Convexity violations for noninteger parameters in certain lattice models

The convexity of the free energy is studied for several lattice models in situations in which a parameter which is normally a positive integer takes on noninteger real values. Examples include the numbern of components in then-vector model, the number of states in the Potts model, and the dimensionality of the lattice. In a typical case there is a critical value of the parameter such that convexity is preserved when the parameter exceeds the critical value, but can be violated for appropriate Hamiltonians whenever the parameter is less than the critical value, but not a positive integer. In several cases the critical value of the parameter increases with the size of the system, thus raising questions about the significance of a continuous variation of the parameter in the thermodynamic limit.     

Misfit dislocation loops in composite core-shell nanoparticles

The critical conditions have been calculated for the generation of circular prismatic loops of misfit dislocations at the interfaces in spherically symmetric composite core-shell nanoparticles. It has been shown that the formation of these loops becomes energetically favorable if the misfit parameter exceeds a critical value, which is determined by the geometry of the system. The most preferred position of the dislocation loop is in the equatorial plane of the nanoparticle. For a given radius of the nanoparticle, there is a minimum value of the critical misfit parameter below which the generation of a misfit dislocation is energetically unfavorable for any ratio of the core and shell radii. For a misfit parameter exceeding the minimum critical value, there are two critical values of the reduced radius of the particle core in the interval between which the generation of a dislocation loop is energetically favorable. This interval increases with increasing misfit parameter for a fixed particle size and decreases with decreasing particle size for a fixed misfit parameter.     

On a leveraging effect of weak azimuthal inhomogeneities explaining the nonuniqueness of critical Maier-Saupe order parameters

The Maier--Saupe theory of liquid crystals predicts a unique critical order parameter for the transition point of isotropic--nematic phase transitions. Experimental studies have shown that critical order parameters are shifted away from that unique value. We show that micro-defects and weak azimuthal inhomogeneities can result due to a leveraging effect in the experimentally observed macro-shifts of the critical order parameter. We demonstrate the leveraging effect by computing the corresponding micro-shifts of the order parameter in the isotropic phase with the help of an effective order parameter. Our approach accounts for critical order parameters observed in experimental studies which is explicitly demonstrated for methyl-deuterated para-azoxyanisole (PAA-d₆), 4-n-pentyl-4′-cyanobiphenyl (5CB), N-(4-methoxybenzylidene)-4-butylaniline (MBBA). Finally, in line with the effective order parameter approach and the dynamical mean field approaches by Hess, Doi, and Edwards, predictions are derived that provide novel testbeds for experimental studies.     

Critical phenomena and relativistic gravitational collapse

Recent calculations have shown the existence of critical phenomena in general relativity associated with the collapse of wavepackets of massless fields that are near, in parameter space, the onset of black hole formation (the critical point). Two physically distinct systems have been explored: collapse of spherically-symmetric massless scalar field and collapse of vacuum, axisymmetric gravitational waves. Nonlinear effects dominate near the critical point. Black-hole mass serves as an order parameter and has a power-law dependence on critical separation in the supercritical region of parameter space. Remarkably, the values of the critical exponent of the power law are nearly identical in the two systems. The nonlinearity induces the fields to oscillate. Each successive oscillation is an echo, obeying a spatial and temporal scaling relation.     

Order parameter of layered superconducting structures

It is suggested that the order parameter and the critical temperature in an S-N structure be described on the basis of the microscopic Ψ theory, modeling the metal in the normal state by the Ginzburg-Landau equation with a negative coefficient a. It is shown that at a contact between layers the order parameter in the layer whose temperature is below the critical temperature induces an order parameter in the other layer whose temperature is higher than the critical temperature.     

Spectral anomalies of diffracted chirped Gaussian pulses and their potential applications

Starting from the Rayleigh-Sommerfeld diffraction integral and without invoking the paraxial approximation, analytical expressions for the field distribution, far-field power spectrum and temporal far-field distribution of chirped Gaussian pulses diffracted at a circular aperture are derived, which enables us to study the spectral anomalous behavior of diffracted chirped Gaussian pulses in the far field. The potential applications of spectral anomalies of ultrashort pulses are discussed. It is found that at the critical angle the spectral switch appears. The frequency difference between the two equal heights of spectral switches increases and the corresponding critical diffraction angle slightly increases as the chirp parameter increases and pulse duration decreases. In a certain region of the truncation parameter, the critical angle decreases with increasing truncation parameter. By suitably varying the pulse duration, chirp parameter and truncation parameter, information encoding and transmission are achievable in the use of chirped Gaussian pulses.     

Properties of parallel upper critical field within continuous Ginzburg–Landau model

In this paper, we employ a continuous Ginzburg–Landau model to study the behaviors of the parallel upper critical field of an intrinsically layered superconductor. Near T_c where the order parameter is nearly homogeneous, the parallel upper critical field is found to vary as (1−T/T_c)^1/2. With a well-localized order parameter, the same field temperature dependence holds over the whole temperature range. The profile of the order parameter at the parallel upper critical field is of a Gaussian type, which is consistent with the usual Ginzburg–Landau theory. In addition, the influences of the unit cell dimension and the average effective masses on the parallel upper critical field and the associated order parameter are also addressed.     

Buckling analysis of nanobeams with exponentially varying stiffness by differential quadrature method

We present the application of differential quadrature(DQ) method for the buckling analysis of nanobeams with exponentially varying stiffness based on four different beam theories of Euler-Bernoulli, Timoshenko, Reddy, and Levison.The formulation is based on the nonlocal elasticity theory of Eringen. New results are presented for the guided and simply supported guided boundary conditions. Numerical results are obtained to investigate the effects of the nonlocal parameter,length-to-height ratio, boundary condition, and nonuniform parameter on the critical buckling load parameter. It is observed that the critical buckling load decreases with increase in the nonlocal parameter while the critical buckling load parameter increases with increase in the length-to-height ratio.     

Characterization of Critical Values of Branching Random Walks on Weighted Graphs through Infinite-Type Branching Processes

We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. Two kinds of survival can be identified: a weak survival (with positive probability there is at least one particle alive somewhere at any time) and a strong survival (with positive probability the colony survives by returning infinitely often to a fixed site). The behavior of the process depends on the value of a certain parameter which controls the birth rates; the threshold between survival and (almost sure) extinction is called critical value. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.     

三维钻石型等级晶格上量子Heisenberg系统的临界性质

利用重整化群方法,研究了三维钻石型等级晶格上的各向异性量子Heisenberg模型,获得了系统的相图和临界性质. 结果表明:对于铁磁系统,在各向同性Heisenberg极限下,系统存在有限温度的相变,并计算了系统的序参量和临界指数; 对于反铁磁系统,在各向同性Heisenberg极限下,临界温度不等于零,在临界线上不存在重入行为.     

Order parameter in the critical phenomena of Kerr–Newman black holes

An order parameter is introduced in describing the singular thermodynamical behaviour of Kerr–Newman black holes in the vicinity of a certain critical temperature. The mean square fluctuation of the order parameter is calculated. The analogy between the critical temperature of a black hole and the Hagedorn temperature in hadronic physics and string theory is discussed.     

On the theory of the magnetization of dimerized magnets

To describe a quantum phase transition in a dimerized antiferromagnet, the formalism of the Landau theory of phase transitions has been applied. The energy of the ground state is minimized by varying the parameter of the mixing of the wavefunctions with different multiplicities. The critical behavior of the parameter of the wavefunction is responsible for the critical behavior of the observables.     

Finite-size scaling analysis of the critical behavior of a general epidemic process in 2D

We investigate the critical behavior of a stochastic lattice model describing a General Epidemic Process. By means of a Monte Carlo procedure, we simulate the model on a regular square lattice and follow the spreading of an epidemic process with immunization. A finite size scaling analysis is employed to determine the critical point as well as some critical exponents. We show that the usual scaling analysis of the order parameter moment ratio does not provide an accurate estimate of the critical point. Precise estimates of the critical quantities are obtained from data of the order parameter variation rate and its fluctuations. Our numerical results corroborate that this model belongs to the dynamic isotropic percolation universality class. We also check the validity of the hyperscaling relation and present data collapse curves which reinforce the accuracy of the estimated critical parameters.     

The structure of the thermal explosion limit with variable pre-exponential factor

An approximate critical explosion parameter, which separates the slow and explosive reactions is determined with respect to the initial data, using the asymptotic technique to analyze a one step reaction mechanism with temperature dependent pre-exponential factor. We obtained and analysed the effect of the variable pre-exponential factor on the critical temperature, critical parameter for thermal explosion and the first order approximation of the slow curve. By introducing a new coordinate and using the Method of Invariant Manifold (MIM), we obtained corrections on the critical temperature. The results obtained are not only general; they also reveal some salient properties of the system hitherto unknown.     

A PHASE TRANSITION MODEL WITH PERPETUAL ORDER

A simple model is given to show there exists a kind of phase transition for which the order parameter never vanishes; its characteristic is the second derivative of the order parameter rather than the order parameter itself, and its critical behavior satisfies the scaling.     

热爆炸临界参数的数值解法

本文提出的热爆炸临界参数的数值解法是一种变分方法.这种方法依据的是文献[1]给出的热爆炸临界参数的一个变分原理。数值计算的结果表明,这种方法是一种简便、经济而又精确度较高的方法。     

Thermal explosion in a stirred medium

The critical conditions of a thermal explosion in a cylindrical chemical reactor with several symmetrically positioned agitators for creating a forced convection of the reaction mixture are examined. The analysis is carried out for an arbitrary Peclet number on the assumption that the flow of the fluid is laminar. The critical value of the thermal explosion parameter (Frank-Kamenetskii parameter) is determined as a function of the number of agitators, Peclet number, and the distances between the axes of the agitators and the reactor axis. In contrast to the previous studies, the dependence of the thermal explosion parameter on the rate of stirring of the medium is considered. Calculations show that, as the number of agitators and mixing rate increases, the thermal explosion probability decreases, so that the critical value of the thermal explosion parameter can be several times greater than its classical value. It is demonstrated that, depending on the values of the parameters, the problem can have four stationary solutions, two of which are stable.     

Phase Transition of Ising Model on a Kind of Periodic Fractal Lattices

In this paper we introduced a periodic fractal structure (PFS) and studied the critical behavior of the nearest-neighbor-interacting Ising model within Migdal-Kadanoff bond-moving approximation. The critical curve equation and the formulas of the critical exponents were derived approximately in a special parameter region.     

Magnetic-field-induced quantum criticality in a spin-<Emphasis Type="Italic">S</Emphasis> planar ferromagnet with single-ion anisotropy

The effects of single-ion anisotropy on quantum criticality in a d-dimensional spin-S planar ferromagnet is explored by means of the two-time Green’s function method. We work at the Tyablikov decoupling level for exchange interactions and the Anderson-Callen decoupling level for single-ion anisotropy. In our analysis a longitudinal external magnetic field is used as the non-thermal control parameter and the phase diagram and the quantum critical properties are established for suitable values of the single-ion anisotropy parameter D. We find that the single-ion anisotropy has sensible effects on the structure of the phase diagram close to the quantum critical point. However, for values of the uniaxial crystal-field parameter below a positive threshold, the conventional magnetic-field-induced quantum critical scenario remains unchanged.     

Nonequilibrium phase transitions in coordinated biological motion: Critical slowing down and switching time

In new experiments on coordinated biological motion we measure relaxation times and switching times as the system evolves from one coordinated state to another at a critical control parameter value. Deviations from the coordinated state are induced by mechanical perturbations and relative phase is used as an order parameter to monitor the dynamics of the collective state. Clear evidence for critical slowing down, a key feature of nonequilibrium phase transitions, is found. The mean and distribution of switching times closely match predictions from a stochastic dynamic theory. Together with earlier results on critical fluctuations these findings strongly favor an interpretation of coordinative change in biological systems as a nonequilibrium phase transition.