Statistical Lattice Model Research Articles

Computing grain boundary “phase” diagrams

AbstractGrain boundaries (GBs) can be treated as two‐dimensional (2‐D) interfacial phases (also called “complexions”) that can undergo interfacial phase‐like transitions. As bulk phase diagrams and calculation of phase diagram (CALPHAD) methods serve as a foundation for modern materials science, we propose to extend them to GBs to have equally significant impacts. This perspective article reviews a series of studies to compute the GB counterparts to bulk phase diagrams. First, a phenomenological interfacial thermodynamic model was developed to construct GB lambda diagrams to forecast high‐temperature GB disordering and related trends in sintering and other properties for both metallic and ceramic materials. In parallel, an Ising‐type lattice statistical thermodynamic model was utilized to construct GB adsorption (segregation) diagrams, which predicted first‐order GB adsorption transitions and critical phenomena. These two simplified thermodynamic models emphasize the GB structural (disordering) and chemical (adsorption) aspects, respectively. Subsequently, hybrid Monte Carlo and molecular dynamics atomistic simulations were used to compute more rigorous and accurate GB “phase” diagrams. Computed GB diagrams of thermodynamic and structural properties were further extended to include mechanical properties. Moreover, machine learning algorithms were combined with atomistic simulations to predict GB properties as functions of four independent compositional variables and temperature in a 5‐D space for a given GB in high‐entropy alloys or as functions of five GB macroscopic (crystallographic) degrees of freedom plus temperature and composition for a binary alloy in a 7‐D space. Other relevant studies are also examined. Future perspective and outlook, including two emerging fields of high‐entropy grain boundaries (HEGBs) and electrically (or electrochemically) induced GB transitions, are discussed.

Critical temperature and magnetic properties of new 2D lattice model with double hexagonal symmetry: Monte Carlo study

Here, we present a theoretical study of a new statistical lattice model based on a double hexagonal structure associated with G2 symmetry. Using Monte Carlo simulation, we study the magnetic properties of the Ising-1/2 model with spin values σ = ±1 residing on the sites of our double hexagonal lattice. In particular, we calculate and analyze the thermal behavior of the total and partial magnetizations as well as the corresponding susceptibilities for different lattice sizes. The present study shows that the total and partial magnetizations vanish at the same critical temperature. This vanishing is continuous, indicating that the type of the transition is a second order. With the help of the finite-size scaling analysis, we estimate the critical transition temperature related to the uniform coupling interaction value equals one. Our findings reflect a good estimation of the critical temperature, TC that is equal to 2.976∓0.004JkB. The obtained critical temperature of our presented model can be found between the critical temperatures of hexagonal and triangular lattice models.

Modulated Phases in a Spin Model with Dzyaloshinskii-Moriya Interactions

We analyze the phase diagram of an elementary statistical lattice model of classical, discrete, spin variables, with nearest-neighbor ferro-magnetic isotropic interactions in competition with chiral interactions along an axis. At the mean-field level, we show the existence of para-magnetic lines of transition to a region of modulated (helimagnetic) structures. We then turn to the analysis of the analogous problem on a Cayley tree. Taking into account the simplicity introduced by the infinite-coordination limit of the tree, we explore several details of the phase diagrams in terms of temperature and a parameter of competition. In particular, we characterize sequences of modulated (helical) structures associated with devil's staircases of a fractal character.

Magnetic Field and Dilution Effects on the Phase Diagrams of Simple Statistical Models for Nematic Biaxial Systems

We use a simple statistical model to investigate the effects of an applied magnetic field and of the dilution of site elements on the phase diagrams of biaxial nematic systems, with an emphasis on the stability of the Landau multicritical point. The statistical lattice model consists of intrinsically biaxial nematogenic units, which interact via a Maier–Saupe potential, and which are characterized by a discrete choice of orientations of the microscopic nematic directors. According to previous calculations at zero field and in the absence of dilution, we regain the well-known sequence of biaxial, uniaxial, and disordered structures as the temperature is increased, and locate the Landau point. We then focus on the topological changes induced in the phase diagram by the application of an external magnetic field, and show that the Landau point is destabilized by the presence of an applied field. On the other hand, in the absence of a field, we show that only a quite strong dilution of nematic sites is capable of destabilizing the Landau point.

Criticality in tissue homeostasis: models and experiments

There is considerable theoretical and experimental support to the proposal that tissue homeostasis in the adult skin can be represented as a critical branching process. The homeostatic condition requires that the proliferation rate of the progenitor (P) cells (capable of cell division) is counterbalanced by the loss rate due to the differentiation of a P cell into differentiated (D) cells, so that the total number of P cells remains constant. We consider the two-branch and three-branch models of tissue homeostasis to establish homeostasis as a critical phenomenon. It is first shown that some critical branching process theorems correctly predict experimental observations. A number of temporal signatures of the approach to criticality are investigated based on simulation and analytical results. The analogy between a critical branching process and mean-field percolation and sandpile models is invoked to show that the size and lifetime distributions of the populations of P cells have power-law forms. The associated critical exponents have the same magnitudes as in the cases of the mean-field lattice statistical models. The results indicate that tissue homeostasis provides experimental opportunities for testing critical phenomena.

Peculiarities in pseudo-transitions of a mixed spin-(1/2, 1) Ising–Heisenberg double-tetrahedral chain in an external magnetic field

Recently, it has been rigorously verified that several one-dimensional (1D) spin models may exhibit a peculiar pseudo-transition accompanied with anomalous response of thermodynamic quantities in a close vicinity of pseudo-critical temperature. In the present work we will introduce and exactly solve a mixed spin-(1/2,1) Ising–Heisenberg double-tetrahedral chain in an external magnetic field as another particular example of 1D lattice-statistical model with short-range interactions that displays a pseudo-transition of this type. The investigated model exhibits at zero temperature three ferrimagnetic phases, three frustrated phases, and one saturated paramagnetic phase. The ground-state phase diagram involves five unusual interfaces (phase boundaries), at which the residual entropy per site equals to a larger entropy of one of two coexisting phases. Four such interfaces are between a non-degenerate ferrimagnetic phase and a macroscopically degenerate frustrated phase, while one interface is between two non-degenerate ferrimagnetic phases. Though thermal excitations typically destroy all fingerprints of zero-temperature phase transitions of 1D lattice-statistical models with short-range forces, the mixed spin-(1/2,1) Ising–Heisenberg double-tetrahedral chain is quite robust with respect to thermal excitations and it displays peculiar pseudo-transitions close to all five aforementioned interfaces.

Inhomogeneous Gaussian free field inside the interacting arctic curve

The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e. models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e. not free, not determinantal) six-vertex model, and provide evidence for the following two claims:(i) the critical fluctuations are given by a Gaussian free field (GFF), as in the free case, but(ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant becomes position-dependent, .The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green’s function of the Laplacian inside the critical domain, but instead, of the Green’s function of a generalized Laplacian parametrized by the function . Surprisingly, we also find that there is a change of regime when the anisotropy parameter Δ is smaller than , with becoming singular at one point.

Simulation of lattice statistical models with defects: Critical Casimir Effect

The influence of lattice structure defects on phase transition phenomena was studied in framework of 2D spin model. Effective mass of defect was proposed for investigation of conformal properties of the model at critical point. The volume dependence of defect’s mass at critical point and the Critical Casimir interaction of two defects were studied. It was shown that this Casimir interaction is attraction for any value of hopping parameter. The confinement of the defect on the defects line and the process of defects line collapse were studied. Applications in nanophysics and biophysics were discussed.

PHENIX results on fluctuations and Bose–Einstein correlations in Au + Au collisions from the RHIC Beam Energy Scan

The RHIC Beam Energy Scan focuses on mapping the QCD phase diagram and pinpointing the location of a possible critical end point. Bose-Einstein correlations and event-by-event fluctuations of conserved quantities, measured as a function of centrality and collision energy, are promising tools in these studies. Recent lattice QCD and statistical thermal model calculations predict that higher-order cumulants of the fluctuations are sensitive indicators of the phase transition. Products of these cumulants can be used to extract the freeze-out parameters [A. Bazavov et al., Phys. Rev. Lett. 109, 192302 (2012)] and to locate the critical point [M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. D 60, 114028 (1999)]. Two-pion interferometry measurements are predicted to be sensitive to potential softening of the equation of state and prolonged emission duration close to the critical point [S. Pratt, Phys. Rev. Lett. 53, 1219 (1984)]. We present recent PHENIX results on fluctuations of net-charge using high-order cumulants and their products in Au+Au collisions at sNN=7.7–200 GeV, and measurement of two-pion correlation functions and emission-source radii in Cu+Cu and Au+Au collisions at several beam energies. The extracted source radii are compared to previous measurements at RHIC and LHC in order to study energy dependence of the specific quantities sensitive to expansion velocity and emission duration. Implications for the search of a critical point and baryon chemical potentials at various collision energies are discussed.

Lattice Statistical Models for the Nematic Transitions in Liquid-Crystalline Systems

We investigate the connections between some simple Maier-Saupe lattice models, with a discrete choice of orientations of the microscopic directors, and a recent proposal of a two-tensor formalism to describe the phase diagrams of nematic liquid-crystalline systems. This two-tensor proposal is used to formulate the statistical problem in terms of fully-connected lattice Hamiltonians, with the local nematic directors restricted to the Cartesian axes. Depending on the choice of interaction parameters, we regain all of the main features of the original mean-field two-tensor calculations. With a standard choice of parameters, we obtain the well-known sequence of isotropic, uniaxial, and biaxial nematic structures, with a Landau multicritical point. With another suitably chosen set of parameters, we obtain two tricritical points, according to some recent predictions of the two-tensor calculations. The simple statistical lattice models are quite easy to work with, for all values of parameters, and the present calculations can be carried out beyond the mean-field level.

Fusion rules for the logarithmic N = 1 superconformal minimal models: I. The Neveu–Schwarz sector

It is now well known that non-local observables in critical statistical lattice models, polymers and percolation for example, may be modelled in the continuum scaling limit by logarithmic conformal field theories. Fusion rules for such theories, sometimes referred to as logarithmic minimal models, have been intensively studied over the last ten years in order to explore the representation-theoretic structures relevant to non-local observables. Motivated by recent lattice conjectures, this work studies the fusion rules of the N = 1 supersymmetric analogues of these logarithmic minimal models in the Neveu–Schwarz sector. Fusion rules involving Ramond representations will be addressed in a sequel.

Fast Physically Accurate Rendering of Multimodal Signatures of Distributed Fracture in Heterogeneous Materials.

This paper proposes a fast, physically accurate method for synthesizing multimodal, acoustic and haptic, signatures of distributed fracture in quasi-brittle heterogeneous materials, such as wood, granular media, or other fiber composites. Fracture processes in these materials are challenging to simulate with existing methods, due to the prevalence of large numbers of disordered, quasi-random spatial degrees of freedom, representing the complex physical state of a sample over the geometric volume of interest. Here, I develop an algorithm for simulating such processes, building on a class of statistical lattice models of fracture that have been widely investigated in the physics literature. This algorithm is enabled through a recently published mathematical construction based on the inverse transform method of random number sampling. It yields a purely time domain stochastic jump process representing stress fluctuations in the medium. The latter can be readily extended by a mean field approximation that captures the averaged constitutive (stress-strain) behavior of the material. Numerical simulations and interactive examples demonstrate the ability of these algorithms to generate physically plausible acoustic and haptic signatures of fracture in complex, natural materials interactively at audio sampling rates.

Bethe-lattice calculations for the phase diagram of a two-state Janus gas

We use a simple lattice statistical model to analyze the effects of directional interactions on the phase diagram of a fluid of two-state Janus particles. The problem is formulated in terms of nonlinear recursion relations along the branches of a Cayley tree. Directional interactions are taken into account by the geometry of this graph. Physical solutions on the Bethe lattice (the deep interior of a Cayley tree) come from the analysis of the attractors of the recursion relations. We investigate a number of situations, depending on the concentrations of the types of Janus particles and the parameters of the potential, and make contact with results from recent numerical simulations.

Determination of freeze-out conditions from fluctuation observables measured at RHIC

We extract chemical freeze-out conditions via a thermal model approach from fluctuation observables measured at RHIC and compare with results from lattice QCD and statistical hadronization model fits. The possible influence of additional critical and non-critical fluctuation sources not accounted for in our analysis is discussed.

New statistical lattice model with double honeycomb symmetry

Inspired from the connection between Lie symmetries and two-dimensional materials, we propose a new statistical lattice model based on a double hexagonal structure appearing in the G2 symmetry. We first construct an Ising-1/2 model, with spin values σ = ±1, exhibiting such a symmetry. The corresponding ground state shows the ferromagnetic, the antiferromagnetic, the partial ferrimagnetic and the topological ferrimagnetic phases depending on the exchange couplings. Then, we examine the phase diagrams and the magnetization using the mean field approximation (MFA). Among others, it has been suggested that the present model could be localized between systems involving the triangular and the single hexagonal lattice geometries.

Reaction efficiency of diffusion-controlled processes on finite, aperiodic planar lattices – III: Supramolecular architectures

Supramolecular architectures provide a reproducible template on which surficial processes can be studied. We consider the irreversible reaction A+B→C where B is a stationary reaction center and A is a coreactant diffusing on a finite, discretized d=2 dimensional surface of a supramolecular assembly. A lattice-statistical model is developed to quantify how the reaction efficiency changes when the template is planar, Euler characteristic Ω=0, or wrapped on the surface of a d=3 host, Ω=2. We find that for aperiodic or regular surfaces of finite spatial extent, dispersed Ω=2 assemblies better optimize surficial reactions than Ω=0 planar hosts.

Transition behavior on lattices with ladders. Part II: Sensitivity to coupling

Quantum–mechanical studies have predicted, and experimental studies on SrCu2O3 have confirmed, that the propagation of bound pairs on even-chain copper oxide ladders is possible, but not on ladders with an odd number of legs. To study whether this quantum–mechanical lattice parity effect has a classical analog, and to document the consequences of assuming different coupling scenarios between the ladder and adjacent sublattices, we develop a classical Markovian lattice-statistical model to monitor the efficiency of migration on a composite lattice. Regions of parameter space where significant departures from results obtained via a symmetrical random walk are identified.

Effect of the on-site interaction on the magnetic properties of an exactly solvablespin–electron system

A hybrid lattice-statistical model on doubly decorated planar lattices, which havelocalized Ising spins at their nodal lattice sites and two itinerant electrons at eachpair of decorating sites, is examined within the framework of an exact analyticalmethod based on the generalized decoration–iteration mapping transformation. Aparticular emphasis is laid on the investigation of the effect of the on-site interactionU on the ground-state properties and critical behaviour of the system. It is shown that theground state of the investigated hybrid system consists of a remarkable four-sublatticequantum antiferromagnetic phase, which manifests itself through a non-trivial criticality atnon-zero temperatures. Moreover, the thermal variations of the staggered magnetizationand the specific heat provide unambiguous evidence that the investigated spin–electronsystem exhibits the critical behaviour from the standard Ising universality class.

Rotational sandpile models: A finite size scaling study

Nontrivial critical behaviour is observed in equilibrium lattice statistical models under rotational constraint. Effect of rotational constraint on lattice statistical models at out of equilibrium situation is important to study. Critical properties of non-equilibrium steady state of sandpile models are studied under rotational constraint via finite size scaling (FSS) analysis. Not only FSS is found to be valid for a wide range of rotational sandpile models but also they are found to belong to the same universality class.

Effect of On-Site Coulomb Repulsion on Phase Transitions in Exactly Solved Spin-Electron Model

A hybrid lattice-statistical model on doubly decorated planar lattices, which have localized Ising spins at their nodal lattice sites and two itinerant electrons at each pair of decorating sites, is exactly solved by the use of a generalized decoration-iteration transformation. Our main attention is focused on an in uence of the on-site Coulomb repulsion on ground-state properties and critical behavior of the investigated system.