressed as a grand canonical average of a nonlinear functional of the catalyst field and can be evaluated exactly. We show that a good approximation is given by a nearest neighbor approach, where only the contribution of the closest source is taken into account for the computation of the random concentration field of the catalyst. We discuss the application of the model to the problem of decay and preservation of marine organic carbon. With minor adaptation the model can be applied to other problems of disordered kinetics, such as spatially distributed heterogeneous catalytic processes.