Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum

Using a recently developed theory of scintillation that is valid under all fluctuation conditions, including the focussing and saturation regimes, we develop general models for predicting power fluctuations (or aperture averaging) over finite-size collecting apertures. Inner-scale effects are introduced using a modified atmospheric spectrum for refractive-index fluctuations that includes a high-wavenumber bump. Where comparisons can be made, these models of aperture averaging are in good agreement with previous asymptotic models and experimental data. In addition to the aperture-averaging factor, we calculate the temporal spectrum associated with power fluctuations over various aperture sizes and conditions of turbulence. These later results clearly show the transfer of power distribution from high to low frequencies as the size of the collecting aperture is increased. The transfer of power is more pronounced in the saturation regime where high frequencies (fastest fluctuations) can be averaged out even for relatively small apertures.     

Dynamical models for fragment formation

The various dynamical models for fragment formation in nuclear collisions are discussed in order to bring out their relative advantages and shortcomings. After discussing the general requirements for dynamical models that aim to describe fragment formation, we consider the various mean-field models that incorporate fluctuations and then turn to models based on molecular dynamics.     

Reheating and entropy perturbations in fibre inflation

We study reheating in some one and two field realizations of fibre inflation.We find that reheating begins with a phase of preheating in which long wavelength fluctuation modes are excited.In two field models there is a danger that the parametric amplification of infrared fluctuations in the second scalar field-associated with an entropy mode-might induce an instability of the curvature fluctuations.We show that,at least in the models we consider,the entropy mode has a sufficiently large mass to prevent this instability.Hence,from the point of view of reheating the models we consider are well-behaved.     

Mesoscopic full counting statistics and exclusion models

We calculate the distribution of current fluctuations in two simple exclusion models. Although these models are classical, we recover even for small systems such as a simple or a double barrier, the same distibution of current as given by traditional formalisms for quantum mesoscopic conductors. Due to their simplicity, the full counting statistics in exclusion models can be reduced to the calculation of the largest eigenvalue of a matrix, the size of which is the number of internal configurations of the system. As examples, we derive the shot noise power and higher order statistics of current fluctuations (skewness, full counting statistics, ....) of various conductors, including multiple barriers, diffusive islands between tunnel barriers and diffusive media. A special attention is dedicated to the third cumulant, which experimental measurability has been demonstrated lately.     

Effects of discretization on viscosity in the necklace model

The consequences of discretization on the resulting viscosity for the necklace model are analyzed to find out the mechanisms by which this model leads to results different from what could be anticipated. In order to accomplish this, various models of increasing complexity are proposed. First, the effects of the jump distance and temporal discretizations are studied. Then, the lack of proportionality between the length and the mass of the chains and the fact that chains do not perform a random walk are analyzed. Knowing of these influences on the viscosity, we finally address the effect of length fluctuations. It is shown that fluctuations can have unexpected effects that lead to larger or smaller values of viscosity. The simplicity of the studied models allows us to quantify the effects of discretization and fluctuations, effects that can be present in other models of similar or larger complexity.     

Soliton generation from randomly modulated return-to-zero pulses

We consider return-to-zero (RZ) pulses with random phase modulation propagating in a nonlinear channel (modelled by the integrable nonlinear Schrödinger equation, NLSE). We suggest two different models for the phase fluctuations of the optical field: (i) Gaussian short-correlated fluctuations and (ii) generalized telegraph process. Using the rectangular-shaped pulse form we demonstrate that the presence of phase fluctuations of both types strongly influences the number of solitons generated in the channel. It is also shown that increasing the correlation time for the random phase fluctuations affects the coherent content of a pulse in a non-trivial way. The result obtained has potential consequences for all-optical processing and design of optical decision elements.     

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     

Constraints on operator ordering from third quantization

In this paper, we analyse the Wheeler–DeWitt equation in the third quantized formalism. We will demonstrate that for certain operator ordering, the early stages of the universe are dominated by quantum fluctuations, and the universe becomes classical at later stages during the cosmic expansion. This is physically expected, if the universe is formed from quantum fluctuations in the third quantized formalism. So, we will argue that this physical requirement can be used to constrain the form of the operator ordering chosen. We will explicitly demonstrate this to be the case for two different cosmological models.     

Surface models with nonlocal potentials: Upper bounds

The behavior of fluctuations in a class of surface models with exponentially decaying nonlocal potentials in studied. Combining a Mayer expansion with a duality transformation, we demonstrate the equivalence of these models to a class of two dimensional spin systems with nonlocal interactions. The expansions give sufficient control over the potentials to allow the fluctuations to be bounded from above by the means of complex translations in the spin representation of the model.The author was an NSF Predoctoral Fellow when this research was begun     

Mesoscopic Modeling for Continuous Spin Lattice Systems: Model Problems and Micromagnetics Applications

In this paper we derive deterministic mesoscopic theories for model continuous spin lattice systems both at equilibrium and non-equilibrium in the presence of thermal fluctuations. The full magnetic Hamiltonian that includes singular integral (dipolar) interactions is also considered at equilibrium. The non-equilibrium microscopic models we consider are relaxation-type dynamics arising in kinetic Monte Carlo or Langevin-type simulations of lattice systems. In this context we also employ the derived mesoscopic models to study the relaxation of such algorithms to equilibrium     

Spin fluctuations and unconventional superconducting pairing in iron-based superconductors

In this article, we review the recent theoretical works on the spin fluctuations and superconductivity in iron-based superconductors. Using the fluctuation exchange approximation and multi-orbital tight-binding models, we study the char- acteristics of the spin fluctuations and the symmetries of the superconducting gaps for different iron-based superconductors. We explore the systems with both electron-like and hole-like Fermi surfaces （FS） and the systems with only the electron-like FS. We argue that the spin-fluctuation theories are successful in explaining at least the essential part of the problems, indicating that the spin fluctuation is the common origin of superconductivity in iron-based superconductors.     

The volatility in a multi-share financial market model

Single index financial market models cannot account for the empirically observed complex interactions between shares in a market. We describe a multi-share financial market model and compare characteristics of the volatility, that is the variance of the price fluctuations, with empirical characteristics. In particular we find its probability distribution is similar to a log normal distribution but with a long power-law tail for the large fluctuations, and that the time development shows superdiffusion. Both these results are in good quantitative agreement with observations.     

Fluctuations in coagulating systems. II

In this paper we consider coagulation processes in large but finite systems, and study the time-dependent behavior of the (nonequilibrium) fluctuations in the cluster size distribution. For this purpose we apply van Kampen's-expansion to a master equation describing coagulation processes, and derive an approximate (Fokker-Planck) equation for the probability distribution of the fluctuations. First we consider two exactly soluble models, corresponding to the choicesK(i, j) =i + jandK(i, j)=1 for the rate constants in the Fokker-Planck equation. For these models and monodisperse initial conditions we calculate the probability distribution of the fluctuations and the equal-time and two-time correlation functions. For general initial conditions we study the behavior of the fluctuations at large cluster sizes, and in the scaling limit. Next we consider, in general, homogeneous rate constants, with the propertyK(i, j) =a ^- K(ai,aj)for alla>0, and we give asymptotic expressions for the equal-time correlation functions at large cluster sizes, and in the scaling limit. In the scaling limit we find that the fluctuations show relatively simple scaling behavior for all homogeneous rate constantsK(i, j).     

Tip models and force definitions in molecular dynamics simulations of scanning force microscopy

The definition of the time varying force on a tip with internal degrees of freedom in atomistic molecular dynamics (MD) simulations of scanning force microscopy experiments is discussed. We show that the static expression for the tip force is inadequate for calculating force fluctuations within the MD simulations and suggest a different method of calculating the tip force. By studying the size of tip force fluctuations for different tip models and various tip positions with respect to the surface, we demonstrate that the new method works equally well in both static and dynamic cases.     

Free-Energy Bounds for Hierarchical Spin Models

In this paper we study two non-mean-field (NMF) spin models built on a hierarchical lattice: the hierarchical Edward–Anderson model (HEA) of a spin glass, and Dyson’s hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the existence of the thermodynamic limit of the free energy and the replica-symmetry-breaking (RSB) free-energy bounds previously derived for the Sherrington–Kirkpatrick model of a spin glass. These RSB mean-field bounds are exact only if the order-parameter fluctuations (OPF) vanish: given that such fluctuations are not negligible in NMF models, we develop a novel strategy to tackle part of OPF in hierarchical models. The method is based on absorbing part of OPF of a block of spins into an effective Hamiltonian of the underlying spin blocks. We illustrate this method for DHM and show that, compared to the mean-field bound for the free energy, it provides a tighter NMF bound, with a critical temperature closer to the exact one. To extend this method to the HEA model, a suitable generalization of Griffith’s correlation inequalities for Ising ferromagnets is needed: since correlation inequalities for spin glasses are still an open topic, we leave the extension of this method to hierarchical spin glasses as a future perspective.     

Density and bond-orientational relaxations in supercooled water

ABSTRACT

Recent computational studies have reported evidence of a metastable liquid–liquid phase transition (LLPT) in molecular models of water under deeply supercooled conditions. A competing hypothesis suggests, however, that non-equilibrium artefacts associated with coarsening of the stable crystal phase have been mistaken for an LLPT in these models. Such artefacts are posited to arise due to a separation of time scales in which density fluctuations in the supercooled liquid relax orders of magnitude faster than those associated with bond-orientational order. Here, we use molecular simulation to investigate the relaxation of density and bond-orientational fluctuations in three molecular models of water (ST2, TIP5P and TIP4P/2005) in the vicinity of their reported LLPT. For each model, we find that density is the slowly relaxing variable under such conditions. We also observe similar behaviour in the coarse-grained mW model of water. Our findings, therefore, challenge the key physical assumption underlying the competing hypothesis.     

Operational definition of (brane-induced) space–time and constraints on the fundamental parameters

First we contemplate the operational definition of space–time in four dimensions in light of basic principles of quantum mechanics and general relativity and consider some of its phenomenological consequences. The quantum gravitational fluctuations of the background metric that comes through the operational definition of space–time are controlled by the Planck scale and are therefore strongly suppressed. Then we extend our analysis to the braneworld setup with low fundamental scale of gravity. It is observed that in this case the quantum gravitational fluctuations on the brane may become unacceptably large. The magnification of fluctuations is not linked directly to the low quantum gravity scale but rather to the higher-dimensional modification of Newton's inverse square law at relatively large distances. For models with compact extra dimensions the shape modulus of extra space can be used as a most natural and safe stabilization mechanism against these fluctuations.     

Letter: An Exact Solution for the Fluctuations of an FRW Cosmology with n Scalar Fields, for an Arbitrary Potential

In this work we rigorously study the fluctuations in FRW models coupled with n neutral scalar fields, minimally coupled to the gravitational field. We find the exact solutions and the asymptotic behavior for the fluctuation around the critical point of the background for an arbitrary potential.