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... A QUBO [see, e.g., 18] problem, as its name indicates, is of the general form min{x ⊺ Qx + c ⊺ x : x ∈ {0, 1}}. QUBO encompasses the max-cut problem and the closely related Ising model [see, e.g., 19,20]. Given that Ising devices [see, e.g. ...
... where E is the set of extreme points of {x ∈ [0, 1] n : Ax = b}. Furthermore, by Theorem 3, there isz ∈ R such that for every z >z, problem (19) is equivalent to ...
... Let z ′ = max{ L 2 ,z}, then, for every z > z ′ , we have that (B 2 z ) is equivalent to (19), and that (19) is equivalent to (20). Thus, (B 2 z ) is equivalent to (20), implying that: ...
In recent years, there has been a surge of interest in studying different ways to reformulate nonconvex optimization problems, especially those that involve binary variables. This interest surge is due to advancements in computing technologies, such as quantum and Ising devices, as well as improvements in quantum and classical optimization solvers that take advantage of particular formulations of nonconvex problems to tackle their solutions. Our research characterizes the equivalence between equality-constrained nonconvex optimization problems and their Lagrangian relaxation, enabling the aforementioned new technologies to solve these problems. In addition to filling a crucial gap in the literature, our results are readily applicable to many important situations in practice. To obtain these results, we bridge between specific optimization problem characteristics and broader, classical results on Lagrangian duality for general nonconvex problems. Further, our approach takes a comprehensive approach to the question of equivalence between problem formulations. We consider this question not only from the perspective of the problem's objective but also from the viewpoint of its solution. This perspective, often overlooked in existing literature, is particularly relevant for problems featuring continuous and binary variables.
... Since it costs 4 bits for binary representation of (8) 10 , the above mechanism is not effective to hide data efficiently. ...
... In this section, we, instead of finding pixel energy, describe a new concept of "Fibonacci Energy" of a pixel inspiring from the Hamiltonian energy estimation of particles in Physics using lattice spin glass model of Ising. The spin (S i ) of any {i : i ∈ τ }-th particle in the lattice/grid structure is either positive (+1) or a negative (-1) and gets influenced by its immediate neighbourhood set {NBD i : i ∈ τ } [10]. The Hamiltonian energy of the particles in the grid structure can be estimated by (3). ...
... ∼ gorr/classes/GeneralGraphics/ imageFormats) image for hiding secret information. Let γ x,y be the intensity value (ranging from (0) 10 to (255) 10 ) of the cover pixel at location (x, y). Difference steps of the process are detailed below. ...
Steganography and Steganalysis are becoming increasingly relevant in information forensics and hiding data in the higher bitplanes without keeping any perceptible signature into the image is a challenging problem in this area. In this paper, we propose a unique solution to this problem using Fibonacci numbers as base. The pixels are selected from the busy part of the image where noticeable changes in pixel intensities occur. The business of the pixels is determined by their Fibonacci energy. The pixels values are converted into Fibonacci base and their corresponding Fibonacci energies are estimated by the Fibonacci expansion of pixel intensities. The set of energetic pixels are considered according to the descending order of their energy values. The binary data are concealed into higher bitplanes (up to 5) of the Fibonacci base of the pixel intensities. We theoretically derive some nice combinatorial properties related to distortion of pixel intensities and also experimentally show that our algorithm withstands against visual, structural and statistical attacks. The average embedding capacity is 3.98 bpp and average PSNR is 39.59 dB. We also demonstrate that our method is capable of resisting from the series of benchmark tests provided by StirMark 4.0.
... We wish to sample configurations from the Boltzmann-Gibbs distribution π(σ) := exp(−βH(σ))/Z on Ω, where Z := σ∈Ω exp(−βH(σ)) is the partition function of the system and β ∈ R + is inverse temperature. (What has been described is the ferromagnetic Ising model, which favors like spins; see [3] for background.) ...
... Suppose the projection chain satisfies a Poincaré inequality with constantλ, and the restriction chains satisfy inequalities with uniform constant λ min . Let γ be defined as in(3). Then the original Markov chain satisfies ...
We consider finite-state Markov chains that can be naturally decomposed into smaller ``projection'' and ``restriction'' chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincare (resp. log-Sobolev) constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev) constants of the projection and restriction chains, together with further a parameter. In the case of the Poincare constant, our bound is always at least as good as existing ones and, depending on the value of the extra parameter, may be much better. There appears to be no previously published decomposition result for the log-Sobolev constant. Our proofs are elementary and self-contained.
... In general, in the Ising model 37,38 , the energy function E(S) of the system with spin vector S is where s i and s j are state of the i-th and j-th spin, w ij is the weight of the edge between the i-th and j-th spins, and h i is a bias term for the i-th spin. Consider defining the probability p of the system with status S as [25][26][27]39 where Z is the sum of all the cases of system energy and given using the equation ...
... In hardware-implemented Ising machines, inputs of p-bits are generated by adding a bias term to the sum of the product of the weight and output of connected spins 37 . However, as the size of the problem is increased, the number of required p-bits increases, and accordingly, the number of p-bit connections dramatically increases. ...
Probabilistic computing has been introduced to operate functional networks using a probabilistic bit (p-bit), broadening the computational abilities in non-deterministic polynomial searching operations. However, previous developments have focused on emulating the operation of quantum computers similarly, implementing every p-bit with large weight-sum matrix multiplication blocks and requiring tens of times more p-bits than semiprime bits. In addition, operations based on a conventional simulated annealing scheme required a large number of sampling operations, which deteriorated the performance of the Ising machines. Here we introduce a prime factorization machine with a virtually connected Boltzmann machine and probabilistic annealing method, which are designed to reduce the hardware complexity and number of sampling operations. From 10-bit to 64-bit prime factorizations were performed, and the machine offers up to 1.2 × 10⁸ times improvement in the number of sampling operations compared with previous factorization machines, with a 22-fold smaller hardware resource.
... As for the applications of these methods to condensed matter physics, most studies focus on investigating the Ising model [44] since it can be treated as a paradigm to solve more complicated problems. The reason for this choice is obvious: it is the simplest model of ferromagnetism that predicts a phase transition in dimension d > 1. ...
... 44 The mean open state probability P op in function of micropipette potential U for three different cellular lines.The data consists of mitoBK current's recordings obtained from the patch-clamp experiment. For analysis, we use traces corresponding to different cellular lines and membrane potentials U m . ...
... Ising problems are closely related to the Ising model studied in physics, which is a statistical model for spin couplings in ferromagnetic materials [12,13]. 1 Here we define a well-studied combinatorial optimization problem: (weighted) Max-Cut [14][15][16] where the objective is to partition the vertices of a weighted graph into two disjoint groups so that the sum of the weights of the edges between the groups is maximized. More formally, given a graph = ( , ) with weighted adjacency matrix , Max-Cut can be written as the following maximization problem: ...
The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm that finds approximate solutions to problems in combinatorial optimization, especially those that can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. In prior work, researchers have considered various ways of "warm-starting" QAOA by constructing an initial quantum state using classically-obtained solutions or information; these warm-starts typically cause QAOA to yield better approximation ratios at much lower circuit depths. For the Max-Cut problem, one warm-start approaches constructs the initial state using the high-dimensional vectors that are output from an SDP relaxation of the corresponding Max-Cut problem. This work leverages these semidefinite warmstarts for a broader class of problem instances by using a standard reduction that transforms any QUBO instance into a Max-Cut instance. We empirically compare this approach to a "QUBO-relaxation" approach that relaxes the QUBO directly. Our results consider a variety of QUBO instances ranging from randomly generated QUBOs to QUBOs corresponding to specific problems such as the traveling salesman problem, maximum independent set, and portfolio optimization. We find that the best choice of warmstart approach is strongly dependent on the problem type.
... This reliance on definitive 0s and 1s poses significant challenges in solving combinatorial optimization (CO) problems, as it necessitates the conventional quadratic unconstrained binary optimization (QUBO) formulation, which inherently leads to data structures that contribute to an exponential increase in computational complexity [14]- [16]. In this scenario, the quantum solution of the Ising model [17] is embedded in a high-dimensional Hilbert space and simulated using variational quantum gate-based circuits [18], [19]. These circuits incorporate unitary operations that evolve over time [20], [21]. ...
Quantum combinatorial optimization algorithms typically face challenges due to complex optimization landscapes featuring numerous local minima, exponentially scaling latent spaces, and susceptibility to quantum hardware noise. In this study, we introduce Direct Entanglement Ansatz Learning (DEAL), wherein we employ a direct mapping from quadratic unconstrained binary problem parameters to quantum ansatz angles for cost and mixer hamiltonians, which improves the convergence rate towards the optimal solution. Our approach exploits a quantum entanglement-based ansatz to effectively explore intricate latent spaces and zero noise extrapolation (ZNE) to greatly mitigate the randomness caused by crosstalk and coherence errors. Our experimental evaluation demonstrates that DEAL increases the success rate by up to 14% compared to the classic quantum approximation optimization algorithm while also controlling the error variance. In addition, we demonstrate the capability of DEAL to provide near optimum ground energy solutions for travelling salesman, knapsack, and maxcut problems, which facilitates novel paradigms for solving relevant NP-hard problems and extends the practical applicability of quantum optimization using noisy quantum hardware.
... Hamiltonian with the PIM is governed by the parameter [28]. Indeed, many energy minimization algorithms, i.e., annealing schemes, tune this parameter with different strategies in order to search for the Ising spin configuration with the lowest energy [1]. ...
Probabilistic computing with pbits is emerging as a computational paradigm for machine learning and for facing combinatorial optimization problems (COPs) with the so-called probabilistic Ising machines (PIMs). From a hardware point of view, the key elements that characterize a PIM are the random number generation, the nonlinearity, the network of coupled pbits, and the energy minimization algorithm. Regarding the latter, in this work we show that PIMs using the simulated quantum annealing (SQA) schedule exhibit better performance as compared to simulated annealing and parallel tempering in solving a number of COPs, such as maximum satisfiability problems, planted Ising problem, and travelling salesman problem. Additionally, we design and simulate the architecture of a fully connected CMOS based PIM able to run the SQA algorithm having a spin-update time of 8 ns with a power consumption of 0.22 mW. Our results also show that SQA increases the reliability and the scalability of PIMs by compensating for device variability at an algorithmic level enabling the development of their implementation combining CMOS with different technologies such as spintronics. This work shows that the characteristics of the SQA are hardware agnostic and can be applied in the co-design of any hybrid analog digital Ising machine implementation. Our results open a promising direction for the implementation of a new generation of reliable and scalable PIMs.
... The Ising model, one of the simplest and most fundamental models in the field of statistical mechanics, was originally developed to describe the interactions between discrete variables in lattice spin systems [55,56]. Over time, its applications have extended far beyond statistical mechanics, making a profound impact in fields such as theoretical physics, artificial intelligence [57][58][59] and combinatorial optimization [60][61][62]. ...
With accelerating urbanization and worsening traffic congestion, optimizing traffic signal systems to improve road throughput and alleviate congestion has become a critical issue. This study proposes a short-term traffic prediction model based on real-world road topologies and a typical four-way, eight-phase traffic signal control scheme. The model accounts for traffic flow disparities across directions and signal phase change frequencies, integrating these factors into an optimization objective for global traffic optimization. The structure of this objective function is similar to spin-glass systems in statistical physics. A Simulated Bifurcation optimization algorithm is introduced, with traditional simulated annealing as a benchmark. The results show that Simulated Bifurcation outperforms simulated annealing in both efficiency and effectiveness. Using real traffic flow and road network data from Beijing, we initialized the model and conducted numerical optimization experiments. The results indicate that Simulated Bifurcation significantly outperforms simulated annealing in computational efficiency, effectively solving combinatorial optimization problems with multiple spin interactions, and reducing the time complexity to . This solution addresses the NP-hard problem of global traffic signal optimization. Importantly, the signal phase patterns generated by Simulated Bifurcation align with the operational requirements of real traffic signal systems, showcasing its potential in optimizing signal control for large, complex urban traffic networks. This work provides solid theoretical and practical foundations for future urban traffic management and intelligent transportation systems.
... 3. OscNet with cosine similarity reaches almost same performance than K-means with Euclidean distance error implemented on CPU. The Ising model [17,38,78,79] was discovered in the context of ferromagnetism in statistical mechanics. The Ising Machine leverages the natural tendency of such systems to minimize their energy, enabling it to efficiently find optimal solutions [18,42,47,53]. ...
Machine learning and AI have achieved remarkable advancements but at the cost of significant computational resources and energy consumption. This has created an urgent need for a novel, energy-efficient computational fabric to replace the current computing pipeline. Recently, a promising approach has emerged by mimicking spiking neurons in the brain and leveraging oscillators on CMOS for direct computation. In this context, we propose a new and energy efficient machine learning framework implemented on CMOS Oscillator Networks (OscNet). We model the developmental processes of the prenatal brain's visual system using OscNet, updating weights based on the biologically inspired Hebbian rule. This same pipeline is then directly applied to standard machine learning tasks. OscNet is a specially designed hardware and is inherently energy-efficient. Its reliance on forward propagation alone for training further enhances its energy efficiency while maintaining biological plausibility. Simulation validates our designs of OscNet architectures. Experimental results demonstrate that Hebbian learning pipeline on OscNet achieves performance comparable to or even surpassing traditional machine learning algorithms, highlighting its potential as a energy efficient and effective computational paradigm.
... Assuming the presence of an external magnetic field denoted by H and considering the interaction of each spin with its four nearest neighbors, the energy of a specific configuration, represented as {s 1 , s 2 , . . . , s N }, is defined using the Hamiltonian function E as follows [24][25][26][27]: ...
Single-pixel imaging leverages a single-pixel detector and structured illumination patterns to reconstruct images, offering a cost-effective solution for imaging across a wide range of wavelengths, such as x-ray and terahertz. However, the technique faces challenges in efficiency due to the need for numerous patterns to achieve high-quality image reconstruction. In this study, we explore the use of spin lattice models from statistical mechanics to design illumination patterns for single-pixel imaging. By employing models like Ising, Potts, XY, and Heisenberg, we generate structured patterns that are adaptable for binary, grayscale, and color imaging. This work creates a direct connection between lattice models and imaging applications, providing a systematic approach to pattern generation that can enhance single-pixel imaging efficiency.
... where, s [ 1 ··· ] ⊤ ∈ {+1, −1} is a coordinate of the hypercubic vertices in the original space but with the Ising spin variable [114][115][116][117]. Instead of binary variables ∈ {1, 0}, we introduce Ising variables = 2 − 1 ∈ {+1, −1} to calculate the inner product because the inner product among the Ising variables reflects the similarity or overlap between the vertices. ...
Projections of hypercubes have been applied for visualizing high-dimensional binary state spaces in various fields of sciences. Conventional methods for projecting hypercubes, however, face practical difficulties. Manual methods require nontrivial adjustments of the projection basis, while optimization methods impose limitations on the interpretation and reproducibility of the resulting plots. Here, we propose using principal component analysis (PCA) for projecting hypercubes, which offers an automated, interpretable, and reproducible solution. We reveal analytically and numerically that PCA effectively captures polarized distributions in the state space. The resulting distribution of projected vertices is shown to be asymptotically the standard Gaussian, a characteristic of high-dimensional orthogonal projections. This method is applied to visualize the hypercubic energy landscapes of Ising spin systems, where the biplots reveal the pathways involving correlated spin flips. Our work demonstrates the potential of PCA for discovering hidden patterns in high-dimensional binary data.
... This model has been extensively studied, leading to profound insights not only in physics but also in fields such as biology, sociology, and computer science. For more information, the reader is invited to check the survey by Cipra [Cip87]. ...
We investigate some previously unexplored (or underexplored) computational aspects of total variation (TV) distance. First, we give a simple deterministic polynomial-time algorithm for checking equivalence between mixtures of product distributions, over arbitrary alphabets. This corresponds to a special case, whereby the TV distance between the two distributions is zero. Second, we prove that unless , it is impossible to efficiently estimate the TV distance between arbitrary Ising models, even in a bounded-error randomized setting.
... The connection between Ising model of statistical mechanics [41] and hard combinatorial optimization problems of mathematics has been known for decades. [42] Boltzmann machines [10,11] and subsequently their restricted version for deep belief networks are Ising models in which the weights of interactions are learned and adjusted with breakthrough algorithms. ...
Belief networks represent a powerful approach to problems involving probabilistic inference, but much of the work in this area is software based utilizing standard deterministic hardware based on the transistor which provides the gain and directionality needed to interconnect billions of them into useful networks. This paper proposes a transistor like device that could provide an analogous building block for probabilistic networks. We present two proof-of-concept examples of belief networks, one reciprocal and one non-reciprocal, implemented using the proposed device which is simulated using experimentally benchmarked models.
... The model we propose is what we call the Seeded Ising Model. It is an Ising model [3] in which bits in certain locations are held fixed throughout Ising model dynamic evolution. This way, our Seeded Ising Model reconstructs a template from a fraction of the information of the whole template. ...
We propose a variant of Ising model, called the Seeded Ising Model, to model probabilistic nature of human iris templates. This model is an Ising model in which the values at certain lattice points are held fixed throughout Ising model evolution. Using this we show how to reconstruct the full iris template from partial information, and we show that about 1/6 of the given template is needed to recover almost all information content of the original one in the sense that the resulting Hamming distance is well within the range to assert correctly the identity of the subject. This leads us to propose the concept of effective statistical degree of freedom of iris templates and show it is about 1/6 of the total number of bits. In particular, for a template of 2048 bits, its effective statistical degree of freedom is about 342 bits, which coincides very well with the degree of freedom computed by the completely different method proposed by Daugman.
... Phase transitions have been observed in a wide variety of studies, such as in physics, chemistry, biology, complex systems, computer science, and random graphs, to list a few. It leads to long term attention in the literature, from physicists such as Ising [1] in the 1920's to mathematicians such as Erdös and Rényi [2] in the 1960's, from complex systems theorists such as Langton [3] in the 1990's to control scientists such as Olfati-Saber [4] in the 2000's. ...
In this paper, we study the phase transition behavior emerging from the interactions among multiple agents in the presence of noise. We propose a simple discrete-time model in which a group of non-mobile agents form either a fixed connected graph or a random graph process, and each agent, taking bipolar value either +1 or -1, updates its value according to its previous value and the noisy measurements of the values of the agents connected to it. We present proofs for the occurrence of the following phase transition behavior: At a noise level higher than some threshold, the system generates symmetric behavior (vapor or melt of magnetization) or disagreement; whereas at a noise level lower than the threshold, the system exhibits spontaneous symmetry breaking (solid or magnetization) or consensus. The threshold is found analytically. The phase transition occurs for any dimension. Finally, we demonstrate the phase transition behavior and all analytic results using simulations. This result may be found useful in the study of the collective behavior of complex systems under communication constraints.
... The Ising model is one of the most famous models in statistical physics. It can be seen as a toy model of ferromagnetism, but it has served as a testbed for a very large number of ideas going far beyond this particular problem [22]. It consists of a finite number N of spins s i ∈ {−1, 1} located on a lattice of arbitrary shape and dimension (although a square lattice is often considered) and interacting through a hamiltonian of the form: ...
Two-dimensional turbulent flows, and to some extent, geophysical flows, are systems with a large number of degrees of freedom, which, albeit fluctuating, exhibit some degree of organization: coherent structures emerge spontaneously at large scales. In this short course, we show how the principles of equilibrium statistical mechanics apply to this problem and predict the condensation of energy at large scales and allow for computing the resulting coherent structures. We focus on the structure of the theory using the language of large deviation theory.
... In this study, we evaluate the ability of diffusion models and GAN to generate configurations of the Ising model which describes the interaction of spins and captures the non-trivial physics involved in first-and second-order phase transitions between a non-magnetic and a ferromagnetic state. 26 Due to its simplicity and the existence of an analytical solution 27 for the non-trivial 2D problem, Ising has played a central role in studies of critical phenomena [28][29][30] and has been applied or extended to a wide range of fields, from biological processes [31][32][33] to crystal plasticity. 34 Recent papers [35][36][37] have applied GAN and diffusion models to model Ising and several studies [38][39][40] have investigated the performance of other generative models. ...
Machine learning has become a central technique for modeling in science and engineering, either complementing or as surrogates to physics-based models. Significant efforts have recently been devoted to models capable of predicting field quantities but the limitations of current state-of-the-art models in describing complex physics are not well understood. We characterize the ability of generative diffusion models and generative adversarial networks (GAN) to describe the Ising model. We find diffusion models trained using equilibrium configurations obtained using Metropolis Monte Carlo for a range of temperatures around the critical temperature can capture average thermodynamic variables across the phase transformation and extrapolate to higher and lower temperatures. The model also captures the overall trends of physical properties associated with fluctuations (specific heat and susceptibility) except at the non-ergodic low temperatures and non-trivial scale-free correlations at the critical temperature, albeit with some difference in the critical exponent compared to Monte Carlo simulations. GANs perform more poorly on thermodynamic properties and are susceptible to mode-collapse without careful training. This investigation highlights the potential and limitations of generative models in capturing the complex phenomena associated with certain physical systems.
... Numerous well-known optimization problems such as boolean formula satisfiability (SAT), knapsack, graph coloring, the traveling salesman problem (TSP), or max clique have already been translated to QUBO form [3], [4], [10], [16], [18]. Note that Ising problems [5] are isomorphic to QUBO problems [32]. ...
QAOA is a quantum algorithm for solving combinatorial optimization problems. It is capable of searching for the minimizing solution vector x of a QUBO problem . The number of two-qubit CNOT gates in the QAOA circuit scales linearly in the number of non-zero couplings of Q and the depth of the circuit scales accordingly. Since CNOT operations have high error rates it is crucial to develop algorithms for reducing their number. We, therefore, present the concept of \textit{semi-symmetries} in QUBO matrices and an algorithm for identifying and factoring them out into ancilla qubits. \textit{Semi-symmetries} are prevalent in QUBO matrices of many well-known optimization problems like \textit{Maximum Clique}, \textit{Hamilton Cycles}, \textit{Graph Coloring}, \textit{Vertex Cover} and \textit{Graph Isomorphism}, among others. We theoretically show that our modified QUBO matrix describes the same energy spectrum as the original Q. Experiments conducted on the five optimization problems mentioned above demonstrate that our algorithm achieved reductions in the number of couplings by up to and in circuit depth by up to .
... This method is executed on quantum processing units (QPU) of D-Wave quantum computing hardware. A D-wave QPU was fundamentally built on the Ising model from statistical mechanics 56 . The Ising model comprises nodes and edges in a lattice structure. ...
Academic achievement is a critical measure of intellectual ability, prompting extensive research into cognitive tasks as potential predictors. Neuroimaging technologies, such as functional near-infrared spectroscopy (fNIRS), offer insights into brain hemodynamics, allowing understanding of the link between cognitive performance and academic achievement. Herein, we explored the association between cognitive tasks and academic achievement by analyzing prefrontal fNIRS signals. A novel quantum annealer (QA) feature selection algorithm was applied to fNIRS data to identify cognitive tasks correlated with CSAT scores. Twelve features (signal mean, median, variance, peak, number of peaks, sum of peaks, range, minimum, kurtosis, skewness, standard deviation, and root mean square) were extracted from fNIRS signals at two time windows (10- and 60-s) to compare results from various feature variable conditions. The feature selection results from the QA-based and XGBoost regressor algorithms were compared to validate the former’s performance. In a two-step validation process using multiple linear regression models, model fitness (adjusted R²) and model prediction error (RMSE) values were calculated. The quantum annealer demonstrated comparable performance to classical machine learning models, and specific cognitive tasks, including verbal fluency, recognition, and the Corsi block tapping task, were correlated with academic achievement. Group analyses revealed stronger associations between Tower of London and N-back tasks with higher CSAT scores. Quantum annealing algorithms have significant potential in feature selection using fNIRS data, and represents a novel research approach. Future studies should explore predictors of academic achievement and cognitive ability.
... A particularly important model that can be constructed from out-of-plane spins is the canonical two-dimensional (2D) Ising model [36][37][38] . Being universally complete, so that any other statistical model 39,40 or Boolean circuit 41,42 can be derived from it, the 2D Ising model is of fundamental significance for statistical physics and has been used to model numerous physical, mathematical and biological processes [43][44][45] . ...
Arrays of coupled nanomagnets have wide-ranging fundamental and practical applications in artificial spin ices, neuromorphic computing and spintronics. However, lacking in these fields are nanomagnets with perpendicular magnetic anisotropy with sufficient magnetostatic interaction. This would not only open up new possibilities for artificial spin ice geometries but also enable novel coupling methods for applications. Here we demonstrate a method to engineer the energy landscape of artificial spin lattices with perpendicular magnetic anisotropy. With this, we are able to realize for the first time magnetostatically-coupled 2D lattices of out-of-plane Ising spins that spontaneously order at room temperature on timescales that can be precisely engineered. We show how this property, together with straightforward electrical interfacing, make this system a promising platform for reservoir computing. Our results open the way to investigate the thermodynamics of out-of-plane magnetostatically coupled nanomagnet arrays with novel spin ice geometries, as well as to exploit such nanomagnet arrays in unconventional computing, taking advantage of the adjustable temporal dynamics and strong coupling between nanomagnets.
... The texture synthesis approach is efficient due to it being image-based (and thus, material agnostic) and this method is known to preserve materials descriptors, especially higher-order correlation functions that cannot be achieved via lower-order feature optimization-based approaches. MRFbased texture synthesis model, in particular, is based on the use of a high-order Ising structure [22] to represent an image of × pixels as × lattice structure. Over the constructed Ising model, a Markov property is applied, and it states that the probability of a pixel coloring ( ) is conditionally independent of all other values in the lattice structure, except its neighbors. ...
... Determining the lowest energy state of a many-body system is one of the most fundamental problems in science and engineering. This type of problem arises in the study of the Ising model [9], graphical modeling [24], sensor network localization [16], and the structure from motion problem [18], to name a few. In these problems, one is usually concerned with minimizing an energy function E. With the exception of some simple cases, the energy landscape of E is plagued with spurious local minima. ...
Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a polynomial-time solvable semidefinite program (SDP) that provides a lower bound for the energy can be derived. In this paper, we propose accelerating the solution of such an SDP relaxation by imposing a hierarchical structure on the positive semidefinite (PSD) primal and dual variables. Furthermore, these matrices can be updated efficiently using the algebra of the compressed representations within an augmented Lagrangian method. We achieve quadratic and even near-linear time per-iteration complexity. Through experimentation on the quantum transverse field Ising model, we showcase the capability of our approach to provide a sufficiently accurate lower bound for the exact ground-state energy.
... The edge weights describe their interaction strengths. 36 The Ising Hamiltonian describes the total energy of this system and can be written as, ...
This paper proposes contributions to the efficient wave digital (WD) modeling of large oscillator networks which are emerging as energy‐efficient alternatives to traditional computers. The WD concept enables in‐operando parameter tuning, real‐time testing, and the associated algorithms are highly parallelizable. We present a general electrical model of N‐shaped nonlinearities that are commonly found in nonlinear oscillators. Our model offers the flexibility to design the current–voltage characteristic based on specific requirements. We show how this model can be used to derive efficient and explicit WD algorithms for nonlinear oscillators. Furthermore, we propose the use of lossless transmission lines between the oscillators and the coupling network to obtain an ideal circuit for an oscillator network that can function as an Ising machine and be efficiently and exactly evaluated in the WD domain. The proposed algorithms are compared against the classical method involving iterative techniques, and their capabilities are evaluated through the emulation of a single FitzHugh‐Nagumo oscillator as well as an Ising machine involving transmission lines. In the latter case, we show that, for large networks, the proposed methods decrease the runtime by up to 75% compared to using iterative techniques.
... To solve an optimization problem on a quantum computer, we need to turn it into a problem of measurement of a quantum Hamiltonian, which is the system's total energy. Finding the maximum cat of the graph can be easily mapped to a problem of maximizing the Hamiltonian of an Ising model [2]. The Ising model is an example of a lattice spin model. ...
This is max-cut related problem which we can solve by using QAOA quantum computing based algorithm.
... In practice, one does not have access to such knowledge but needs to estimate the likelihood of W, given a single sample W, and then plug in (4) Accounting for the dependencies of binary random variables turns out to be even more challenging in our context because all the binary random variables in W are embedded in a tensor grid with ultra-high dimensionality. Fortunately, the Ising model (Cipra 1987) provides one way of modeling dependent binary random variables on a lattice grid. The binary random variables here are the indicators of data missingness instead of atomic spins in ferromagnetism but a similar idea applies to our modeling context. ...
Tensor data, or multi-dimensional array, is a data format popular in multiple fields such as social network analysis, recommender systems, and brain imaging. It is not uncommon to observe tensor data containing missing values and tensor completion aims at estimating the missing values given the partially observed tensor. Sufficient efforts have been spared on devising scalable tensor completion algorithms but few on quantifying the uncertainty of the estimator. In this paper, we nest the uncertainty quantification (UQ) of tensor completion under a split conformal prediction framework and establish the connection of the UQ problem to a problem of estimating the missing propensity of each tensor entry. We model the data missingness of the tensor with a tensor Ising model parameterized by a low-rank tensor parameter. We propose to estimate the tensor parameter by maximum pseudo-likelihood estimation (MPLE) with a Riemannian gradient descent algorithm. Extensive simulation studies have been conducted to justify the validity of the resulting conformal interval. We apply our method to the regional total electron content (TEC) reconstruction problem.
... The Ising model is a statistical description of the random spin phase transition in a ferromagnetic lattice [42]. Assuming that there are N lattice sites in the Ising model, the Hamiltonian of the ensemble without the influence of an external magnetic field can be written as: ...
Gain-dissipative Ising machines (GIMs) are dedicated devices that can rapidly solve combinatorial optimization problems. The noise intensity in traditional GIMs should be significantly smaller than its saturated fixed-point amplitude, indicating a lower noise margin. To overcome the existing limit, this work proposes an overdamped bistability-based GIM (OBGIM). Numerical test on uncoupled spin network show that the OBGIM has a different bifurcation dynamics from that of the traditional GIM. Moreover, the domain clustering dynamics on non-frustrated network proves that the overdamped bistability enables the GIM to suppress noise-induced random spin-state switching effectively; thus, it can function normally in an environment with a relatively large noise level. Besides, some prevalent frustrated graphs from the SuiteSparse Matrix Collection were adopted as MAXCUT benchmarks. The results show that the OBGIM can induce stochastic resonance phenomenon when solving difficult benchmarks. Compared with the traditional GIM, this characteristic makes the OBGIM achieve comparable solution accuracy in larger noise environment, thus achieving strong noise robustness.
... D-Wave QPU is built based on the Ising model from statistical mechanics. This model consists of lattice structures with nodes and edges 42 . In each node, spins are located as a binary variable. ...
Diverse cases regarding the impact, with its related factors, of the COVID-19 pandemic on mental health have been reported in previous studies. In this study, multivariable datasets were collected from 751 college students who could be easily affected by pandemics based on the complex relationships between various mental health factors. We utilized quantum annealing (QA)-based feature selection algorithms that were executed by commercial D-Wave quantum computers to determine the changes in the relative importance of the associated factors before and after the pandemic. Multivariable linear regression (MLR) and XGBoost models were also applied to validate the QA-based algorithms. Based on the experimental results, we confirm that QA-based algorithms have comparable capabilities in factor analysis research to the MLR models that have been widely used in previous studies. Furthermore, the performance of the QA-based algorithms was validated through the important factor results from the algorithms. Pandemic-related factors (e.g., confidence in the social system) and psychological factors (e.g. decision-making in uncertain situations) were more important in post-pandemic conditions. Although the results should be validated using other mental health variables or national datasets, this study will serve as a reference for researchers regarding the use of the quantum annealing approach in factor analysis with validation through real-world survey dataset analysis.
... The realizability of the 1D critical state is unusual because it is commonly thought that phase transitions cannot exist in 1D systems, but this is only true when the interactions are sufficiently short-ranged. 97,98 Indeed, in the Appendix, we show that the effective pair potential has a long-ranged attractive part, i.e., v(r) ∼ −r −(d−1/2) as r → ∞. This long-range behavior in the interaction explains why our 1D, 2D, and 3D critical-point states do not belong to the Ising universality class that is characterized by short-ranged interactions. ...
The knowledge of exact analytical functional forms for the pair correlation function g2(r) and its corresponding structure factor S(k) of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing database of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform, and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperatures that precisely match targeted forms for both g2(r) and S(k). Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve many-particle systems with a wide range of translational order and self-diffusion coefficients D, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics.
... In order to quantify the cation ordering of the dolomite surface configuration, we use the antiferromagnetic Ising model (75) and define 'cation ordering' as Eqs. [3] and [4] where N is the total number of cation sites in the surface. ...
Crystals grow in supersaturated solutions. A mysterious counterexample is dolomite CaMg(CO 3 ) 2 , a geologically abundant sedimentary mineral that does not readily grow at ambient conditions, not even under highly supersaturated solutions. Using atomistic simulations, we show that dolomite initially precipitates a cation-disordered surface, where high surface strains inhibit further crystal growth. However, mild undersaturation will preferentially dissolve these disordered regions, enabling increased order upon reprecipitation. Our simulations predict that frequent cycling of a solution between supersaturation and undersaturation can accelerate dolomite growth by up to seven orders of magnitude. We validated our theory with in situ liquid cell transmission electron microscopy, directly observing bulk dolomite growth after pulses of dissolution. This mechanism explains why modern dolomite is primarily found in natural environments with pH or salinity fluctuations. More generally, it reveals that the growth and ripening of defect-free crystals can be facilitated by deliberate periods of mild dissolution.
... Applying the calculated total energy of the DFT to the Ising model will allow us to estimate the magnetic exchange interaction. For this purpose, the magnetic exchange interaction can be calculated as follow [47]: ...
Strong correlated materials are emerging materials in low magnetocaloric cooling applications. The development of these fields necessitates materials with appealing magnetic, and magnetocaloric properties. In this context, Density Functional Theory (DFT) combined with Monte Carlo simulations (MCs) are performed to comprehensively examine various physical, and magnetic features of the orthorhombic perovskite-type RTiO3. A metallic behavior of (Ho, Er)TiO3 has been found via the density of state using GGA+U approach. X-ray magnetic circular dichroism (XMCD) was carried out to identify the magnetic contributions of Ho, Er, and Ti atoms in HoTiO3, and ErTiO3 compounds. The spin-orbit coupling was added in our calculations to study and evaluate the magnetocrystalline anisotropies of RTiO3 with (R= Ho, Er). We noticed an easy magnetization axis along the b-direction for HoTiO3, and along the c-direction for ErTiO3. Low magnetocrystalline anisotropy energies were found compared to other perovskite oxides such as RVO3, and RMnO3 indicating that a weak rotating magnetocaloric effect (RMCE) is expected in (Ho, Er)TiO3. Moreover, the exchange coupling interactions were calculated using Ising model were the exchange interaction coupling between transition metal elements (Ti-Ti) is ferromagnetic, and larger than the other exchange interactions in (Ho, Er)TiO3. Furthermore, these exchange coupling interactions were utilized in our model for MCs within the Heisenberg model, allowing us to understand the driving mechanisms underlying magnetocaloric and magnetic behaviors in (Ho, Er)TiO3.
... The 3R3XOR problem is a methodology for generating benchmark problem sets for Ising machines devices designed to solve discrete optimization problems cast as Ising models introduced by Hen (2019). The Ising model, named after Ernst Ising, is concerned with the physics of magnetic-driven phase transitions (Cipra 1987). The Ising model is defined on a lattice, where a spin s i ∈ {−1, 1} is located on each lattice site (Block and Preis 2012). ...
Simulated Annealing using Metropolis steps at decreasing temperatures is widely used to solve complex combinatorial optimization problems (Kirkpatrick et al. in Science 220(4598):671–680, 1983). To improve its efficiency, we can use the Rejection-Free version of the Metropolis algorithm, which avoids the inefficiency of rejections by considering all the neighbors at every step (Rosenthal et al. in Comput Stat 36(4):2789–2811, 2021). To prevent the algorithm from becoming stuck in local extreme areas, we propose an enhanced version of Rejection-Free called Partial Neighbor Search, which only considers random parts of the neighbors while applying Rejection-Free. We demonstrate the superior performance of the Rejection-Free Partial Neighbor Search algorithm compared to the Simulation Annealing and Rejection-Free with several examples, such as the QUBO question, the Knapsack problem, the 3R3XOR problem, and the quadratic programming.
... The (classical) Ising model was first introduced as a mathematical model of ferromagnetism [112]. The variables take values in a discrete (binary) set σ i = {±1}, and are typically referred to as spin, since in the physical model they describe the atomic spin of the particles. ...
Collateral optimization refers to the systematic allocation of financial assets to satisfy obligations or secure transactions, while simultaneously minimizing costs and optimizing the usage of available resources. This involves assessing number of characteristics, such as cost of funding and quality of the underlying assets to ascertain the optimal collateral quantity to be posted to cover exposure arising from a given transaction or a set of transactions. One of the common objectives is to minimise the cost of collateral required to mitigate the risk associated with a particular transaction or a portfolio of transactions while ensuring sufficient protection for the involved parties. Often, this results in a large-scale combinatorial optimization problem. In this study, we initially present a Mixed Integer Linear Programming (MILP) formulation for the collateral optimization problem, followed by a Quadratic Unconstrained Binary optimization (QUBO) formulation in order to pave the way towards approaching the problem in a hybrid-quantum and NISQ-ready way. We conduct local computational small-scale tests using various Software Development Kits (SDKs) and discuss the behavior of our formulations as well as the potential for performance enhancements. We find that while the QUBO based approaches fail to find the global optima in the small scale experiments, they are reasonably close suggesting their potential for large instances. We further survey the recent literature that proposes alternative ways to attack combinatorial optimization problems suitable for collateral optimization.
... First, a large combinatorial optimization problem is formulated using an undirected graphical representation, with vertices representing the spin states and edges representing the coupling weights. The state of spins s i (s i = ±1) determine the objective function to be minimized, given by the Ising Hamiltonian function 7,8 ...
Quantum-inspired computing systems can be used to efficiently solve combinatorial optimization problems. In developing such systems, a key challenge is the creation of large hardware topologies with all-to-all node connectivity that allow arbitrary problem graphs to be directly mapped to the hardware. Here we report a physics-based Ising solver chip fabricated in a standard 1.2 V, 65 nm complementary metal–oxide–semiconductor technology. The chip features an all-to-all architecture with 48 spins and a highly uniform coupling circuit with integer weights ranging from −14 to +14. The all-to-all architecture strongly couples a horizontal oscillator with a vertical oscillator so that each horizontal–vertical oscillator pair intersects with all the other pairs in a crossbar-style array and allows any graph with up to 48 nodes to be directly mapped to the hardware. We use the Ising solver chip to carry out statistical measurements for different problem sizes, graph densities, operating temperatures and problem instances.
... There exist embeddings for many classical combinatorial optimization problems [29]. Because of their similarity to the Ising Model [13], QUBO formulations are often used to solve problems on Ising machines such as D-Wave's Quantum Annealer [24] and Fujitsu's Digital Annealer [22]. These solvers can outperform many classical algorithms implemented on general-purpose computers [4,5,30]. ...
... There are also clear similarities with theoretical physics studies of spin-glass systems, such as the Ising-Lenz model (see e.g. Cipra (1987)), and with studies of cascades and epidemics on abstract networks (e.g. Watts (2002)). ...
... The precision matrix of a GMRF determines the underlying graph structure and is often estimated using the well-known graphical Lasso algorithm (Friedman et al., 2008). Other PGMs, such as different variants of GMRFs, discrete Markov random fields and Bayesian networks, also have found their applications in many fields, including but not limited to physics (Cipra, 1987), genetics (Yang et al., 2015;Park et al., 2017), microbial ecology (Kurtz et al., 2015;Biswas et al., 2016;Chiquet et al., 2019) and causal inference (Zhang & Poole, 1996;Yu et al., 2004). ...
A common challenge in applying graph machine learning methods is that the underlying graph of a system is often unknown. Although different graph inference methods have been proposed for continuous graph signals, inferring the graph structure underlying other types of data, such as discrete counts, is under-explored. In this paper, we generalize a graph signal processing (GSP) framework for learning a graph from smooth graph signals to the exponential family noise distribution to model various data types. We propose an alternating algorithm that estimates the graph Laplacian as well as the unobserved smooth representation from the noisy signals. We demonstrate in synthetic and real-world data that our new algorithm outperforms competing Laplacian estimation methods under noise model mismatch.
... To overcome these limitations, we need a system analysis method based on a system perspective, analogous to the synchronization model [11] or the Ising model [12], rather than a variable perspective like PID. Furthermore, this method should capture the nature and characteristics of the system without specifying or introducing any special variable, and also take into account all the interactive relationships among all variables in the system, including pairwise and higher-order relationships. ...
In order to characterize complex higher-order interactions among variables in a system, we introduce a new framework for decomposing the information entropy of variables in a system, termed System Information Decomposition (SID). Diverging from Partial Information Decomposition (PID) correlation methods, which quantify the interaction between a single target variable and a collection of source variables, SID extends those approaches by equally examining the interactions among all system variables. Specifically, we establish the robustness of the SID framework by proving all the information atoms are symmetric, which detaches the unique, redundant, and synergistic information from the specific target variable, empowering them to describe the relationship among variables. Additionally, we analyze the relationship between SID and existing information measures and propose several properties that SID quantitative methods should follow. Furthermore, by employing an illustrative example, we demonstrate that SID uncovers a higher-order interaction relationships among variables that cannot be captured by current measures of probability and information and provide two approximate calculation methods verified by this case. This advance in higher-order measures enables SID to explain why Holism posits that some systems cannot be decomposed without loss of characteristics under existing measures, and offers a potential quantitative framework for higher-order relationships across a broad spectrum of disciplines.
... The (classical) Ising model was first introduced as a mathematical model of ferromagnetism [112]. The variables take values in a discrete (binary) set σ i = {±1}, and are typically referred to as spin, since in the physical model they describe the atomic spin of the particles. ...
Collateral optimization refers to the systematic allocation of financial assets to satisfy obligations or secure transactions, while simultaneously minimizing costs and optimizing the usage of available resources. {This involves assessing number of characteristics, such as cost of funding and quality of the underlying assets to ascertain the optimal collateral quantity to be posted to cover exposure arising from a given transaction or a set of transactions. One of the common objectives is to minimise the cost of collateral required to mitigate the risk associated with a particular transaction or a portfolio of transactions while ensuring sufficient protection for the involved parties}. Often, this results in a large-scale combinatorial optimization problem. In this study, we initially present a Mixed Integer Linear Programming (MILP) formulation for the collateral optimization problem, followed by a Quadratic Unconstrained Binary optimization (QUBO) formulation in order to pave the way towards approaching the problem in a hybrid-quantum and NISQ-ready way. We conduct local computational small-scale tests using various Software Development Kits (SDKs) and discuss the behavior of our formulations as well as the potential for performance enhancements. We further survey the recent literature that proposes alternative ways to attack combinatorial optimization problems suitable for collateral optimization.
Advances in next‐generation sequencing technology have enabled the high‐throughput profiling of metagenomes and accelerated microbiome studies. Recently, there has been a rise in quantitative studies that aim to decipher the microbiome co‐occurrence network and its underlying community structure based on metagenomic sequence data. Uncovering the complex microbiome community structure is essential to understanding the role of the microbiome in disease progression and susceptibility. Taxonomic abundance data generated from metagenomic sequencing technologies are high‐dimensional and compositional, suffering from uneven sampling depth, over‐dispersion, and zero‐inflation. These characteristics often challenge the reliability of the current methods for microbiome community detection. To study the microbiome co‐occurrence network and perform community detection, we propose a generalized Bayesian stochastic block model that is tailored for microbiome data analysis where the data are transformed using the recently developed modified centered‐log ratio transformation. Our model also allows us to leverage taxonomic tree information using a Markov random field prior. The model parameters are jointly inferred by using Markov chain Monte Carlo sampling techniques. Our simulation study showed that the proposed approach performs better than competing methods even when taxonomic tree information is non‐informative. We applied our approach to a real urinary microbiome dataset from postmenopausal women. To the best of our knowledge, this is the first time the urinary microbiome co‐occurrence network structure in postmenopausal women has been studied. In summary, this statistical methodology provides a new tool for facilitating advanced microbiome studies.
Repulsive colloids can form liquid crystals (LCs) that combine fluidity and structural order. Screening repulsion leads to abrupt collapse of LC, losing fluidity either by gelation or frustrating structural order by coagulation. However, the evolution of liquid crystallinity before the transition from overall repulsion to attraction remains unclear. Here we find an intermediate LC state of graphene oxide (GO), named as fragmentated LC (FLC). FLC features fragmented domains down to single entity size but keeps good fluidity contrary to gel and coagulation. In FLC, the balanced interaction keeps single-layer dispersed state of GO and triggers transient networks. GO FLCs surprisingly serve as a peculiar processing state towards amorphous but compact structures with both high strength and toughness, beyond brittle crystalline structures assembled from ordinary nematic LCs. Our findings enable precisely modulating LC directors down to the molecular size limit and provide a new design method for exotic amorphous materials.
The emergence of quantum computing proposes a revolutionary paradigm that can radically transform numerous scientific and industrial application domains. However, realizing this promise in industrial applications is far from being practical today. In this paper, we discuss industry experiences with respect to quantum computing, and the gap between quantum software engineering research and state-of-the-practice in industry-scale quantum computing.
Ising machines have received growing interest as efficient and hardware-friendly solvers for combinatorial optimization problems (COPs). They search for the absolute or approximate ground states of the Ising model with a proper annealing process. In contrast to Ising machines built with superconductive or optical circuits, complementary metal–oxide–semiconductor (CMOS) Ising machines offer inexpensive fabrication, high scalability, and easy integration with mainstream semiconductor chips. As low-energy and CMOS-compatible emerging technologies, spintronics and phase-transition devices offer functionalities that can enhance the scalability and sampling performance of Ising machines. In this article, we survey various approaches in the process flow for solving COPs using CMOS, hybrid CMOSspintronic, and phase-transition devices. First, the methods for formulating COPs as Ising problems and embedding Ising formulations to the topology of the Ising machine are reviewed. Then, Ising machines are classified by their underlying operational principles and reviewed from a perspective of hardware implementation. CMOS solutions are advantageous with denser connectivity, whereas hybrid CMOS-spintronic and phase-transition device-based solutions show great potential in energy efficiency and high performance. Finally, the challenges and prospects are discussed for the Ising formulation, embedding process, and implementation of Ising machines.
In this work, an innovative design model aimed at enhancing the efficacy of ground-state probabilistic logic with a binary energy landscape (GSPL-BEL) is presented. This model enables the direct conversion of conventional CMOS-based logic circuits into corresponding probabilistic graphical representations based on a given truth table. Compared to the conventional approach of solving the configuration of Ising model-basic probabilistic gates through linear programming, our model directly provides configuration parameters with embedded many-body interactions. For larger-scale probabilistic logic circuits, the GSPL-BEL model can fully utilize the dimensions of many-body interactions, achieving minimal node overhead while ensuring the simplest binary energy landscape and circumventing additional logic synthesis steps. To validate its effectiveness, hardware implementations of probabilistic logic gates were conducted. Probabilistic bits were introduced as Ising cells, and cascaded conventional XNOR gates along with passive resistor networks were precisely designed to realize many-body interactions. HSPICE circuit simulation results demonstrate that the probabilistic logic circuits designed based on this model can successfully operate in free, forward, and reverse modes, exhibiting the simplest binary probability distributions. For a 2-bit × 2-bit integer factorizer involving many-body interactions, compared to the logic synthesis approach, the GSPL-BEL model significantly reduces the number of consumed nodes, the solution space (in the free-run mode), and the number of energy levels from 12, 4096, and 9–8, 256, and 2, respectively. Our findings demonstrate the significant potential of the GSPL-BEL model in optimizing the structure and performance of probabilistic logic circuits, offering a new robust tool for the design and implementation of future probabilistic computing systems.
Inspired by statistical thermodynamics, we presume that neuron system has equilibrium condition with respect to neural firing. We show that, even with dynamically changeable neural connections, it is inevitable for neural firing to evolve to equilibrium. To study the dynamics between neural firing and neural connections, we propose an extended communication system where noisy channel has the tendency towards fixed point, implying that neural connections are always attracted into fixed points such that equilibrium can be reached. The extended communication system and its mathematics could be useful back in thermodynamics.
A land pattern change represents a globally significant trend with implications for the environment, climate, and societal well-being. While various methods have been developed to predict land change, our understanding of the underlying change processes remains inadequate. To address this issue, we investigate the suitability of the 2D kinetic Ising model (IM), an idealized model from statistical mechanics, for simulating land change dynamics. We test the IM on a variety of diverse thematic contexts. Specifically, we investigate four sites characterized by distinct patterns, presumably driven by different physical processes. Each site is observed on eight occasions between 2001 and 2019. Given the observed pattern at times t i , i = 1, …, 7, we find two parameters of the IM such that the model-evolved land pattern at t i +1 resembles the observed land pattern at that time. Our findings indicate that the IM produces approximate matches to the observed patterns in terms of layout, composition, texture, and patch size distributions. Notably, the IM simulations even achieve a high degree of cell-scale pattern accuracy in two of the sites. Nevertheless, the IM has certain limitations, including its inability to model linear features, account for the formation of new large patches, and handle pattern shifts.
A few minor modifications are made in the Peierls argument that a two-dimensional Ising ferromagnet possesses a spontaneous moment at sufficiently low temperatures, in order to make the proof quite rigorous.
This article is an interdisciplinary review of lattice gauge theory
and spin systems. It discusses the fundamentals, both physics and
formalism, of these related subjects. Spin systems are models of
magnetism and phase transitions. Lattice gauge theories are cutoff
formulations of gauge theories of strongly interacting particles.
Statistical mechanics and field theory are closely related subjects,
and the connections between them are developed here by using the
transfer matrix. Phase diagrams and critical points of continuous
transitions are stressed as the keys to understanding the character
and continuum limits of lattice theories. Concepts such as duality,
kink condensation, and the existence of a local, relativistic field
theory at a critical point of a lattice theory are illustrated in
a thorough discussion of the two-dimensional Ising model. Theories
with exact local (gauge) symmetries are introduced following Wegner's
Ising lattice gauge theory. Its gauge-invariant "'loop"' correlation
function is discussed in detail. Three?dimensional Ising gauge theory
is studied thoroughly. The renormalization group of the two dimensional
planar model is presented as an illustration of a phase transition
driven by the condensation of topological excitations. Parallels
are drawn to Abelian lattice gauge theory in four dimensions. Non-Abelian
gauge theories are introduced and the possibility of quark confinement
is discussed. Asymptotic freedom of O(n) Heisenberg spin systems
in two dimensions is verified for n >= and is explained in simple
terms. The direction of present-day research is briefly reviewed.
In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the two-dimensional square net as the matrix order gets large. It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at JkTc=0.8814 if we denote by J the coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at Tc, as, e.g., a jump in the specific heat. However, the information thus gathered by rigorous analytic methods remains incomplete.
Many physico-chemical systems can be represented more or less accurately by a lattice arrangement of molecules with nearest-neighbor interactions. The simplest and most popular version of this theory is the so-called "Ising model," discussed by Ernst Ising in 1925 but suggested earlier (1920) by Wilhelm Lenz. Major events in the subsequent history of the Lenz-Ising model are reviewed, including early approximate methods of solution, Onsager's exact result for the two-dimensional model, the use of the mathematically equivalent "lattice gas" model to study gas-liquid and liquid-solid phase transitions, and recent progress in determining the singularities of thermodynamic and magnetic properties at the critical point. Not only is there a wide range of possible physical applications of the model, there is also an urgent need for the application of advanced mathematical techniques in order to establish its exact properties, especially in the neighborhood of phase transitions where approximate methods are unreliable.
The spontaneous magnetization of a two-dimensional Ising model is calculated exactly. The result also gives the long-range order in the lattice.
One of the earliest proofs that a physical model exhibits a spontaneous magnetization was Peierls's [Cambridge Phil. Soc. 32, 477 (1986)] proof for the classical, two-dimensional, spin one-half Ising [Z. Physik 31, 253 (1925)] ferromagnet. The simplicity and beauty of this argument makes it an ideal topic for discussion in a statistical mechanics course. Recent work has extended the application of the Peierls technique to more sophisticated models. An extension to the classical two-dimensional, spin one Ising ferromagnet is discussed here. A review of the original Peierls argument as made rigorous by Griffiths [Phys. Rev. 136, A437 (1964)] is also presented. Other models to which the technique has been applied are also discussed briefly.
I The Construction, and Other General Results.- 1. Markov Processes and Their Semigroups.- 2. Semigroups and Their Generators.- 3. The Construction of Generators for Particle Systems.- 4. Applications of the Construction.- 5. The Martingale Problem.- 6. The Martingale Problem for Particle Systems.- 7. Examples.- 8. Notes and References.- 9. Open Problems.- II Some Basic Tools.- 1. Coupling.- 2. Monotonicity and Positive Correlations.- 3. Duality.- 4. Relative Entropy.- 5. Reversibility.- 6. Recurrence and Transience of Reversible Markov Chains.- 7. Superpositions of Commuting Markov Chains.- 8. Perturbations of Random Walks.- 9. Notes and References.- III Spin Systems.- 1. Couplings for Spin Systems.- 2. Attractive Spin Systems.- 3. Attractive Nearest-Neighbor Spin Systems on Z1.- 4. Duality for Spin Systems.- 5. Applications of Duality.- 6. Additive Spin Systems and the Graphical Representation.- 7. Notes and References.- 8. Open Problems.- IV Stochastic Ising Models.- 1. Gibbs States.- 2. Reversibility of Stochastic Ising Models.- 3. Phase Transition.- 4. L2 Theory.- 5. Characterization of Invariant Measures.- 6. Notes and References.- 7. Open Problems.- V The Voter Model.- 1. Ergodic Theorems.- 2. Properties of the Invariant Measures.- 3. Clustering in One Dimension.- 4. The Finite System.- 5. Notes and References.- VI The Contact Process.- 1. The Critical Value.- 2. Convergence Theorems.- 3. Rates of Convergence.- 4. Higher Dimensions.- 5. Notes and References.- 6. Open Problems.- VII Nearest-Particle Systems.- 1. Reversible Finite Systems.- 2. General Finite Systems.- 3. Construction of Infinite Systems.- 4. Reversible Infinite Systems.- 5. General Infinite Systems.- 6. Notes and References.- 7. Open Problems.- VIII The Exclusion Process.- 1. Ergodic Theorems for Symmetric Systems.- 2. Coupling and Invariant Measures for General Systems.- 3. Ergodic Theorems for Translation Invariant Systems.- 4. The Tagged Particle Process.
We present a relatively simple derivation, based on Burgoyne's combinational method, of the Onsager formula for the partition function of the two-dimensional square Ising model.
An elementary method which yields the partition function of a
two-dimensional Ising model is described. The method is purely
combinatorial and does not involve any of the algebraic apparatus used
in this connection by Onsager and Kaufman.
The study of two-dimensional matter would seem to be a completely theoretical branch of physics, were it not that such matter actually exists. There are two-dimensional gases, liquids, and crystalline solids; the analogues of conventional bulk phases, two-dimensional lattice gases, realizations of important statistical models; two-dimensional magnets, solutions, and even amorphous glasses. Indeed, much of the variety of states of bulk substances has been created in two-dimensional matter and is intensively studied in many laboratories throughout the world.
A selection of recent work by various authors concerning spin ; Ising modls with pair couplings is reviewed. Topics addressed include: the universality of corrections-to-scaling and nonlinear scaling fields in planar Ising models; aspects of many-particle correlation functions; the testing and verification of scaling hypotheses for local magnetizations in the presence of walls and boundaries, including the singular nature of the critical region scaling functions and the decay of the influence of a far wall; a soluble model of a surface roughening transition; nonuniversal critical effects of one-dimensional perturbations in planar Ising models; the effects of spatially anisotropic further neighbor interactions, in particular ANNNI models in d 2 dimensions and soluble mock ANNNI models in 2 and 3 dimensions.RésuméOn examine un choix d'oeuvres récents par auteurs diverses, concernant modèles d'Ising au spin-; avec coupling de paires. Parmis les sujets inclus sont: l'universalité des ajustements au “scaling”, et champs non-linéaire dans modèles planaire d'Ising: aspects des fonctions de corrélation des plusiers-corps; l'épreuve et la vérification d'hypothèses de “scaling”, en cas d'aimantation locale en présence des parois et limites, comprenant la nature singulière des fonctions d'échelle dans la région critique, et la diminuation des effets d'une parois lointaine; une modèle soluble d'un transition d'une surface rugueuse; les effets non-universale dans la région critique des perturbations une dimensionelle dans modèles d'Ising planaire; les effets des interactions anisotropique en espace avec voisins lointains, notament des modèles ANNNI en dimensions d 2, et une modèle ANNNI simulée en 2 et 3 dimensions.
A theory of equations of state and phase transitions is developed
that describes the condensed as well as the gas phases and the transition
regions. The thermodynamic properties of an infinite sample are studied
rigorously and Mayer's theory is re-examined.
The partition function of a two-dimensional "ferromagnetic" with scalar
"spins" (Ising model) is computed rigorously for the case of vanishing
field. The eigenwert problem involved in the corresponding computation
for a long strip crystal of finite width (n atoms), joined straight
to itself around a cylinder, is solved by direct product decomposition;
in the special case n=â an integral replaces a sum. The choice of
different interaction energies (±J,±Jâ²) in the (0 1) and (1 0) directions
does not complicate the problem. The two-way infinite crystal has
an order-disorder transition at a temperature T=Tc given by the condition
sinh(2J / kTc) sinh(2Jâ² / kTc)=1. The energy is a continuous function
of T; but the specific heat becomes infinite as -log |T-Tc|. For
strips of finite width, the maximum of the specific heat increases
linearly with log n. The order-converting dual transformation invented
by Kramers and Wannier effects a simple automorphism of the basis
of the quaternion algebra which is natural to the problem in hand.
In addition to the thermodynamic properties of the massive crystal,
the free energy of a (0 1) boundary between areas of opposite order
is computed; on this basis the mean ordered length of a strip crystal
is (exp (2J / kT) tanh(2Jâ² / kT))n.
The problems of an Ising model in a magnetic field and a lattice gas
are proved mathematically equivalent. From this equivalence an example
of a two-dimensional lattice gas is given for which the phase transition
regions in the p-v diagram is exactly calculated.
A theorem is proved which states that under a class of general conditions
the roots of the grand partition function always lie on a circle.
Consequences of this theorem and its relation with practical approximation
methods are discussed. All the known exact results about the two-dimensional
square Ising lattice are summarized, and some new results are quoted.
The partition function for a two-dimensional binary lattice is evaluated
in terms of the eigenvalues of the 2n-dimensional matrix V characteristic
for the lattice. Use is made of the properties of the 2n-dimensional
"spin"-representation of the group of rotations in 2n-dimensions.
In consequence of these properties, it is shown that the eigenvalues
of V are known as soon as one knows the angles of the 2n-dimensional
rotation represented by V.
Together with the eigenvalues of V, the matrix Ψ which diagonalizes
V is obtained as a spin-representation of a known rotation. The determination
of Ψ is needed for the calculation of the degree of order.
The approximation, in which all the eigenvalues of V but the largest
are neglected, is discussed, and it is shown that the exact partition
function does not differ much from the approximate result.
- L D Landau
- E M Lifshitz
L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon Press, 1980.
Cooperative Phenomena near Phase Transitions (A bibliography with selected readings), M.I.T. Press, 1973. 41. , Introduction to Phase Transitions and Critical Phenomena
- H E Stanley
H. E. Stanley, ed., Cooperative Phenomena near Phase Transitions (A bibliography with selected
readings), M.I.T. Press, 1973.
41.
, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1985.
- G H Wannier
G. H. Wannier, Statistical Physics, Wiley, 1966.
- S.-K Ma
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