Showing papers by "Luciano Pietronero published in 1998"

Nonperturbative Renormalization of the Kardar-Parisi-Zhang Growth Dynamics

Abstract: We introduce a non-perturbative renormalization approach which identifies stable fixed points in any dimension for the Kardar-Parisi-Zhang dynamics of rough surfaces. The usual limitations of real space methods to deal with anisotropic (self-affine) scaling are overcome with an indirect functional renormalization. The roughness exponent $\alpha$ is computed for dimensions $d=1$ to 8 and it results to be in very good agreement with the available simulations. No evidence is found for an upper critical dimension. We discuss how the present approach can be extended to other self-affine problems.

Isotope effect on m* in high-Tc materials due to the breakdown of Migdal's theorem

Abstract: We show that the inclusion of effects beyond Migdal's limit in the electron-phonon interaction naturally leads to an isotope effect for the effective mass m* of the charge carriers even much before reaching the polaron limit. This is the situation already considered in our approach to nonadiabatic superconductivity (Phys. Rev. Lett., 75 (1995) 1158). Such a result provides a scenario different from the polaronic one for the interpretation of the recently observed isotope effect on m* in YBa2Cu3O6 + x and La2 − xSrxCuO4 (Zhao et al., Phys. Rev. B, 51 (1995) 16487 and Nature, 385 (1997) 236).

On the fractal structure of galaxy distribution and its implications for cosmology.

Abstract: Two fundamental empirical laws have been established in the analysis of galaxy space distribution. First, recent analyses have revealed that the three-dimensional distribution of galaxies and clusters is characterized by large-scale structures and huge voids: such a distribution shows fractal correlations up to the limits of the available samples. This has confirmed the earlier de Vaucouleurs power-law density — distance relation, now corresponding to a fractal structure with dimension D ≈ 2, at least, in the range of scales ~1 ÷ 200 Mpc (H0=55 km/sec/Mpc). An eventual cut-off towards homogenization has not been yet identified. Second, since Huble's discovery, the linear redshift-distance law has been well established within 200 Mpc and also much deeper. The co-existence of these laws within the same scales is a challenge for the standard cosmology, where the linear Hubble law is a strict consequence of homogeneity of the expanding universe. This puzzle is now sufficiently strong to raise doubts for the standard cosmology.

High dimensional behavior of the kardar-parisi-zhang growth dynamics

Abstract: We investigate analytically the large dimensional behavior of the Kardar-Parisi-Zhang (KPZ) dynamics of surface growth using a recently proposed nonperturbative renormalization for self-affine surface dynamics. Within this framework, we show that the roughness exponent \ensuremath{\alpha} decays not faster than $\ensuremath{\alpha}\ensuremath{\sim}1/d$ for large $d.$ This implies the absence of a finite upper critical dimension.

Nonadiabatic pairing effects for tight-binding electrons interacting with phonons

Abstract: The nonadiabatic electron-phonon corrections for the superconducting pairing are investigated for a specific tight-binding model corresponding to a 2d square lattice. This permits to investigate the role of various specific properties like the band filling, nesting effects and a realistic van Hove singularity on the superconducting effective pairing beyond Migdal's limit. The main results are: (i) Starting from a momentum independent electron-phonon coupling the nonadiabatic effects lead to an effective pairing which is strongly dependent on frequency and momentum. (ii) If instead the electron-phonon coupling is mainly forward (as due to correlation effects) the resulting pairing results to be strongly enhanced. These results confirm but also extend the simplified scheme used up to now to compute these properties. In this respect our results link the nonadiabatic effects to the specific properties of realistic materials.

Hierarchical model of slow constrained dynamics

Abstract: We introduce a simple hierarchically constrained model of slow relaxation. The configurational energy has a simple form as there is no coupling among the spins defining the system; the associated stationary distribution is an equilibrium, Gibbsian one. However, due to the presence of hierarchical constraints in the dynamics the system is found to relax to its equilibrium distribution in an extremely slow fashion when suddenly cooled from an initial temperature ${T}_{0}$ to a final one ${T}_{f}.$ The relaxation curve in that case can be fit by a stretched-exponential curve. On the other hand, the relaxation function is found to be exponential when ${T}_{f}g{T}_{0}$, with characteristic times depending on both ${T}_{f}$ and ${T}_{0}$, with characteristic times obeying an Arrhenius law. Numerical results as well as some analytical studies are presented. In particular, we introduce a simple equation that captures the essence of the slow relaxation.

The physical origin of the electron-phonon vertex correction

Abstract: The electron-phonon vertex correction has a complex structure both in momentum and frequency. We explain this structure on the basis of physical considerations and we show how the vertex correction can be decomposed into two terms with different physical origins. In particular, the first term describes the lattice polarization induced by the electrons and it is essentially a single-electron process whereas the second term is governed by the particle-hole excitations due to the exchange part of the phonon-mediated electron-electron interaction. We show that by weakening the influence of the exchange interaction the vertex takes mostly positive values giving rise to an enhanced effective coupling in the scattering with phonons. This weakening of the exchange interaction can be obtained by lowering the density of the electrons, or by considering only long-ranged (small q) electron-phonon couplings. These findings permit to understand why in the High-Tc materials the small carrier density and the long ranged electron-phonon interaction may play a positive role in enhancing Tc.

The Uneven Distribution of Numbers in Nature

Abstract: Suppose you look at today's stock prices and bet on the value of the first digit One could guess that a fair bet should correspond to the frequency of $1/9 = 1111%$ for each digit from 1 to 9 This is by no means the case, and one can easily observe a strong prevalence of the small values over the large ones The first three integers 1,2 and 3 alone have globally a frequency of 60% while the other six values 4, 5, 6, 7, 8 and 9 appear only in 40% of the cases This situation is actually much more general than the stock market and it occurs in a variety of number catalogs related to natural phenomena The first observation of this property traces back to S Newcomb in 1881 but a more precise account was given by F Benford in 1938 In this note we illustrate these observations with the enlightening specific example of the stock market We also identify the general mechanism for the origin of this uneven distribution in the multiplicative nature of fluctuations in economics and in many natural phenomena This provides a natural explanation for the ubiquitous presence of the Benford's law in many different phenomena with the common element that their fluctuations refer to a fraction of their values This brings us close to the problem of the spontaneous origin of scale invariant properties in various phenomena which is a debated question at the frontier of different fields

Comment on the paper by L. Guzzo "Is the Universe homogeneous ?"

Abstract: This comment is in response to the paper by L. Guzzo recently appeared in “New Astronomy” related to our work. The subject of the discussion concerns the correlation properties of galaxy distribution in the available 3-d samples. There is a general agreement that galaxy structures exhibit fractal properties, at least up to some scale. However the presence of an eventual crossover towards homogenization, as well as the exact value of the fractal dimension, are still matter of debate. Here we briefly summarize our point of view by discussing three main topics. The first one is methodological, i.e. we clarify which are the correct methods to detect the real correlations properties of the 3-d galaxy distribution. Then we discuss the results of the analysis of several samples in two ranges of scales. In the first range of scale, below 100 ÷200h 1 Mpc, the statistical quality of the data is rather good, and we find that galaxy distribution has fractal properties with D ≈ 2. At larger distances the statistical robustness of the present data is weaker, but, we show that there is evidence for a continuation of the fractal behavior, without any tendency towards homogenization.

Comment on the paper ``The ESO Slice Project galaxy redshift survey: V. Evidence for a D=3 sample dimensionality''

Abstract: In a recent analysis of number counts in the ESP survey Scaramella et al. (1998) claim to find evidence for a cross-over to homogeneity at large scales, and against a fractal behaviour with dimension $D \approx 2$. In this comment we note firstly that, if such a cross-over exists as described by the authors, the scale characterizing it is ~ 100 - 300 Mpc/h. This invalidates the ``standard'' analysis of the same catalogue given elsewhere by the authors which results in a ``correlation length'' of only r_0 = 4 Mpc/h. Furthermore we show that the evidences for a cross-over to homogeneity rely on the choice of cosmological model, and most crucially on the so called K corrections. We show that the D ~ 3 behaviour seen in the K-corrected data of Scaramella et al. is in fact unstable, increasing systematically towards D=4 as a function of the absolute magnitude limit. This behaviour can be quantitatively explained as the effect of an unphysical K-correction in the relevant range of red-shift (z ~ 0.1- 0.3). A more consistent interpretation of the number counts is that D is in the range 2 - 2.5, depending on the cosmological model, consistent with the continuation of the fractal D ~ 2 behaviour observed at scales up to ~100 Mpc/h. This implies a smaller K-correction. Given, however, the uncertainty in the effect of intrinsic fluctuations on the number counts statistic, and its sensitivity on these large scales to the uncertain K corrections, we conclude that it is premature to put a definitive constraint on the galaxy distribution using the ESP data alone.

Role of the superconducting gap opening on vertex corrections

Abstract: The nonadiabatic effects due to the breakdown of Migdal's theorem in high-$T_c$ superconductors are strongly affected by the opening of the superconducting gap. Here we report how the momentum-frequency dependence of vertex corrections is modified by the gap. A general effect is that the positive region of the vertex corrections is increased leading to an enhancement of their relevance. This has a number of physical consequences that we discuss in details in relation to specific experiments.

On the fractal structure of galaxy distribution and its implications for cosmology

Abstract: Two fundamental empirical laws have been established in the analysis of galaxy space distribution. First, recent analyses have revealed that the three dimensional distribution of galaxies and clusters is characterized by large scale structures and huge voids: such a distribution shows fractal correlations up to the limits of the available samples. This has confirmed the earlier de Vaucouleurs power-law density - distance relation, now corresponding to a fractal structure with dimension $D \approx 2$, at least, in the range of scales $ \sim 1 \div 200 Mpc$ ($H_0 = 55 km/sec/Mpc$). An eventual cut-off towards homogenization has not been yet identified. Second, since Hubble's discovery, the linear redshift-distance law has been well established within $200 Mpc$ and also much deeper. The co-existence of these laws within the same scales is a challenge for the standard cosmology where the linear Hubble law is a strict consequence of homogeneity of the expanding universe. This puzzle is now sufficiently strong to raise doubts for the standard cosmology.

Scale invariance in galaxy structures: general concepts and application to the most recent data a

Abstract: The debate on the correlation properties of galaxy structures has having an increas-ing interest during the last year In this lecture we discuss the claims of differentauthors who have criticized our approach and results In order to have a clearcut of the situation, we focus mainly on galaxy distribution in the intermediaterange of distances ∼ 100 ÷ 200h

Theory of Extremal Dynamics with Quenched Disorder: Self-Organization, Avalanche Dynamics and Critical Exponents

Abstract: In this paper we discuss a general theoretical scheme, that we have recently proposed, for a class of phenomena characterized by extremal dynamics with quenched disorder. The approach is based on a transformation of the quenched dynamics into a stochastic dynamics with cognitive memory. This transformation, together with other concepts, permits a mathematical characterization of the self-organized nature of the avalanche type dynamics. By combining the mapping with real space methods, like the fixed scale transformation (FST), it is also possible to compute the relevant critical exponents directly from the microscopic model. A specific application to Invasion Percolation is presented but the approach can be easily extended to various other problems with quenched disorder.