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     

(TTT)2I3+δ compounds--superconducting fluctuations?

We suggest that the high conductivity observed in non-stoichiometric (TTT)₂I_3+δ samples is a consequence of the suppression of 3D corrections by the incommensurate iodine chains which produce a random potential on the conducting TTT chains. As a result, the TTT chains can act as an array of independent 1D conductors for which it is known that superconducting fluctuations, suppressed in the 3D case, are strong and lead to high conductivity.     

Under what kind of parametric fluctuations is spatiotemporal regularity the most robust?

It was observed that the spatiotemporal chaos in lattices of coupled chaotic maps was suppressed to a spatiotemporal fixed point when some fractions of the regular coupling connections were replaced by random links. Here we investigate the effects of different kinds of parametric fluctuations on the robustness of this spatiotemporal fixed point regime. In particular we study the spatiotemporal dynamics of the network with noisy interaction parameters, namely fluctuating fraction of random links and fluctuating coupling strengths. We consider three types of fluctuations: (i) noisy in time, but homogeneous in space; (ii) noisy in space, but fixed in time; (iii) noisy in both space and time. We find that the effect of different kinds of parametric noise on the dynamics is quite distinct: quenched spatial fluctuations are the most detrimental to spatiotemporal regularity; spatiotemporal fluctuations yield phenomena similar to that observed when parameters are held constant at the mean value, and interestingly, spatiotemporal regularity is most robust under spatially uniform temporal fluctuations, which in fact yields a larger fixed point range than that obtained under constant mean-value parameters.     

Effect of the Berendsen thermostat on the dynamical properties of water

The effect of the Berendsen thermostat on the dynamical properties of bulk SPC/E water is tested by generating power spectra associated with fluctuations in various observables. The Berendsen thermostat is found to be very effective in preserving temporal correlations in fluctuations of tagged particle quantities over a very wide range of frequencies. Even correlations in fluctuations of global properties, such as the total potential energy, are well preserved for time periods shorter than the thermostat time constant. Deviations in dynamical behaviour from the microcanonical limit do not, however, always decrease smoothly with increasing values of the thermostat time constant, but may be somewhat larger for some intermediate values of τ_B, especially in the supercooled regime, which are similar to time scales for slow relaxation processes in bulk water.     

Acoustic waves in random fields

The effect of space- and time-dependent random mass density, velocity, and pressure fields on frequencies and amplitudes of acoustic waves is considered by means of the analytical perturbative method. The analytical results, which are valid for weak fluctuations and long wavelength sound waves, reveal frequency and amplitude alteration, the effect of which depends on the type of random field. In particular, the effect of a random mass density field is to increase wave frequencies. Space-dependent random velocity and pressure fields reduce wave frequencies. While space-dependent random fields attenuate wave amplitudes, their time-dependent counterparts lead to wave amplification. In another example, sound waves that are trapped in the vertical direction but are free to propagate horizontally are affected by a space-dependent random mass density field. This effect depends on the direction along which the field is varying. A random field, which varies along the horizontal direction, does not couple vertically standing modes but increases their frequencies and attenuates amplitudes. These modes are coupled by a random field which depends on the vertical coordinate, but the dispersion relation remains the same as in the case of the deterministic medium.     

Analytical Study of Giant Fluctuations and Temporal Intermittency

We study analytically giant fluctuations and temporal intermittency in a stochastic one-dimensional model with diffusion and aggregation of masses in the bulk, along with influx of single particles and outflux of aggregates at the boundaries. We calculate various static and dynamical properties of the total mass in the system for both biased and unbiased movement of particles and different boundary conditions. These calculations show that (i) in the unbiased case, the total mass has a non-Gaussian distribution and shows giant fluctuations which scale as system size (ii) in all the cases, the system shows strong intermittency in time, which is manifested in the anomalous scaling of the dynamical structure functions of the total mass. The results are derived by taking a continuum limit in space and agree well with numerical simulations performed on the discrete lattice. The analytic results obtained here are typical of the full phase of a more general model with fragmentation, which was studied earlier using numerical simulations.     

Frequency dependent conductivity for asymmetric hopping in one dimension

The frequency dependent conductivity for one dimensional disordered classical systems is considered in the presence of an external static electric field which acts as a bias for the hopping rates. In the high frequency limit one obtains an expansion in inverse frequencies for the conductivity, whereas for low frequencies an expansion in the inverse moments of the distribution of the random transition rates is derived. The lowest order contributions of these expansions are explicitely evaluated and compared with recent calculations by Derrida and Orbach. The method is restricted to systems with nonsingular distributions of the transfer rates.     

Random sound waves in a weakly stratified atmosphere

The influence of random mass density and velocity fields on the frequencies and amplitudes of the sound waves that propagate along a constant gravity field is examined in the limit of weak random fields, small amplitude oscillations and a weakly stratified medium. Using a perturbative method, we derive dispersion relations from which we conclude that the effect of a space-dependent random mass density field is to attenuate sound waves. Frequencies of these waves are higher than in the case of a coherent medium. A time-dependent random mass density field increases frequencies and amplifies the sounds waves. On the other hand, a space-dependent random flow reduces the wave frequencies and attenuates the sound waves. The time-dependent random flow raises the frequencies of the sound waves and amplifies their amplitudes. In the limit of the gravity-free medium the above results are in an agreement with the former findings.     

A Simple Mean Field Model for Social Interactions: Dynamics,Fluctuations, Criticality

We study the dynamics of a spin-flip model with a mean field interaction. The system is non reversible, spacially inhomogeneous, and it is designed to model social interactions. We obtain the limiting behavior of the empirical averages in the limit of infinitely many interacting individuals, and show that phase transition occurs. Then, after having obtained the dynamics of normal fluctuations around this limit, we analyze long time fluctuations for critical values of the parameters. We show that random inhomogeneities produce critical fluctuations at a shorter time scale compared to the homogeneous system.     

Mass hierarchy, mass gap and corrections to Newton’s law on thick branes with Poincaré symmetry

We consider a scalar thick brane configuration arising in a 5D theory of gravity coupled to a self-interacting scalar field in a Riemannian manifold. We start from known classical solutions of the corresponding field equations and elaborate on the physics of the transverse traceless modes of linear fluctuations of the classical background, which obey a Schrödinger-like equation. We further consider two special cases in which this equation can be solved analytically for any massive mode with $m^2\ge 0$ , in contrast with numerical approaches, allowing us to study in closed form the massive spectrum of Kaluza–Klein (KK) excitations and to analytically compute the corrections to Newton’s law in the thin brane limit. In the first case we consider a novel solution with a mass gap in the spectrum of KK fluctuations with two bound states—the massless 4D graviton free of tachyonic instabilities and a massive KK excitation—as well as a tower of continuous massive KK modes which obey a Legendre equation. The mass gap is defined by the inverse of the brane thickness, allowing us to get rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It is shown that due to this lucky circumstance, the solution of the mass hierarchy problem is much simpler and transparent than in the thin Randall–Sundrum (RS) two-brane configuration. In the second case we present a smooth version of the RS model with a single massless bound state, which accounts for the 4D graviton, and a sector of continuous fluctuation modes with no mass gap, which obey a confluent Heun equation in the Ince limit. (The latter seems to have physical applications for the first time within braneworld models). For this solution the mass hierarchy problem is solved with positive branes as in the Lykken–Randall (LR) model and the model is completely free of naked singularities. We also show that the scalar–tensor system is stable under scalar perturbations with no scalar modes localized on the braneworld configuration.     

Exactly solvable toy model for the pseudogap state

We present an exactly solvable toy model which describes the emergence of a pseudogap in an electronic system due to a fluctuating off-diagonal order parameter. In one dimension our model reduces to the fluctuating gap model (FGM) with a gap that is constrained to be of the form , where A and Q are random variables. The FGM was introduced by Lee, Rice and Anderson [Phys. Rev. Lett. 31, 462 (1973)] to study fluctuation effects in Peierls chains. We show that their perturbative results for the average density of states are exact for our toy model if we assume a Lorentzian probability distribution for Q and ignore amplitude fluctuations. More generally, choosing the probability distributions of A and Q such that the average of vanishes and its covariance is , we study the combined effect of phase and amplitude fluctuations on the low-energy properties of Peierls chains. We explicitly calculate the average density of states, the localization length, the average single-particle Green's function, and the real part of the average conductivity. In our model phase fluctuations generate delocalized states at the Fermi energy, which give rise to a finite Drude peak in the conductivity. We also find that the interplay between phase and amplitude fluctuations leads to a weak logarithmic singularity in the single-particle spectral function at the bare quasi-particle energies. In higher dimensions our model might be relevant to describe the pseudogap state in the underdoped cuprate superconductors. Received 15 March 2000     

A statistical test of Anderson localization

Numerical demonstrations of localization in random systems are difficult to obtain and interpret because of statistical fluctuations in the electron probability density. This difficulty can be avoided through the use of correlation functions defined in terms of the electron probability density. The fluctuations can then be eliminated by averaging over a large number of Anderson Hamiltonians. The resulting averaged correlation functions clearly show that electrons are exponentially localized. The localization demonstrated here is sufficient to insure a zero dc conductivity in the limit of large systems.     

Space-time clustering and correlations of major earthquakes

Earthquake occurrence in nature is thought to result from correlated elastic stresses, leading to clustering in space and time. We show that the occurrence of major earthquakes in California correlates with time intervals when fluctuations in small earthquakes are suppressed relative to the long term average. We estimate a probability of less than 1% that this coincidence is due to random clustering.     

The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension

We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n ^1/4, where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form invariant distributions, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n ^1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.     

Fluctuations in the Weakly Asymmetric Exclusion Process with Open Boundary Conditions

We investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state. We show that these fluctuations are given, in the macroscopic limit, by a centered Gaussian field and we compute explicitly its covariance function. We use two approaches. The first method is dynamical and based on fluctuations around the hydrodynamic limit. We prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation, and we search for the stationary distribution of this evolution. The second approach, which is based on a representation of the steady state as a sum over paths, allows one to write the density fluctuations in the steady state as a sum over two independent processes, one of which is the derivative of a Brownian motion, the other one being related to a random path in a potential.     

Backscattering of Dirac fermions in HgTe quantum wells with a finite gap

The density-dependent mobility of n-type HgTe quantum wells with inverted band ordering has been studied both experimentally and theoretically. While semiconductor heterostructures with a parabolic dispersion exhibit an increase in mobility with carrier density, high-quality HgTe quantum wells exhibit a distinct mobility maximum. We show that this mobility anomaly is due to backscattering of Dirac fermions from random fluctuations of the band gap (Dirac mass). Our findings open new avenues for the study of Dirac fermion transport with finite and random mass, which so far has been hard to access.     

Response functions of hot and dense matter in the Nambu-Jona-Lasino model

We investigate current-current correlation functions, or the so-called response functions of a two-flavor Nambu-Jona-Lasino model at finite temperature and density. The linear response is investigated introducing the conjugated gauge fields as external sources within the functional path integral approach. The response functions can be obtained by expanding the generational functional in powers of the external sources. We derive the response functions parallel to two well-established approximations for equilibrium thermodynamics, namely mean-field theory and a beyond-mean-field theory, taking into account mesonic contributions. Response functions based on the mean-field theory recover the so-called quasiparticle random phase approximation. We calculate the dynamical structure factors for the density responses in various channels within the random phase approximation, showing that the dynamical structure factors in the baryon axial vector and isospin axial vector channels can be used to reveal the quark mass gap and the Mott dissociation of mesons, respectively. Noting that the mesonic contributions are not taken into account in the random phase approximation, we also derive the response functions parallel to the beyond-mean-field theory. We show that the mesonic fluctuations naturally give rise to three kinds of famous diagrammatic contributions: the Aslamazov-Lakin contribution, the self-energy or density-of-state contribution, and the Maki-Thompson contribution.Unlike the equilibrium case, in evaluating the fluctuation contributions, we need to carefully treat the linear terms in external sources and the induced perturbations. In the chiral symmetry breaking phase, we find an additional chiral order parameter induced contribution, which ensures that the temporal component of the response functions in the static and long-wavelength limit recovers the correct charge susceptibility defined using the equilibrium thermodynamic quantities. These contributions from mesonic fluctuations are expected to have significant effects on the transport properties of hot and dense matter around the chiral phase transition or crossover, where the mesonic degrees of freedom are still important.     

On the low-frequency limit of the Schönfeld pulse 1/f law

On the basis of propositions of the common fluctuation theory, peculiarities of small fluctuations in real physical systems with limited sizes are analyzed. It is established that small fluctuations should necessarily be divided into two types of fluctuations: “small” and “very small”. It is shown that the damping process of “small” fluctuations has relaxation character, while the damping process of “very small” fluctuations is of random character, i.e., it represents a random rectangular signal. The probability density of “very small” fluctuations is shown to be Gaussian. The agreement of the obtained results with experimental data acquired from semiconductor-based devices is analyzed. A relation between the generation–recombination noise and phonon number fluctuations in semiconductors is studied. On the basis of this consideration it is shown that the Schönfeld pulse spectrum preserves its well-known 1/f form only in the range of intermediate frequencies; at lower frequencies the spectrum gets saturated. An expression for the low-frequency limit of Schönfeld pulse 1/f law is obtained.     

Perturbation theory for large Stokes number particles in random velocity fields

We derive a perturbative approach to study, in the large inertia limit, the dynamics of solid particles in a smooth, incompressible and finite-time correlated random velocity field. We carry on an expansion in powers of the inverse square root of the Stokes number, defined as the ratio of the relaxation time for the particle velocities and the correlation time of the velocity field. We describe in this limit the residual concentration fluctuations of the particle suspension, and determine the contribution to the collision velocity statistics produced by clustering. For both concentration fluctuations and collision velocities, we analyze the differences with the compressible one-dimensional case.     

Localization of particles in a two-dimensional random potential

A recently proposed theory of the density response of particles moving in a random potential is applied to two-dimensional systems. The particles are found to be localized for arbitrary small disorder. By decreasing the potential fluctuations we find an abrupt transition from an insulating state to a quasi-conducting state exhibiting exceedingly small values for the inverse susceptibility, the inverse localization length and the excitation gap of the dynamical conductivity.