SK Model Research Articles

Correction to: Free Energy Fluctuations of the Bipartite Spherical SK Model at Critical Temperature

Correction to: Free Energy Fluctuations of the Bipartite Spherical SK Model at Critical Temperature

Spin Glass to Paramagnetic Transition and Triple Point in Spherical SK Model

Spin Glass to Paramagnetic Transition and Triple Point in Spherical SK Model

Free Energy Fluctuations of the Bipartite Spherical SK Model at Critical Temperature

Free Energy Fluctuations of the Bipartite Spherical SK Model at Critical Temperature

A new model for compressor surge and stall control

This paper compares the bifurcations and closed-loop performances of two compressor models, Moore-Greitzer (MG) and a developed model based on MG (Shahriyari Khaleghi, SK). First, both models are linearized about two equilibrium points (pure surge and fully-developed rotating stall), and the perturbed state-space dynamics and input matrices are obtained. The compressor unstable regions are then identified using an eigenvalue and global bifurcation analysis. Furthermore, optimal LQR controllers are designed, and the performances of closed-loop systems are compared. The LQRs are designed to control the compression system near the peak pressure rise by suppressing surge or stall. Results reveal that if the initial operating point is in the positive slope region of the compressor characteristic and the initial amplitude of the disturbances is small, the LQR controller can stabilize the compressor in both models. However, when the disturbances are intensive, the two models respond differently: although the SK model damps a fair range of disturbances and predicts instability for excessively powerful disturbances, the MG model always damps them, even when extremely intense. Without a controller in the MG model, initial disturbances (even very large) can never grow and are always damped in the compressor’s negative slope region (obviously, the same applies to the controller). However, pending the amplitude of the disturbances (in the absence of a controller), the disturbances in the SK model may be damped or grow. The SK model can successfully control the instabilities if the disturbances are small. Nonetheless, the controller fails to dampen the instabilities for extreme disturbances, which is consistent with reality.

The two point function of the SK model without external field at high temperature

The two point function of the SK model without external field at high temperature

On the concavity of the TAP free energy in the SK model

We analyze the Hessian of the Thouless–Anderson–Palmer (TAP) free energy for the Sherrington–Kirkpatrick model, below the de Almeida–Thouless line, evaluated in Bolthausen’s approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, and that the support includes zero on the AT line. In this “macroscopic” sense, we show that TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetizations. This proves a spectral interpretation of the AT line. We also find different magnetizations than Bolthausen’s approximate solutions at which the Hessian of the TAP free energy has positive outlier eigenvalues. In particular, when the magnetizations are assumed to be independent of the disorder, we prove that Plefka’s second condition is equivalent to all eigenvalues being negative. On this occasion, we extend the convergence result of Capitaine et al. (Electron. J. Probab. 16, no. 64, 2011) for the largest eigenvalue of perturbed complex Wigner matrices to the GOE.

Fluctuations of the overlap at low temperature in the 2-spin spherical SK model

We describe the fluctuations of the overlap between two replicas in the 2-spin spherical SK model about its limiting value in the low temperature phase. We show that the fluctuations are of order $N^{-1/3}$ and are given by a simple, explicit function of the eigenvalues of a matrix from the Gaussian Orthogonal Ensemble. We show that this quantity converges and describe its limiting distribution in terms of the Airy1random point field (i.e., the joint limit of the extremal eigenvalues of the GOE) from random matrix theory.

The replica symmetric formula for the SK model revisited

We provide a simple extension of Bolthausen’s Morita-type proof of the replica symmetric formula [E. Bolthausen, “A Morita type proof of the replica-symmetric formula for SK,” in Statistical Mechanics of Classical and Disordered Systems, Springer Proceedings in Mathematics and Statistics (Springer, Cham., 2018), pp. 63–93; arXiv:1809.07972] for the Sherrington–Kirkpatrick model and prove the replica symmetry for all (β, h) that satisfy β2Esech2(βqZ+h)≤1, where q=E⁡tanh2(βqZ+h). Compared to the work of Bolthausen [“A Morita type proof of the replica-symmetric formula for SK,” in Statistical Mechanics of Classical and Disordered Systems, Springer Proceedings in Mathematics and Statistics (Springer, Cham., 2018), pp. 63–93; arXiv:1809.07972], the key of the argument is to apply the conditional second moment method to a suitably reduced partition function.

Free energy fluctuations of the two-spin spherical SK model at critical temperature

We investigate the fluctuations of the free energy of the two-spin spherical Sherrington–Kirkpatrick model at critical temperature βc = 1. When β = 1, we find asymptotic Gaussian fluctuations with variance 16N2log(N), confirming in the spherical case a physics prediction for the SK model with Ising spins. We, furthermore, prove the existence of a critical window on the scale β=1+αlog(N)N−1/3. For any α∈R, we show that the fluctuations are at most order log(N)/N in the sense of tightness. If α → ∞ at any rate as N → ∞, then, properly normalized, the fluctuations converge to the Tracy–Widom1 distribution. If α → 0 at any rate as N → ∞ or α < 0 is fixed, the fluctuations are asymptotically Gaussian as in the α = 0 case. In determining the fluctuations, we apply a recent result of G. Lambert and E. Paquette [“Strong approximation of Gaussian beta-ensemble characteristic polynomials: The edge regime and the stochastic airy function,” arXiv:2009.05003 (2020)] on the behavior of the Gaussian-β-ensemble at the spectral edge.

TAP equations are repulsive

We show that for low enough temperatures, but still above the AT line, the Jacobian of the TAP equations for the SK model has a macroscopic fraction of eigenvalues outside the unit interval. This provides a simple explanation for the numerical instability of the fixed points, which thus occurs already in high temperature. The insight leads to some algorithmic considerations on the low temperature regime.

Comparison of Spiegler–Kedem combined with film theory model and original SK model

Comparison of Spiegler–Kedem combined with film theory model and original SK model

Experimental Investigation of the Effect of Fluid-Solid Mixture Flow by Pneumatic System in Textile Industry

The transport of multiphase flows of slurry (fluid-solid) through pipelines is usually encountered through several hurdles in various industries such as cement, textile and chemical industries. The flow parameters such as solids concentration, pipe directions and so on, add to these complexities and concepts associated with transport of slurries seems to be uncertain which has resulted in significant research being conducted on flow of slurries through pipelines. The investigation has been carried out in Textile and garment factory at Ethiopia in eastern Africa. The objective of the investigation is to study the effects of transporting of fluid-solid mixture (air-cotton ball) through horizontal, vertical and inclined pipelines in a textile factory. For this purpose one generalized mathematical model developed by Shrivastava and Kar (SK Model) was applied to predict more accurate results of pressure drop. The results of pressure drop obtained using SK model were compared with experimental results. The results showed that the percentage of head losses is increased in the smaller diameter pipe line about 48% and 67% with air velocity 12 m/s and 24 m/s respectively as compared to that of larger diameter pipeline. In the proposed pipeline system, it was observed that the energy saving can be achieved about 65% and 77% with air velocity 12 m/s and 24 m/s respectively.

Quantum echo dynamics in the Sherrington-Kirkpatrick model

Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a well-defined chaotic semi-classical (large-N) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins N. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on N and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.

The SK Model Is Infinite Step Replica Symmetry Breaking at Zero Temperature

We prove that the Parisi measure of the mixed p-spin model at zero temperature has infinitely many points in its support. This establishes Parisi's prediction that the functional order parameter of the Sherrington-Kirkpatrick model is not a step function at zero temperature. As a consequence, we show that the number of levels of broken replica symmetry in the Parisi formula of the free energy diverges as the temperature goes to zero.

Central limit theorem near the critical temperature for the overlap in the 2-spin spherical SK model

We prove a central limit theorem for the normalized overlap in the spherical SK model in the high temperature phase. The convergence holds almost surely with respect to the disorder variables, and the inverse temperature can approach the critical value at a polynomial rate with any exponent strictly greater than −1/3.

A very simple proof of the LSI for high temperature spin systems

We present a very simple proof that the O(n) model satisfies a uniform logarithmic Sobolev inequality (LSI) if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n. This condition applies in particular to the SK spin glass model at inverse temperature β<1/4. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry–Emery theory for the renormalised measure.

Investigation on Pressure Drop of Fluid-Solid Mixture Flow through Pipes Using CFD and SK Model

The carrier fluid (air or water) is used to transport solid material from the source place to its destination point through pipeline. Using air as carrier fluid to transport solid material through pipeline is called Pneumo transport, whereas transporting material with water or any other liquid through pipeline is called as hydraulic transport. A large number of installations are now available globally to transport solid materials to short, medium, and long distances using water/air as carrier fluid. However, the design of such system of pipeline is still an empirical art. In the present investigation, one generalized mathematical model developed by Shrivastava and Kar (SK Model) and CFD models were used and compared with experimental results for pneumatic and hydraulic transport of granular solids. The motivation of present work is to find the accuracy of SK model based on analytical, empirical and semi-empirical for the prediction of pressure drop and comparing the result with CFD based on mathematical equation for the mixture flow in the horizontal and vertical pipe lines. The comparison of pressure drop results obtained by using SK model and CFD model were validated with the experimental results for pneumatic and hydraulic transport of solids through. From the comparison results, it was observed that the results of pressure drop predicted by SK model are more accurate than the CFD models for all the cases considered.

Order of fluctuations of the free energy in the SK model at critical temperature

We present an elementary approach to the order of fluctuations for the free energy in the Sherrington-Kirkpatrick mean field spin glass model at and near the critical temperature. It is proved that at the critical temperature the variance of the free energy is of $O((\log N)^2).$ In addition, we show that if one approaches the critical temperature from the low temperature regime at the rate $O(N^{-\alpha})$ for some $\alpha>0,$ then the variance is of $O((\log N)^2+N^{1-\alpha}).$

Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound

The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [18]. The central argument therein relied on a very dedicated study of the coupled free energy via the two-dimensional Guerra-Talagrand (GT) replica symmetry breaking bound. It is believed that this bound and its higher dimensional generalization are highly related to the conjectures of temperature chaos and ultrametricity in the SK model, but a complete investigation remains elusive. Motivated by Bovier-Klimovsky [2] and Auffinger-Chen [3], the aim of this paper is to present a novel approach to analyzing the Parisi functional and the two-dimensional GT bound in the mixed $p$-spin models in terms of optimal stochastic control problems. We compute the directional derivative of the Parisi functional and derive equivalent criteria for the Parisi measure. We demonstrate how our approach provides a simple and efficient control for the GT bound that yields several new results on Talagrand's positivity of the overlap [20,Section 14.12] and disorder chaos in Chatterjee [5] and Chen [6]. In particular, we provide some examples of the models containing odd $p$-spin interactions.

Manganite/Cuprate Superlattice as Artificial Reentrant Spin Glass

Emerging physical phenomena at the unit‐cell‐controlled interfaces of transition‐metal oxides have attracted lots of interest because of the rich physics and application opportunities. This work reports a reentrant spin glass behavior with strong magnetic memory effect discovered in oxide heterostructures composed of ultrathin manganite La0.7Sr0.3MnO3 (LSMO) and cuprate La2CuO4 (LCO) layers. These heterostructures are featured with enhanced ferromagnetism before entering the spin glass state: a Curie temperature of 246 K is observed in the superlattice with six‐unit‐cell LSMO layers, while the reference LSMO film with the same thickness shows much weaker magnetism. Furthermore, an insulator–metal transition emerges at the Curie temperature, and below the freezing temperature the superlattices can be considered as a glassy ferromagnetic insulator. These experimental results are closely related to the interfacial spin reconstruction revealed by the first‐principles calculations, and the dependence of the reentrant spin glass behavior on the LSMO layer thickness is in line with the general phase diagram of a spin system derived from the infinite‐range SK model. The results of this work underscore the manganite/cuprate superlattices as a versatile platform of creating artificial materials with tailored interfacial spin coupling and physical properties.