Potts Model Research Articles

Holliday junctions in the set of DNA molecules for new translation-invariant Gibbs measures of the Potts model

Holliday junctions in the set of DNA molecules for new translation-invariant Gibbs measures of the Potts model

Holliday junctions in the set of DNA molecules for new translation-invariant Gibbs measures of the Potts model

Молекула ДНК рассматривается как конфигурация модели Поттса на путях дерева Кэли. Для этой модели изучаются новые трансляционно-инвариантные меры Гиббса. Найдены точные значения параметра, устанавливающего единственность трансляционно-инвариантных мер Гиббса. Каждая такая мера описывает состояния (фазу) множества молекул ДНК. Эти меры Гиббса применяются для изучения вероятностных распределений соединения Холлидея в молекулах ДНК.

Emergence of compact disordered phase in a polymer Potts model.

One of the central problems in epigenetics is how epigenetic modification patterns and chromatin structure are regulated in the cell nucleus. The polymer Potts model, a recently studied model of chromatins, is introduced with an offset in the interaction energy as a parameter, and the equilibrium properties are investigated using the mean-field analysis of the lattice model and molecular dynamics simulations of the off-lattice model. The results show that in common with both models, a phase emerges, which could be called the compact-disordered phase, in which the polymer conformation is compact and the epigenetic modification pattern is disordered, depending on the offset in the interaction energy and the fraction of the modified nucleosomes.

Complex dynamics approach to dynamical quantum phase transitions: The Potts model

Complex dynamics approach to dynamical quantum phase transitions: The Potts model

Non-equilibrium dynamics and phase transitions in Potts model and interacting Ehrenfest urn model

We show that the recently proposed interacting Ehrenfest M-urn model at equilibrium (Cheng et al., 2020) can be exactly mapped to a mean-field M-state Potts model. By exploiting this correspondence, we show that the M-state Potts model with M≥3, with transition rates motivated by the non-equilibrium urn model, can exhibit rich non-equilibrium spin dynamics such as non-equilibrium steady states and non-equilibrium periodic states. Monte Carlo simulations of the 3-state Potts model are performed to demonstrate explicitly the first-order transitions for the equilibrium and non-equilibrium steady states, as well as the far-from-equilibrium periodic states.

Energy landscape and metastability of stochastic Ising and Potts models on three-dimensional lattices without external fields

Energy landscape and metastability of stochastic Ising and Potts models on three-dimensional lattices without external fields

CMOS Single-Photon Avalanche Diode Circuits for Probabilistic Computing.

Intrinsically random hardware devices are increasingly attracting attention for their potential use in probabilistic computing architectures. One such device is the single-photon avalanche diode (SPAD) and an associated functional unit, the variable-rate SPAD circuit (VRSC), recently proposed by us as a source of randomness for sampling and annealing Ising and Potts models. This work develops a more advanced understanding of these VRSCs by introducing several VRSC design options and studying their tradeoffs as implemented in a 65-nm CMOS process. Each VRSC is composed of a SPAD and a processing circuit. Combinations of three different SPAD designs and three different types of processing circuits were evaluated on several metrics such as area, speed, and variability. Measured results from the SPAD design space show that even extremely small SPADs are suitable for probabilistic computing purposes, and that high dark count rates are not detrimental either, so SPADs for probabilistic computing are actually easier to integrate in standard CMOS processes. Results from the circuit design space show that the time-to-analog-based designs introduced in this work can produce highly exponential and analytical transfer functions, but that the less analytically tractable output of the previously proposed filter-based designs can achieve less variability in a smaller footprint. Probabilistic bits (P-bits) composed of the fabricated VRSCs achieve bit flip rates of 50 MHz and allow at least one order of magnitude of control over their simulated annealing temperature.

The cellular Potts model on disordered lattices.

The cellular Potts model, also known as the Glazier-Graner-Hogeweg model, is a lattice-based approach by which biological tissues at the level of individual cells can be numerically studied. Traditionally, a square or hexagonal underlying lattice structure is assumed for two-dimensional systems, and this is known to introduce artifacts in the structure and dynamics of the model tissues. That is, on regular lattices, cells can assume shapes that are dictated by the symmetries of the underlying lattice. Here, we developed a variant of this method that can be applied to a broad class of (ir)regular lattices. We show that on an irregular lattice deriving from a fluid-like configuration, two types of artifacts can be removed. We further report on the transition between a fluid-like disordered and a solid-like hexagonally ordered phase present for monodisperse confluent cells as a function of their surface tension. This transition shows the hallmarks of a first-order phase transition and is different from the glass/jamming transitions commonly reported for the vertex and active Voronoi models. We emphasize this by analyzing the distribution of shape parameters found in our state space. Our analysis provides a useful reference for the future study of epithelia using the (ir)regular cellular Potts model.

Bidirectionally validated in silico and in vitro formation of specific depth zone-derived chondrocyte spheroids and clusters.

3D multicellular self-organized cluster models, e.g., organoids are promising tools for developing new therapeutic modalities including gene and cell therapies, pharmacological mechanistic and screening assays. Various applications of these models have been used extensively for decades, however, the mechanisms of cluster formation, maintenance, and degradation of these models are not even known over in-vitro-life-time. To explore such advantageous models mimicking native tissues or organs, it is necessary to understand aforementioned mechanisms. Herein, we intend to clarify the mechanisms of the formation of cell clusters. We previously demonstrated that primary chondrocytes isolated from distinct longitudinal depth zones in articular cartilage formed zone-specific spherical multicellular clusters in vitro. To elucidate the mechanisms of such cluster formation, we simulated it using the computational Cellular Potts Model with parameters were translated from gene expression levels and histological characteristics corresponding to interactions between cell and extracellular matrix. This simulation in silico was validated morphologically with cluster formation in vitro and vice versa. Since zone specific chondrocyte cluster models in silico showed similarity with corresponding in vitro model, the in silico has a potential to be used for prediction of the 3D multicellular in vitro models used for development, disease, and therapeutic models.

Phase transitions of the four-state Potts model on the square and the honeycomb lattices

Abstract It is widely believed that the phase transition for the four-state ferromagnetic Potts model on the square lattice is of the pseudo-first order. Specifically, it is expected that first-order phase transition behavior is found on small lattices and that the true nature of second-order phase transition only emerges with large system sizes. It is also intuitively expected that for other geometries, the types of the associated phase transitions should be identical to that of the square lattice. However, after simulating more than 16 million spins for the four-state Pott model, we observe that a feature of first-order phase transition persists on the square lattice. Additionally, a characteristic of second-order phase transition already appears on a small honeycomb lattice. Indications of a pseudo-first-order phase transition were not found in our investigation. This suggests that a thorough analytic calculation may be required to develop a better understanding of the presented results.

Engineering Proteins Using Statistical Models of Coevolutionary Sequence Information.

Homologous protein sequences are wonderfully diverse, indicating many possible evolutionary "solutions" to the encoding of function. Consequently, one can construct statistical models of protein sequence by analyzing amino acid frequency across a large multiple sequence alignment. A central premise is that covariance between amino acid positions reflects coevolution due to a shared functional or biophysical constraint. In this review, we describe the implementation and discuss the advantages, limitations, and recent progress on two coevolution-based modeling approaches: (1) Potts models of protein sequence (direct coupling analysis [DCA]-like), and (2) the statistical coupling analysis (SCA). Each approach detects interesting features of protein sequence and structure-the former emphasizes local physical contacts throughout the structure, while the latter identifies larger evolutionarily coupled networks of residues. Recent advances in large-scale gene synthesis and high-throughput functional selection now motivate additional work to benchmark model performance across quantitative function prediction and de novo design tasks.

Translation-invariant Gibbs measures for Potts model with competing interactions with a countable set of spin values on Cayley tree

Translation-invariant Gibbs measures for Potts model with competing interactions with a countable set of spin values on Cayley tree

Q-state Potts model from the nonperturbative renormalization group.

We study the q-state Potts model for q and the space dimension d arbitrary real numbers using the derivative expansion of the nonperturbative renormalization group at its leading order, the local potential approximation (LPA and LPA^{'}). We determine the curve q_{c}(d) separating the first [q>q_{c}(d)] and second [q<q_{c}(d)] -order phase transition regions for 2.8<d≤4. At small ε=4-d and δ=q-2 the calculation is performed in a double expansion in these parameters, and we find q_{c}(d)=2+aε^{2} with a≃0.1. For finite values of ε and δ, we obtain this curve by integrating the LPA and LPA^{'} flow equations. We find that q_{c}(d=3)=2.11(7), which confirms that the transition is of first order in d=3 for the three-state Potts model.

Learning the phase transitions of two-dimensional Potts model with a pre-trained one-dimensional neural network

Conventionally, the training of a neural network for learning phases of matter uses real physical quantities as the training set. However, it has been demonstrated in several studies that this may not be required. Here we investigate the phase transitions of the two-dimensional (2D) q-state Potts models on the square lattice using a pre-trained neural network (NN). The employed NN was trained previously using two artificially made configurations as the training set. Hence no training is conducted for the present study. Remarkably, the used NN not only calculates the critical points of the considered phase transitions precisely, but also determines the nature of these phase transitions definitely without ambiguity. Our results as well as the outcomes found previously suggest that training a NN using real physical quantities as the training set is not required to obtain a working NN. These unconventional studies also imply that if the configuration space of a system can be classified into two categories, then the NN considered here is likely applicable to study the phase transition of that system. Comparison between our approach and other known NN methods for studying the 2D q-state Potts models is briefly discussed as well.

Dimensional reduction of dynamical systems by machine learning: automatic generation of the optimum extensive variables and their time-evolution map

A framework is proposed to generate a phenomenological model that extracts the essence of a dynamical system (DS) with large degrees of freedom using machine learning. For a given microscopic DS, the optimum transformation to a small number of macroscopic variables, which is expected to be extensive, and the rule of time evolution that the variables obey are simultaneously identified. The utility of this method is demonstrated through its application to the nonequilibrium relaxation of the three-state Potts model.

The exotic ground state of the decorated honeycomb lattice

The study is focusing on the exploration of the magnetic properties of the frustrated decorated honeycomb lattices. The presence of geometrical frustration and C3 symmetry leads to an exotic ground state. Monte Carlo simulations and analytical calculations are used to analyze the system’s behavior. The dependence of the magnetization on the external field of the Ising model exhibits a step-like behavior, while the magnetization of the classical Heisenberg model has no plateau in the isotropic case. An efficient Hamiltonian is proposed to describe the properties of this system on the unfrustrated hexagonal lattice within the framework of the chiral Potts model. Within a specific range of fields, the state of the effective Hamiltonian aligns with that of the original Hamiltonian. The ground state configurations and degeneracy of the system are explored, revealing fractured stripe patterns separated by spins with opposite orientations. These findings contribute to the knowledge of the properties of decorated lattices, offering valuable insights for potential experimental and practical applications.

Reducing rejection exponentially improves Markov chain Monte Carlo sampling

The choice of transition kernel critically influences the performance of the Markov chain Monte Carlo method. Despite the importance of kernel choice, guiding principles for optimal kernels have not been established. Here, we propose a one-parameter rejection control transition kernel that can be applied to various Monte Carlo samplings and demonstrate that the rejection process plays a major role in determining the sampling efficiency. Varying the rejection probability, we examine the autocorrelation time of the order parameter in the two- and three-dimensional ferromagnetic Potts models. Our results reveal that reducing the rejection rate leads to an exponential decrease in autocorrelation time in sequential spin updates and an algebraic reduction in random spin updates. The autocorrelation times of conventional algorithms almost fall on a single curve as a function of the rejection rate. The present transition kernel with an optimal parameter provides one of the most efficient samplers for general cases of discrete variables.

Probabilistic cell/domain-type assignment of spatial transcriptomics data with SpatialAnno.

In the analysis of both single-cell RNA sequencing (scRNA-seq) and spatially resolved transcriptomics (SRT) data, classifying cells/spots into cell/domain types is an essential analytic step for many secondary analyses. Most of the existing annotation methods have been developed for scRNA-seq datasets without any consideration of spatial information. Here, we present SpatialAnno, an efficient and accurate annotation method for spatial transcriptomics datasets, with the capability to effectively leverage a large number of non-marker genes as well as 'qualitative' information about marker genes without using a reference dataset. Uniquely, SpatialAnno estimates low-dimensional embeddings for a large number of non-marker genes via a factor model while promoting spatial smoothness among neighboring spots via a Potts model. Using both simulated and four real spatial transcriptomics datasets from the 10x Visium, ST, Slide-seqV1/2, and seqFISH platforms, we showcase the method's improved spatial annotation accuracy, including its robustness to the inclusion of marker genes for irrelevant cell/domain types and to various degrees of marker gene misspecification. SpatialAnno is computationally scalable and applicable to SRT datasets from different platforms. Furthermore, the estimated embeddings for cellular biological effects facilitate many downstream analyses.

Effect of boundary grooving on grain growth by Potts model simulations

Grain growth in thin films is a technologically important and scientifically fascinating topic: As the grain size influences many mechanical and optical properties of a material, any change in the grain structure and therewith in the grain size changes also the materials properties. There are many factors influencing coarsening of the microstructure such as grain boundary properties or the presence of particles; additionally the film thickness becomes rather important. It has been observed that when the grain size reaches the magnitude of the layer thickness, coarsening slows down and may even stagnate. For thin films, surface effects such as grain boundary grooving become important. In general, grooves are totally left out of implementations in the standard Potts model. In the current investigation, we introduce grain boundary grooves in the Potts model as physical entities. In the analysis, we show how the mere presence of such grooves reduces the driving force of grain boundary migration at the surface tremendously at any stage of coarsening.

Spatial mixing and the random‐cluster dynamics on lattices

AbstractAn important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high‐temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random‐cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this article, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random‐cluster dynamics in finite subsets of for general . These imply a host of new results, including optimal mixing for the random cluster dynamics on torii and boxes on vertices in at all high temperatures and at sufficiently low temperatures, and for large values of quasi‐polynomial (or quasi‐linear when ) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst‐case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on for general .