Two-loop Critical Fluctuation-Dissipation Ratio for the Relaxational Dynamics of the O(N) Landau-Ginzburg Hamiltonian

The off-equilibrium purely dissipative dynamics (Model A) of the O(N) vector model is considered at criticality in an $\epsilon = 4- d > 0$ up to O($\epsilon^2$). The scaling behavior of two-time response and correlation functions at zero momentum, the associated universal scaling functions, and the nontrivial limit of the fluctuation-dissipation ratio are determined in the aging regime.Comment: 21 pages, 6 figures. Discussion enlarged and two figures added. Final version accepted for publication in Phys. Rev.

Ageing Properties of Critical Systems

In the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest. Ageing phenomena are observed during this ever-lasting non-equilibrium evolution. A simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high-temperature phase to the critical point. The aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case. Particular emphasis is put to the field-theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities. Fluctuation-dissipation relations are discussed and in particular the concept of fluctuation-dissipation ratio (FDR) is reviewed, emphasizing its connection with the definition of a possible effective temperature. The Renormalization-Group approach to critical dynamics is summarized and the scaling forms of the time-dependent non-equilibrium correlation and response functions of a generic observable are discussed. From them the universality of the associated FDR follows as an amplitude ratio. It is then possible to provide predictions for ageing quantities in a variety of different models. In particular the results for Model A, B, and C dynamics of the O(N) Ginzburg-Landau Hamiltonian, and Model A dynamics of the weakly dilute Ising magnet and of a \phi^3 theory, are reviewed and compared with the available numerical results and exact solutions. The effect of a planar surface on the ageing behaviour of Model A dynamics is also addressed within the mean-field approximation.Comment: rvised enlarged version, 72 Pages, Topical Review accepted for publication on JP

Aging and fluctuation-dissipation ratio for the diluted Ising Model

We consider the out-of-equilibrium, purely relaxational dynamics of a weakly diluted Ising model in the aging regime at criticality. We derive at first order in a $\sqrt{\epsilon}$ expansion the two-time response and correlation functions for vanishing momenta. The long-time limit of the critical fluctuation-dissipation ratio is computed at the same order in perturbation theory.Comment: 4 pages, 2 figure

Ictal epileptic headache. an old story with courses and appeals

The term "ictal epileptic headache" has been recently proposed to classify the clinical picture in which headache is the isolated ictal symptom of a seizure. There is emerging evidence from both basic and clinical neurosciences that cortical spreading depression and an epileptic focus may facilitate each other, although with a different degree of efficiency. This review address the long history which lead to the 'migralepsy' concept to the new emerging pathophysiological aspects, and clinical and electroencephalography evidences of ictal epileptic headache. Here, we review and discuss the common physiopathology mechanisms and the historical aspects underlying the link between headache and epilepsy. Either experimental or clinical measures are required to better understand this latter relationship: the development of animal models, molecular studies defining more precise genotype/phenotype correlations as well as multicenter clinical studies with revision of clinical criteria for headache-/epilepsy-related disorders represent the start of future research. Therefore, the definition of ictal epileptic headache should be used to classify the rare events in which headache is the only manifestation of a seizure. Finally, using our recently published criteria, we will be able to clarify if ictal epileptic headache represents an underestimated phenomenon or not

The critical behavior of magnetic systems described by Landau-Ginzburg-Wilson field theories

We discuss the critical behavior of several three-dimensional magnetic systems, such as pure and randomly dilute (anti)ferromagnets and stacked triangular antiferromagnets. We also discuss the nature of the multicritical points that arise in the presence of two distinct O(n)-symmetric order parameters and, in particular, the nature of the multicritical point in the phase diagram of high-T_c superconductors that has been predicted by the SO(5) theory. For each system, we consider the corresponding Landau-Ginzburg-Wilson field theory and review the field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the epsilon and of the fixed-dimension d=3 expansions.Comment: 29 pages, 6 fig

Multicritical behavior in frustrated spin systems with noncollinear order

We investigate the phase diagram and, in particular, the nature of the the multicritical point in three-dimensional frustrated $N$-component spin models with noncollinear order in the presence of an external field, for instance easy-axis stacked triangular antiferromagnets in the presence of a magnetic field along the easy axis. For this purpose we study the renormalization-group flow in a Landau-Ginzburg-Wilson \phi^4 theory with symmetry O(2)x[Z_2 +O(N-1)] that is expected to describe the multicritical behavior. We compute its MS \beta functions to five loops. For N\ge 4, their analysis does not support the hypothesis of an effective enlargement of the symmetry at the multicritical point, from O(2) x [Z_2+O(N-1)] to O(2)xO(N). For the physically interesting case N=3, the analysis does not allow us to exclude the corresponding symmetry enlargement controlled by the O(2)xO(3) fixed point. Moreover, it does not provide evidence for any other stable fixed point. Thus, on the basis of our field-theoretical results, the transition at the multicritical point is expected to be either continuous and controlled by the O(2)xO(3) fixed point or to be of first order.Comment: 28 pages, 3 fig

Spin models with random anisotropy and reflection symmetry

We study the critical behavior of a general class of cubic-symmetric spin systems in which disorder preserves the reflection symmetry $s_a\to -s_a$, $s_b\to s_b$ for $b\not= a$. This includes spin models in the presence of random cubic-symmetric anisotropy with probability distribution vanishing outside the lattice axes. Using nonperturbative arguments we show the existence of a stable fixed point corresponding to the random-exchange Ising universality class. The field-theoretical renormalization-group flow is investigated in the framework of a fixed-dimension expansion in powers of appropriate quartic couplings, computing the corresponding $\beta$-functions to five loops. This analysis shows that the random Ising fixed point is the only stable fixed point that is accessible from the relevant parameter region. Therefore, if the system undergoes a continuous transition, it belongs to the random-exchange Ising universality class. The approach to the asymptotic critical behavior is controlled by scaling corrections with exponent $\Delta = - \alpha_r$, where $\alpha_r\simeq -0.05$ is the specific-heat exponent of the random-exchange Ising model.Comment: 28 pages, 1 fi

Crossover from random-exchange to random-field critical behavior in Ising models

We compute the crossover exponent $\phi$ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as Fe$_x$Zn$_{1-x}$F$_2$ and Mn$_x$Zn$_{1-x}$F$_2$, when applying an external magnetic field. Our result $\phi=1.42(2)$ for the crossover exponent is in good agreement with the available experimental estimates.Comment: 5 page

Gini estimation under infinite variance

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index $\alpha\in(1,2)$). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. We start by discussing how the nonparametric estimator of the Gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution, as the data distribution shifts from the domain of attraction of a light-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of $\alpha$. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution