Decay Of Correlation Function Research Articles

A remark on the decay of superconducting correlations in one- and two-dimensional hubbard models

Upper bounds on the decay of various correlation functions are derived for a general class of itinerant fermion models with long-range hopping matrix. These bounds extend prevous results of Koma and Tasaki and rule out the possibility of magnetic ordering and condensation of superconducting electron pairs in one and two dimensions for finite temperatures.

The velocity autocorrelation function for a random walk on percolation clusters

Abstract We apply the random Lorentz model to diffusion on percolation clusters on a square lattice. In this model a walker performs a random walk on cluster sites and can visit sites occupied by impurities which play the role of reflectors. The velocity autocorrelation function and the mean-square displacement of a random walker are calculated with high accuracy using a moment-propagation numerical technique. Exponents describing the decay of correlation function are calculated at the percolation threshold and above it.

Phase diagram and correlation function decay of bipolarons in 2D lattices

The phase diagram and local pair mechanism of superconductivity with strong electron-phonon interaction for 2p electrons of oxygen in Cu 2 O planes is investigated in the generalized negative U Hubbard model. The upper bound restrictions for the superconducting correlation function at finite temperature on square 2D lattices is obtained taking into account orbital degeneracy.

Percolation through self-affine surfaces

We study the percolation transition in a long-range correlated system: a self-affine surface. For all relevant physical cases (i.e. positive roughness exponents), it is found that the onset of percolation is governed by the largest wavelength of the height distribution, and thus self-averaging breaks down. Self-averaging is recovered for negative roughness exponents (i.e. power-law decay of the height pair correlation function) and, in this case, the critical exponents that characterize the transition are explicitly dependent on the roughness exponent above a threshold value. Below this threshold, the spatial correlations are no longer relevant. The problem is analytically investigated for a hierarchical network and by means of numerical simulations in two dimensions. Finally, we discuss the application of those properties to mercury porosimetry in cracks.

Evaluation of thermal rates for reactions with intermediate wells: Removal of bound state contributions to quantum flux correlation functions

We consider the calculation of rates of chemical reactions that have bound intermediate states (i.e., wells along the reaction path) using flux correlation function methods. When time-dependent wave packets are used to evaluate the propagator matrix elements, and the dividing surface is located at a point where bound states have nonzero probability density, the standard expression for the flux correlation function shows infinitely long lived oscillations due to these bound states, making the evaluation of rate constants numerically ill-behaved. However, if the bound part of the initial wave packet is projected out, the resulting continuum-only propagators produce rapidly decaying correlation functions, and numerically well-behaved rate constants. We illustrate this projection operator approach by considering a one-dimensional reaction path model in which the potential is taken to be an Eckart well, and the dividing surface is located at the minimum. In another application, we consider a two degree of freedom model of H2 dissociative chemisorption on a rigid metal surface. This application is sufficiently complex that it is impractical to calculate the chemisorption rate using conventional flux correlation function methods, but with the projected wavepacket approach, the problem is made relatively easy. We also consider a second approach to the treatment of bound states in which the flux correlation function is altered to remove implicitly the bound state contributions to the propagator at long times. This second expression can be used with the full propagator, eliminating the need to construct and project explicitly the bound states. This should be advantageous when many bound states are present.

Faceting, roughness, and step disordering of vicinal Si(111) surfaces: An x-ray-scattering study.

The equilibrium and nonequilibrium behavior of the step structure of vicinal Si(111) surfaces has been studied in synchrotron x-ray-scattering experiments. Below the 7×7 reconstruction transition temperature of 1100 K, the surface phase separates into macroscopic (111) facets larger than 0.4×0.1 μm 2 and stepped regions with increased step density. At temperatures above the transition up to 1275 K where there is significant sublimation, the surface is thermally rough, characterized by a uniform step lattice with an algebraically decaying correlation function. The algebraic decay exponent η was measured to be 0.18±0.13, in agreement with the theory of entropically interacting steps

Solvation dynamics in a Brownian dipolar lattice. Comparison between computer simulation and various molecular theories of solvation dynamics

Several recent theoretical and computer simulation studies have considered solvation dynamics in a Brownian dipolar lattice which provides a simple model solvent for which detailed calculations can be carried out. In this article a fully microscopic calculation of the solvation dynamics of an ion in a Brownian dipolar lattice is presented. The calculation is based on the non-Markovian molecular hydrodynamic theory developed recently. The main assumption of the present calculation is that the two-particle orientational correlation functions of the solid can be replaced by those of the liquid state. It is shown that such a calculation provides an excellent agreement with the computer simulation results. More importantly, the present calculations clearly demonstrate that the frequency-dependent dielectric friction plays an important role in the long time decay of the solvation time correlation function. We also find that the present calculation provides somewhat better agreement than either the dynamic mean spherical approximation (DMSA) or the Fried–Mukamel theory which use the simulated frequency-dependent dielectric function. It is found that the dissipative kernels used in the molecular hydrodynamic approach and in the Fried–Mukamel theory are vastly different, especially at short times. However, in spite of this disagreement, the two theories still lead to comparable results in good agreement with computer simulation, which suggests that even a semiquantitatively accurate dissipative kernel may be sufficient to obtain a reliable solvation time correlation function. A new wave vector and frequency-dependent dissipative kernel (or memory function) is proposed which correctly goes over to the appropriate expressions in both the single particle and the collective limits. This form is expected to lead to better results than all the existing descriptions.

Structural disorder in two-dimensional random magnets: Very thin films of rare earths and transition metals.

The low-temperature isothermal magnetization curves, M(H), of ${\mathrm{SmCo}}_{4}$ and ${\mathrm{Fe}}_{3}$Tb thin films are studied according to the two-dimensional correlated spin-glass model of Chudnovsky. We have calculated the magnetization law in approach to saturation and shown that the M(H) data fit well the theory at high and low fields. In our fit procedure we have used three different correlation functions. The Gaussian decay correlation function fits well the experimental data for both samples.

Optical absorption of molecular aggregates under the influence of coloured noise

Optical absorption lineshapes of electronic excitations are calculated for dimers and molecular chains. The interaction with phonons is described by dichotomic and Gaussian stochastic processes with exponentially decaying correlation functions (coloured noise) generalizing in this way the Haken-Strobl-Reineker treatment. A closed system of equations is obtained for dimers with dichotomic fluctuations of the excitation energy and the lineshape can be determined exactly. In the case of Gaussian fluctuations an approximate treatment is used, which, when applied to dichotomic fluctuations, can be compared with the exact solution. The lineshape for electronic excitations is calculated for a linear chain within this approximation scheme for Gaussian fluctuations. It is shown that the lineshapes are asymmetric for slow fluctuations and symmetric in the case of fast fluctuations. Finally, the question of whether the properties of exciton transport can be derived from the absorption lineshape is briefly considered.

Copolymers as amphiphiles in ternary mixtures: An analysis employing disorder, equimaxima, and Lifshitz lines

We study the phase diagram and scattering functions of a ternary mixture of A and B homopolymers, in equal concentrations, and AB diblock copolymer. We locate the disorder line, which marks the appearance of damped oscillatory decaying correlation functions in the system, and also the Lifshitz and equimaxima lines of the structure functions. At these lines, the oscillatory component becomes dominant in a given correlation so that the peak in scattering intensity is no longer at the zero wave vector. We find that while the Lifshitz line of the structure function of all A monomers is quite close to the disorder line, that of A monomers in the homopolymer only is far from it. This shows that the copolymer orders the homopolymers inefficiently, a behavior which is also reflected in the copolymer’s weak ability to solubilize them, and which contrasts with the amphiphilic solubilizers of oil and water. The disorder line and all Lifshitz lines meet at a Lifshitz tricritical point. As these ternary mixtures provide a rare opportunity to study this unusual point, we discuss its effects on the structure functions and surface tensions.

Dynamic slowing down in dense percolating microemulsions.

The dynamic slowing down of the first cumulant and the stretched exponential decay of the droplet density-time correlation function at long time, previously observed for the one-phase sodium-di-2-ethylhexylsulfosuccinate (AOT)-water-decane microemulsion system, are attributed to the approach to a percolation threshold as the volume fraction of the microemulsion is increased at constant temperature. A model for dynamic light scattering, formulated along the line of the scattering from a system of transient polydispersed fractal clusters, quantitatively accounts for all the light-scattering data reported so far and also for some additional results

Anomalous decay of pair correlations for two-dimensional critical wetting

Develops a scaling theory for the decay of the spin-spin correlation function at two-dimensional (d=2) wetting transitions in systems with short-ranged forces. For the critical wetting transition effective Hamiltonian results show that the scaling theory is obeyed but the decay of correlations is anomalous. In contrast, for the complete wetting transition, scaling and the Ornstein-Zernike theory are both valid. The author argues that the anomalous decay is specific to zero bulk field critical wetting transitions ad d=2.

Relevance of level repulsion to quantum correlations for classically chaotic systems

The decay of correlation functions for the kicked rotator model is shown to be characterized by the level repulsion of the initially excited states. The effective repulsion parameter is numerically found to be of the order of 0.05-0.3. The universal character of the level spacing distribution is discussed.

Deterministic Brownian motion.

The goal of this thesis is to contribute to the ambitious program of the foundation of developing statistical physics using chaos. We build a deterministic model of Brownian motion and provide a microscpoic derivation of the Fokker-Planck equation. Since the Brownian motion of a particle is the result of the competing processes of diffusion and dissipation, we create a model where both diffusion and dissipation originate from the same deterministic mechanism - the deterministic interaction of that particle with its environment. We show that standard diffusion which is the basis of the Fokker-Planck equation rests on the Central Limit Theorem, and, consequently, on the possibility of deriving it from a deterministic process with a quickly decaying correlation function. The sensitive dependence on initial conditions, one of the defining properties of chaos insures this rapid decay. We carefully address the problem of deriving dissipation from the interaction of a particle with a fully deterministic nonlinear bath, that we term the booster. We show that the solution of this problem essentially rests on the linear response of a booster to an external perturbation. This raises a long-standing problem concerned with Kubo's Linear Response Theory and the strong criticism against it by van Kampen. Kubo's theory is based on a perturbation treatment of the Liouville equation, which, in turn, is expected to be totally equivalent to a first-order perturbation treatment of single trajectories. Since the boosters are chaotic, and chaos is essential to generate diffusion, the single trajectories are highly unstable and do not respond linearly to weak external perturbation. We adopt chaotic maps as boosters of a Brownian particle, and therefore address the problem of the response of a chaotic booster to an external perturbation. We notice that a fully chaotic map is characterized by an invariant measure which is a continuous function of the control parameters of the map. Consequently if the external perturbation is made to act on a control parameter of the map, we show that the booster distribution undergoes slight modifications as an effect of the weak external perturbation, thereby leading to a linear response of the mean value of the perturbed variable of the booster. This approach to linear response completely bypasses the criticism of van Kampen. The joint use of these two phenomena, diffusion and friction stemming from the interaction of the Brownian particle with the same booster, makes the microscopic derivation of a Fokker-Planck equation and Brownian motion, possible.

Correlation function decay in linear continuum systems

In this paper, written for the volume commemorating the sixtieth birthday of Michael Fisher, I reexamine the long-range decay of the pair correlation function in one-dimensional continuum models where attractive and repulsive potentials are present together. I argue that the correlation function decay goes effectively from monotonic to oscillatory at a pseudo-phase-transition from a gas-like phase of clusters to a liquid-like phase. Though there is no actual discontinuity, this pseudo-phase-transition resembles an “anomalous first order phase transition”.

Exact treatment of the transport of a quantum particle in a dimer under the influence of dichotomic colored noise.

The transport of a quantum particle in a dimer is investigated for the case in which the influence of the phonons is modeled by a dichotomic stochastic process with exponentially decaying correlation functions. An exact numerical solution of the density-matrix equation of the problem is carried out, and the transition from coherent to incoherent transport is discussed.

Statistical properties of dynamically generated anomalous diffusion.

We consider a one-dimensional Lorenz-type mapping with slowly decaying correlation function, which produces anomalous diffusion with squared displacement growing faster then linearly. The statistical properties of the process are investigated by means of diffusion-index calculation. A simple approximation for probability distribution gives a good estimate for these indices.

CORRELATED FERMIONS IN ONE DIMENSION

A brief introduction to the bosonization method for interacting one-dimensional fermion systems is given. Using these results, the long-distance decay of correlation functions in the one-dimensional Hubbard model is determined exactly for arbitrary bandfilling and correlation strength, using the exact solution of Lieb and Wu. For infinite U the results are generalized to the case of nonzero nearest-neighbour interaction. The behaviour of thermodynamic quantities, of the frequency-dependent conductivity, and of the thermopower is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half- and quarter-filling. The one-dimensional Luttinger liquid is shown to be unstable in the presence of interchain hopping. The results for the metal-insulator transition are compared with other scenarios developed in higher dimensions.

THE DISAPPEARANCE OF TWO-DIMENSIONAL LONG RANGE ORDER IN THE HEISENBERG MODEL WITH COMPETING INTERACTIONS

In the frustrated Heisenberg model with first (J1) and second (J2) nearest neighbours interactions on a square lattice the transition from the long range order state (LROS) to spin liquid state (SLS) is found at α = J1/J2 ≅ 0.25. SLS is characterized by the gap in spin excitation spectrum at T = 0 and, hence, by exponential decay of spin correlation function at large distance. As a result, correlation length is temperature independent in SLS in accordance with neutron experiments on doped La 2 CuO 4.

A many-body stochastic approach to rotational motions in liquids: complex decay times in highly viscous fluids

A two-body Smoluchowski equation, including a solute molecule and a collective solvent mode, is developed for studying reorientational dynamics in complex fluids. Multiexponential decay correlation functions for first- and second-rank observables are computed. They exhibit bifurcation and other properties related to typical observations on glassy and supercooled liquids.