SK Model Research Articles

Chronoamperometry Study of the Electrodeposited Ni<sub>100-X</sub>P<sub>x</sub> Alloy Thin Films

Elecrodeposited Ni-P alloy thin films have been the subject of extended investigations, since the pioneering works of Brenner et al, in the late 1940s. It is well known, that the physical and chemical properties of the Ni-P deposit composition are strongly influenced by the preparation conditions. In our experimental procedure, we have used a Parstat 2253 potensiostat equipped with Power-Suite software. All the electrochemical experiments were performed in a three electrode cell in which the volume of the bath was 150ml, in this experimental technique, we can measure one or more of three parameters: the potential (V), current (i), and time (t). The aim of our work consists to study the nucleation and growth process and given morphology and composition of electrodeposited Nix-P100-x thin films on Copper substrates. For this purpose, cyclic voltametry and chnonoamperomaty have been used in order to determine the previous cited properties of thin films Ni-P, [0.10 of NaPH2O2 solutions. The chrnoamperogramms can be interpreted by the use of one of three models called: Growth mode (Me layer by layer formation); Franck Van der Merwe, FM model, Growth mode 3D (Me island formation on the top of predeposited), 2D Meads overlayers on substrat and Stranski-Krastanov; (SK model).

Introduction to the SK model

Introduction to the SK model

Relaxational Modes and Aging in the Glauber Dynamics of the Sherrington-Kirkpatrick Model

The relaxational modes and aging of the Glauber dynamics of the mean-field spin-glass model, SK model, are studied by a numerical diagonalization technique and Monte Carlo simulations. We found that the aging process of the model is understood as hierarchical growth of quasi-equilibrium domain in the phase-space.

Convergence of clock processes in random environments and ageing in the $p$-spin SK model

We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [(2010), (2011), forthcoming], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [Ann. Probab. 6 (1978) 829-846]. We demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the p-spin SK model. We prove that on a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent results of Ben Arous, Bovier and Cerny [Comm. Math. Phys. 282 (2008) 663-695] that obtained similar convergence results in law, with respect to the random environment.

Aplikasi Sistem Informasi Geografis untuk Zonasi Kawasan Hutan Berdasarkan Kementrian Kehutanan dan Kemampuan Lahan di Kabupaten Bandung

The main objectives in the geographic information system applications for zoning of forest areas based on the ministries of forestry and land capability is to determine the broad allocation of forest land in Bandung Regency based on Ministerial Decree 79/2001 on the basis of analysis with analytical capabilities of BPDAS land so as to provide a common perception in spatial planning and resource management natural resources in Bandung Regency. Research is done by quantitative method using Geographic Information System. Area difference of forest zone based on Ministerial Decree 79/2001 to BPDAS land capability is 69,709.009 ha or 26.042 kms and 26.042 kms. The result of research providing recommendations to decision makers in Bandung Regency associated with the analytical model and the Ministerial Decree 79/2001 SK model BPDAS analysis of land capability so that spatial planning and management of natural resources can be more synergistic to achieve sustainable development that takes into account the principles sustainable principles, optimal, harmonious and balanced.

Bethe–Peierls approximation and the inverse Ising problem

We apply the Bethe–Peierls approximation to the inverse Ising problem and show how thelinear response relation leads to a simple method for reconstructing couplings and fields ofthe Ising model. This reconstruction is exact on tree graphs, yet its computational expenseis comparable to those of other mean-field methods. We compare the performance of thismethod to the independent-pair, naive mean-field, and Thouless–Anderson–Palmerapproximations, the Sessak–Monasson expansion, and susceptibility propagation on theCayley tree, SK model and random graph with fixed connectivity. At low temperatures,Bethe reconstruction outperforms all of these methods, while at high temperatures it iscomparable to the best method available so far (the Sessak–Monasson method).The relationship between Bethe reconstruction and other mean-field methods isdiscussed.

Magnetization densities as replica parameters: The dilute ferromagnet

In this paper we compute exactly the ground state energy and entropy of the dilute ferromagnetic Ising model. The two thermodynamic quantities are also computed when a magnetic field with random locations is present. The result is reached in the replica approach frame by a class of replica order parameters introduced by Monasson (1998) [5]. The strategy is first illustrated considering the SK model, for which we will show the complete equivalence with the standard replica approach. Then, we apply to the diluted ferromagnetic Ising model with a random located magnetic field, which is mapped into a Potts model.

Cavity method in the spherical SK model

Nous developpons la methode de la cavite pour le modele spherique de Sherrington-Kirkpatrick a haute temperature et champs externe faible. Nous illustrons la methode par le calcul de la matrice de covariance des fluctuations des recouvrements et de la magnetisation.

A comparison of methods to estimate nutritional requirements from experimental data

1. Research papers use a variety of methods for evaluating experiments designed to determine nutritional requirements of poultry. Growth trials result in a set of ordered pairs of data. Often, point-by-point comparisons are made between treatments using analysis of variance. This approach ignores that response variables (body weight, feed efficiency, bone ash, etc.) are continuous rather than discrete. Point-by-point analyses harvest much less than the total amount of information from the data. Regression models are more effective at gleaning information from data, but the concept of “requirements” is poorly defined by many regression models. 2. Response data from a study of the lysine requirements of young broilers was used to compare methods of determining requirements. In this study, multiple range tests were compared with quadratic polynomials (QP), broken line models with linear (BLL) or quadratic (BLQ) ascending portions, the saturation kinetics model (SK) a logistic model (LM) and a compartmental (CM) model. 3. The sum of total residuals squared was used to compare the models. The SK and LM were the best fit models, followed by the CM, BLL, BLQ, and QP models. A plot of the residuals versus nutrient intake showed clearly that the BLQ and SK models fitted the data best in the important region where the ascending portion meets the plateau. 4. The BLQ model clearly defines the technical concept of nutritional requirements as typically defined by nutritionists. However, the SK, LM and CM models better depict the relationship typically defined by economists as the “law of diminishing marginal productivity”. The SK model was used to demonstrate how the law of diminishing marginal productivity can be applied to poultry nutrition, and how the “most economical feeding level” may replace the concept of “requirements”.

Is the Sherrington–Kirkpatrick model relevant for real spin glasses?

I describe the results of numerical simulations which test whether the Sherrington–Kirkpatrick model, which was solved by Parisi, applies to short-range spin glasses. I conclude that it probably does for dimension d greater than 6, but there appear to be some differences in lower dimensions. In particular, there does not appear to be the line of transitions in a magnetic field (AT line) which is seen in the SK model.

Dynamics of an Ising spin glass on the Bethe lattice

We study the dynamical low-temperature behaviour of the Ising spin glass on the Bethe lattice. Starting from Glauber dynamics we propose a cavity such as an ansatz that allows for the treatment of the slow (low temperature) part of dynamics. Assuming a continuous phase transition and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature. The theory is formulated in terms of correlation–response functions of arbitrary order. They can, however, be broken down completely to products of pair functions depending on two time arguments only. For binary couplings J = ±I a spin glass solution is found which approaches the corresponding solution for the SK model in the limit of high connectivity. For more general distributions P(J) no stable or marginal solution of this type appears to exist. The nature of the low-temperature phase in this more general case is unclear.

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ β ,p > 0, such that for all exponential time scales, exp(γ N), with γ < γ β ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β 2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

Some Observations for Mean-Field Spin Glass Models

We obtain bounds to show that the pressure of a two-body, mean-field spin glass is a Lipschitz function of the underlying distribution of the random coupling constants, with respect to a particular semi-norm. This allows us to re-derive a result of Carmona and Hu, on the universality of the SK model, by a different proof, and to generalize this result to the Viana–Bray model. We also prove another bound, suitable when the coupling constants are not independent, which is what is necessary if one wants to consider “canonical” instead of “grand canonical” versions of the SK and Viana–Bray models. Finally, we review Viana–Bray type models, using the language of Lévy processes, which is natural in this context.

Effect of Tree Spacing on Vegetative Growth and Reproduction in an Early Growth Stage in Two Cultivars of Ficus carica L.

The nursery stocks of two fig varieties, ‘Masui dauphine’ and ‘Houraishi’ (Ficus carica L.), were raised by cutting in a nursery, transplanted with changing tree density to an experiment field on July 8, 2004, and cultivated using straight-line training in the Agricultural Food and Environmental Science Research Center of Osaka Prefecture, Japan. The density grades per 1.0-a land area (100 m2) were 200, 66.7, 40, and 6.06. The growth of the aboveground vegetative parts of each tree was monitored at irregular intervals in 2005 by measuring the length and diameter of three organs, i.e. trunk, primary scaffold limb and fruit-bearing shoots. The linear dimensions of the three organs were transformed into dry weight values of woody organs and leaves, giving individual vegetative weight per tree. The fruit were harvested daily during the fruiting season from August to December 2005. The mean values of individual vegetative weight at different densities were approximated well by the reciprocal equation of the C-D effect (hereafter, the SK model), which was proposed by Shinozaki and Kira ca. 50 years ago. Mean annual fruit production per tree at a given density grade was also covered by the SK model, suggesting the clear effect of density on vegetative growth and fruit production in young fig monocultures. The coefficients of the equation conformed to conventional recognition about the poorer vegetative growth and richer fruit harvest in ‘Masui dauphine’ than in ‘Houraishi’. The observed density dependency of the reproductive allocation (RA) represented the decrease of RA with the increase of density in two cultivars although the predicted density dependency of RA was contradictory to that of the observation of the ‘Masui dauphine’.

Exponential Control of Overlap in the Replica Method for p-Spin Sherrington-Kirkpatrick Model

Recently, Michel Talagrand computed the large deviations limit $\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For $a\geq 1$ the limit is given by Guerra's inverse bound and this result extends the classical physicist's replica method that corresponds to integer $a.$ We give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that provides a strong exponential control of the overlap.

The Percolation Signature of the Spin Glass Transition

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin–Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering—both in short-range (EA) and infinite-range (SK) models—within a two-replica FK representation and also within the different Chayes–Machta–Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.

Scaling relationships between sizes of nucleation regions and eventual sizes of microearthquakes

We investigate the initial rupture process of microearthquakes to reveal relationships between nucleation region sizes and eventual earthquake sizes. In order to obtain high quality waveform data, we installed a trigger recording system with a sampling frequency of 10 kHz at the base of a deep borehole at the Nojima Fault, Japan. We analyze waveform data of 31 events around the borehole, with seismic moment ranging from 4.2 × 10 9 Nm to 7.1 × 10 11 Nm. We use both a circular crack model with an accelerating rupture velocity (SK model) [Sato, T., Kanamori, H., 1999. Beginning of earthquakes modeled with the Griffith's fracture criterion, Bull. Seism. Soc. Am., 89, 80-93.], which generates a slow initial phase of velocity pulse, and a circular crack model with a constant rupture velocity (SH model) [Sato, T, Hirasawa, T., 1973. Body wave spectra from propagating shear cracks, J. Phys. Earth, 21, 415-431.], which generates a ramp-like velocity pulse. Source parameters of these two models are estimated by waveform inversion of the first half cycle of the observed velocity pulse applying both a grid search and a non-linear least squares method. 14 of 31 events are never reproduced by the SH model with a constant Q operator. But SK model with a constant Q operator provides a size of the pre-existing crack, corresponding to the size of the nucleation regions, and a size of the eventual crack. We recognize that (i) the eventual seismic moment is approximately scaled as the cube of the size of pre-existing cracks, (ii) the eventual seismic moment is scaled as the cube of the size of eventual cracks, and (iii) the size of eventual cracks is roughly proportional to the size of pre-existing cracks. We, thus, conclude that the size of eventual earthquakes is controlled by the size of the nucleation regions.

A note on Talagrand's positivity principle

Talagrand's positivity principle states that one can slightly perturb a Hamiltonian in the Sherrington-Kirkpatrick model in such a way that the overlap of two configurations under the perturbed Gibbs' measure will become typically nonnegative. In this note we observe that abstracting from the setting of the SK model only improves the result and does not require any modifications in Talagrand's argument. In this version, for example, positivity principle immediately applies to the setting of replica symmetry breaking interpolation. Also, abstracting from the SK model improves the conditions in the Ghirlanda-Guerra identities and as a consequence results in a perturbation of smaller order necessary to ensure positivity of the overlap.

On the multiple overlap function of the SK model

In this note, we prove an asymptotic expansion and a central limit theorem for the multiple overlap $R_{1,\dotsc,s}$ of the SK model, defined for given $N, s\ge 1$ by $R_{1,\dotsc, s}=N^{-1}\sum_{i\le N}\sigma_i^{1}\ldots\sigma_i^{ s}$. These results are obtained by a careful analysis of the terms appearing in the cavity derivation formula, as well as some graph induction procedures. Our method could hopefully be applied to other spin glasses models.

Spin Glass Computations and Ruelle’s Probability Cascades

We study the Parisi functional, appearing in the Parisi formula for the pressure of the SK model, as a functional on Ruelle's Probability Cascades (RPC). Computation techniques for the RPC formulation of the functional are developed. They are used to derive continuity and monotonicity properties of the functional retrieving a theorem of Guerra. We also detail the connection between the Aizenman-Sims-Starr variational principle and the Parisi formula. As a final application of the techniques, we rederive the Almeida-Thouless line in the spirit of Toninelli but relying on the RPC structure.