Exact Phase Transition Research Articles

Hysteresis Elimination for an Anisotropic Liquid-Crystal Model via Molecule Design and Replica-Exchange Optimization.

The phenomenon of hysteresis in simulations, in which a system's current state is correlated to previous states and inhibits the transition to a more stable phase, may often lead to misleading results in physical chemistry. In this study, in addition to the replica exchange method (REM), a novel approach was taken by combining an evolution strategy based on the evolutionary principles of nature to predict phase transitions for the Hess-Su liquid-crystal model. In this model, an anisotropy term is added to the simple 6-12 Lennard-Jones model to intuitively reproduce the behavior of liquid crystals. We first applied the pressure-temperature REM to the Hess-Su model and optimized the replica spacing for the energy distribution to gain the maximum advantage from the REM. We then used the same approach as for the Hamiltonian REM, seeking to optimize the replica spacing in the same way. Based on both results, we attempted to predict this coarse-grained liquid-crystal model's exact phase transition point. In the Hamiltonian REM, replicas were prepared with different molecular aspect ratios corresponding to the values of the anisotropy terms in the potential function. The Hess-Su liquid-crystal model, which undergoes a direct transition from the nematic to the solid phase without going through a smectic phase, is a challenging research target for understanding phase transitions. Despite the tremendous computational difficulty in overcoming the strong hysteresis present in this system, our method could predict the phase transition point clearly and significantly reduce the extent of hysteresis. Our approach is beneficial when simulating more complex systems and, above all, shows great potential for more accurate and efficient phase transition predictions in the field of molecular simulation in the future.

Eigenvalue Distribution of a High-Dimensional Distance Covariance Matrix With Application

We introduce a new random matrix model called distance covariance matrix in this paper, whose normalized trace is equivalent to the distance covariance. We first derive a deterministic limit for the eigenvalue distribution of the distance covariance matrix when the dimensions of the vectors and the sample size tend to infinity simultaneously. This limit is valid when the vectors are independent or weakly dependent through a finite-rank perturbation. It is also universal and independent of the details of the distributions of the vectors. Furthermore, the top eigenvalues of this distance covariance matrix are shown to obey an exact phase transition when the dependence of the vectors is of finite rank. This finding enables the construction of a new detector for such weak dependence where classical methods based on large sample covariance matrices or sample canonical correlations may fail in the considered high-dimensional framework.

Detection of phase transitions in liquid crystals through optical, thermal and electrical techniques

A ferroelectric hydrogen bond liquid crystal is chosen for documentation of phase transition temperature by various techniques. In this article, the identification of phase transition from smectic C* to smectic G* through ten various techniques which includes optical, thermal, dielectric and electrical methods are discussed. Optical textures are the first evidence for the phase transition to occur. Smectic C* is manifested by broken focal conic texture while smectic G* is identified by smooth mosaic texture. An intense peak in the DSC thermogram provided the exact phase transition temperature along with enthalpy value of C* to G* phase. The variation of permittivity and dielectric loss with temperature also evinced C* to G* phase transition temperature. In addition to these, thermo-electric profiles of this sample are constructed between voltage and temperature at a fixed magnitude of current. An anomaly in thermo-electric profiles indicated detection of the phase transition temperature. Lastly thermo-resistive profiles are constructed for different values of temperature. An anomaly in the plot between resistance of the mesogen with temperature clearly indicates the first order C* to G* phase transition temperature.

The thermal transport characterization of borophene: A molecular dynamics study

Over recent years, two-dimensional (2D) boron sheets (borophene) have been reported to have unique properties, but studies on the thermal properties of borophene using molecular dynamics simulation are still few and far between. In this research, phase transition temperature, heat capacity (Cp), thermal expansion coefficients, and the thermal conductivity of borophene were investigated using the molecular dynamics (MD) simulation method via the reactive force field (ReaxFF). The present work is the first to report the exact phase transition temperature for the borophene sheets. The effects of temperature, point defects, and chirality angle on the thermal conductivity of these sheets were further studied. In addition, an approximate structure parameter on the relaxed system was obtained. Although a theoretical study predicted the structure parameter of borophene, it can be inferred that the ReaxFF was incapable of accurately predicting the borophene lattice parameters. Our simulation results were well in line with the available values reported in other studies where the thermal conductivity of borophene decreased with the increase in temperature. Moreover, the chirality angle effect showed that the thermal conductivity of borophene was anisotropic, and the thermal conductivity in the armchair direction was noticeably larger than that in the zigzag direction. Furthermore, it was confirmed that the thermal conductivity was extremely dependent on the vacancy defects. Understanding the thermal properties of borophene may be crucial for expanding its application in novel nanosystems.

Exact plaquette singlet phases in an orthogonal-plaquette model

We introduce a quantum spin-1/2 model hosting two exact plaquette singlet ground states in extended parameter regimes. There is an exact phase transition between both phases, at which the system has an extensive ground-state degeneracy. Further, the model exhibits an extensive number of conserved quantities, which allow the prediction of additional phases. We exploit this feature and explore the phase diagram in detail for two specific parameter regimes. The model is based solely on Heisenberg interactions and seems sufficiently simple to be realized in quantum materials. A general scheme to determine exact singlet product states is briefly discussed.

Reduced thermodynamics and an exact phase transition of five-dimensional Kerr-AdS black holes with equal spin parameters

Multi-spinning higher dimensional Kerr-AdS black holes admit the stable small-large black hole phase transition of van der Waals type. In this paper, we study the exact critical phenomena and phase structure in five-dimensional spacetime. First, we examine the thermodynamic laws in the reduced parameter space and find that they are quite different from the conventional thermodynamic laws. Then based on the reduced laws, the phase structure in different parameter spaces is investigated. The stable and metastable black hole phases are clearly displayed. We present a highly accurate fitting formula for the coexistence curve of small and large black hole phases. Using this fitting formula, we examine the critical exponents when the black hole system approaches the critical point along the coexistence curve. Moreover, employing the dimensional analysis and symmetry analysis, we also give a numerical study of the critical point for the unequal spinning black holes. These results are very useful on further understanding the microstructure of the multi-spinning black holes in higher dimensional spacetime.

The scaling window of the model d-k-CSP

Consider the model d-k-CSP which consists of a set of n variables with the domain size d, and a set of t=rnln⁡d−ln⁡p constraints where each constraint binds k distinct variables and permits pdk tuples of values. We prove that the exact phase transition of the model d-k-CSP occurs at rc=1 as long as kln⁡d=αln⁡n, where α satisfies that 12+12(2k−1)+ε≤α≤1 (ε>0 is a sufficiently small constant). More importantly, we establish the finite-size scaling about this transition. Specifically, let δ>0 be a given constant, the probability of satisfiability of the model d-k-CSP is greater than 1−δ if r≤1−Θ(1n1−νln⁡d) (ν is dependent on the control parameters) and is less than δ if r≥1+Θ(1nln⁡d).

A general model and thresholds for random constraint satisfaction problems

In this paper, we study the relation among the parameters in their most general setting that define a large class of random CSP models d-k-CSP where d is the domain size and k is the length of the constraint scopes. The model d-k-CSP unifies several related models such as the model RB and the model k-CSP. We prove that the model d-k-CSP exhibits exact phase transitions if klnd increases no slower than the logarithm of the number of variables. A series of experimental studies with interesting observations are carried out to illustrate the solubility phase transition and the hardness of instances around phase transitions.

Exact Phase Transitions and Approximate Algorithm of #CSP

The study of phase transition phenomenon of NP complete problems plays an important role in understanding the nature of hard problems. In this paper, we follow this line of research by considering the problem of counting solutions of Constraint Satisfaction Problems (#CSP). We consider the random model, i.e. RB model. We prove that phase transition of #CSP does exist as the number of variables approaches infinity and the critical values where phase transitions occur are precisely located. Preliminary experimental results also show that the critical point coincides with the theoretical derivation. Moreover, we propose an approximate algorithm to estimate the expectation value of the solutions number of a given CSP instance of RB model.

Threshold behaviors of a random constraint satisfaction problem with exact phase transitions

We consider a random constraint satisfaction problem named model RB, which exhibits a sharp satisfiability phase-transition phenomenon when the control parameters pass through the critical values denoted by r c r and p c r . Using finite-size scaling analysis, we bound the width of the transition region for finite problem size n, which might be the first rigorous study on the threshold behaviors of NP-complete problems.

Stationary-point approach to the phase transition of the classicalXYchain with power-law interactions

The stationary points of the potential energy function V of the classical XY chain with power-law pair interactions (i.e., interactions decaying like r{-α} with the distance) are analyzed. For a class of "spinwave-type" stationary points, the asymptotic behavior of the Hessian determinant of V is computed analytically in the limit of large system size. The computation is based on the Toeplitz property of the Hessian and makes use of a Szegö-type theorem. The results serve to illustrate a recently discovered relation between phase transitions and the properties of stationary points of classical many-body potentials. In agreement with this relation, the exact phase transition potential energy of the model can be read off from the behavior of the Hessian determinant for exponents α between zero and one. For α between one and two, the phase transition is not manifest in the behavior of the determinant, and it might be necessary to consider larger classes of stationary points.

Determination of the Maximal Lyapunov Exponent through the Effective Potential Energy: Exact Phase Transition Temperature of Few Particle System CF4

We calculate the maximal Lyapunov exponent of carbon tetra fluoride using the classical molecular dynamics (MD) simulation for different temperatures. The system is composed of 108 particles interacting either through an effective intermolecular potential energy in the vicinity of its liquid–gas phase transition point. As a main result, we find that at the transition temperature T c , the Lyapunov exponent exhibits a power-law behavior with a functional form λ= | T - T c | -ω , where the exponent ω is 0.155. This confirms that the maximal Lyapunov exponent behaves as an exact order parameter to signal the phase transition point for a system even with the infinite particles if the effective potential energy has been used in simulation. The applied effective intermolecular potential-energy function for carbon tetra fluoride has been obtained directly from the extended law of corresponding states for viscosity data using the inversion method. Then results were used to obtain a best Morse-Split-van der Waals ...

Random constraint satisfaction: Easy generation of hard (satisfiable) instances

In this paper, we show that the models of random CSP instances proposed by Xu and Li [K. Xu, W. Li, Exact phase transitions in random constraint satisfaction problems, Journal of Artificial Intelligence Research 12 (2000) 93–103; K. Xu, W. Li, Many hard examples in exact phase transitions with application to generating hard satisfiable instances, Technical report, CoRR Report cs.CC/0302001, Revised version in Theoretical Computer Science 355 (2006) 291–302] are of theoretical and practical interest. Indeed, these models, called RB and RD, present several nice features. First, it is quite easy to generate random instances of any arity since no particular structure has to be integrated, or property enforced, in such instances. Then, the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness. In that case, a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold, i.e., to have an exponential tree-resolution complexity. Next, a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones. This analysis is supported by some representative results taken from an intensive experimentation that we have carried out, using complete and incomplete search methods.

Phase separation of a Ge2Sb2Te5 alloy in the transition from an amorphous structure to crystalline structures

Changes in the structural and electrical properties of a Ge2Sb2Te5 alloy thin film induced by phase transition were investigated using various analytical techniques. X-ray diffraction and scanning photoelectron microscopy showed that the phase separation occurred in a local area of the film during a phase transition when the amorphous structure was being transformed into crystalline structures. It was found that the heterogeneous distribution of Sb atoms that diffused during the phase transition accompanied the phase separation. Atomic force microscopy was used to examine the changes in surface morphology and roughness. The electrical conductance of the film was dramatically improved after the phase transition from an amorphous structure to crystalline structures as evidenced by the sheet resistance measurements. The sheet resistance changed from ∼109to∼102Ω∕sq. during the phase transition. Differential scanning calorimetry was used to determine the exact phase transition temperature (160–170°C) and the effective activation energy for the phase transition (2.5±0.11eV) on the basis of Kissinger’s equation.

Many hard examples in exact phase transitions

This paper analyzes the resolution complexity of two random constraint satisfaction problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.

Exact Phase Transitions in Random Constraint Satisfaction Problems

In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems are satisfiable to a region where almost all problems are unsatisfiable do exist for Model RB as the number of variables approaches infinity. Moreover, the critical values at which the phase transitions occur are also known exactly. By relating the hardness of Model RB to Model B, it is shown that there exist a lot of hard instances in Model RB.

Exact phase transitions in random constraint satisfaction problems

In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems ar...

Exact results of a general spin-1 model on 2D decorated lattices

By the decoration-iteration transformation, we find a general relation between interaction parameters of a general spin-1 model on any lattice with unique coordination number and on its decorated lattice. We obtain three exact phase transition points of the general spin-1 model on two-dimensional decorated lattices in terms of the exact results of the model on corresponding two-dimensional regular lattices, including the triangle, square, kagome and honeycomb lattices. We find the critical temperatures have no exact relation with the average coordination number and the type of coordination number for both the Ising model and the general spin-1 model.

A heuristic principle for the phase boundaries of pure and quenched bond-diluted spin models

Results from high-temperature series for a class of discrete, classical spin models on the square lattice suggest that the phase boundary is, at least in part, given by a simple heuristic principle. It is shown how this principle can be extended to triangular and hexagonal lattices by making use of duality and the star-triangle transformation. For the Potts model, the exact phase transition is recovered, not only for integer number of states, but also in the percolation limit. For more complicated models, a number of phase diagrams are derived, which are presumably exact except in regions where partially broken symmetry phases or massless phases intervene. The principle can also be applied to quenched bond-diluted spin models. The results for the Potts model are obtained for all three lattices and shown to be exact in certain limiting cases. The M to 1 limit of the wreath product model S(M):S(M), which corresponds to 'double' percolation, is studied to provide an example for which the principle is completely incorrect.

Exact phase transition points for vertex models on even-coordinated lattices

The authors study the manifold of fixed points of the generalized weak-graph transformation on lattices of even coordination number for the most general vertex model respecting spin-flip symmetry. They conjecture these fixed points to be the loci of phase transitions. As an example, they turn to coordination number six and find phase transitions in certain regions of the manifold of fixed points: first by investigating a gauge-invariant Ising model on a three-dimensional SC lattice with the help of Monte-Carlo simulations; and second for the ice-type zero-field ferroelectric model, in which the transition between frozen ordered and disordered phase is of first order.