Polyconvex double well functions

A point-defect model of structural phase transitions in YBa2Cu3O6 + x is proposed. The model treats the basal-plane oxygen atoms as point defects. The treatment parallels the work of Alefeld and coworkers on ferroelasticity due to hydrogen in metals. The model predicts that the tetragonal-to-orthorhombic phase transition should be accompanied by precipitation into regions of high and low oxygen density, as well as high and low oxygen ordering. This is in agreement with experiment, as well as with the predictions of Khachaturyan's concentration-wave model. A new feature of the present model is the prediction of two characteristic temperatures related, respectively, to Snoek-type and Gorsky-type diffusive relaxations. Practical implications of the high diffusivity of the basal-plane oxygens include the easy fabricability of a bicrystal of Y-Ba-Cu-O such that the two components of the bicrystal are rotated by 90° with respect to each other, and are separated by a non-superconducting layer. Such a bicrystal can find applications as a Josephson-tunneling device.

Two cellular automata models of nonequilibrium phase transitions with one adsorbing state are studied in one and two dimensions. New evidence is found against the conjecture according to which all the one-component models with a single adsorbing state belong to the universality class of Reggeon field theory or directed percolation.

We consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problem \(\inf_{\int_{\Omega} u\,dx=m} \int_{\Omega} \frac{1}{\varepsilon}W(u)+\frac{\varepsilon}{2} \big|\nabla u\big|^2 dx.\) Here W is a double‐well potential and \(\Omega\subset\R^n\) is a strictly convex domain. For e small, this is closely related to the problem of partitioning Ω into two subdomains of fixed volume, where the subdomain boundaries correspond to the transitional boundary between phases. Motivated by this geometry problem, we show that in a strictly convex domain, stable critical points of the original variational problem have a connected, thin transition layer separating the two phases. This relates to work in [GM] where special geometries such as cylindrical domains were treated, and is analogous to the results in [CHo] which show that in a convex domain, stable critical points of the corresponding unconstrained problem are constant. The proof of connectivity employs tools from geometric measure theory including the co‐area formula and the isoperimetric inequality on manifolds. The thinness of the transition layer follows from a separate calculation establishing spatial decay of solutions to the pure phases.

The phase transition in information spreading was investigated. Two phase transitions were studied, respectively, from the point of view of network information and the node information, and the corresponding simulations were implemented on high-order networks. In order to explore the phase transition in information dissemination, this paper analyzes the characteristics of social networks and constructs a high-order network model with explicit, subexplicit, and implicit relationships. The factors affecting interpersonal communication were analyzed, the social cumulative effect was introduced, and two transmission models with characteristics of permanent forgetting and temporary forgetting were proposed. The comparison between simulated experiment and real case shows that the network mode accord with the characteristics of the actual network, and the two models of phase transitions are closer to the actual.

The analysis from X-ray or neutron diffraction data, of the librational motion around the long molecular axis of the p-terphenyl central ring permits resolution of its disorder: the phenyl rotation angle on either side of the average molecular plane is ¢ = _+ 13.3 c. This disorder is associated with a double-well potential between two twisted conformations, and the overall libration on each site, at the bottoms of the double well, has the mean-square amplitude (02) = 52.5 deg 2 at room temperature. The analysis also permits separation in (02) of the mean-square amplitude of the torsional g mode (0~) = 35 deg 2 from that of the external mode (0~) = 17.5deg 2. Thus it becomes possible to scale the parameters of a simple model describing inter- and intramolecular interactions in the whole family of polyphenyls. It is shown that the intramolecular potential between two adjacent phenyl rings cannot be described by a simple sinusoidal function but exhibits a steeper gradient near the planar conformation. This double potential well model accounts for disorder and libration in crystalline p-terphenyl and p-quaterphenyl. It gives

Phase transition in organic–inorganic perovskite (C9H19NH3)2 PbI2Br2 of long-chain alkylammonium

A possibility of pattern formation in the models of the first order phase transitions with coupling between the order parameter and its gradient is discussed. We use the standard model of phase transitions extended to the higher derivatives of the order parameter that makes possible to describe the formation of various spatial distributions of the order parameter after phase transition. An example of the simple model with coupling between the order parameter and its gradient is considered. The proposed model is analogical to the mechanical nonlinear oscillator with the coordinate-dependent mass or velocity-dependent elastic module. The exact solution of this model is obtained that can be used to predict the order parameter distribution in the case of a spinodal decomposition.

We have now seen how powerful the tools of statistical mechanics can be in the study of complex systems in thermal equilibrium and how they can be used to determine exactly the behaviour of some simplified, but nevertheless interesting, models of phase transitions. However, these exact solutions are sometimes lengthy and difficult and there is no guarantee that any particular model will yield exact results. For example, we were able to solve the two-dimensional, but not the three-dimensional, Ising model, and that only in zero magnetic field. So, it is natural to consider whether we can use a computer to simulate these systems. In this chapter we shall see that this is indeed possible; a great deal of our understanding of critical phenomena is based wholly or partly on the results of such numerical work. However, as we shall also see, such simulations are not entirely straightforward and a number of tricks are needed to obtain accurate results. This is partly a consequence of the very physics that makes the phase transitions interesting to start with. One should probably think of numerical simulation as a whole subject, almost an art, of its own; we shall have time only to scratch at the surface and discuss a few of the most important ideas.

The paper addresses the question of asymptotic stability for front solutions corresponding to certain models of phase transition, invasions in population genetics, or nonlinear dynamics of perturbations of partial differential equations. The structure of front solutions for these equations is discussed, with emphasis on the relationship between the monotone front with minimum velocity and known front speed results. For a class of scalar reaction-diffusion equations, a Lyapunov functional in a travelling frame of reference is derived. Solutions which are minimal for the Lyapunov functional in certain directions of function space are stable for perturbations in those directions. The well-known minimal monotonic front solution turns out to be a minimum for the Lyapunov functional. A description of the class of perturbations to which this front is stable is derived.

Abstract

A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions

Contribution to the solution of Heisenberg's model for phase transitions: The A2→B2 disorder-order transition N binary alloys of AB stoichiometry

A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.

The structure of the bismuth electronic spectrum suggested in Ref. 4 is discussed from the viewpoint of stability of the chosen initial state of a simple cubic lattice with respect to electron and phonon interaction. The types of various instabilities are considered, and the difficulties of the fulfilment of the Luttinger theorem on the number of carriers are discussed. The initial state is shown to be absolutely unstable with respect to the doubling of the period, the dielectric arising as a result. In order to obtain a semimetallic spectrum, an acoustic deformation is necessary; its existence, in principle, may be specified by the instability mentioned above due to nonlogarithmic terms. In this case the arising state has, apparently, a superstructure with the period ∼ 100 A. A model of phase transitions under pressure is discussed.

The disadvantages of the “strategy” scan-lists for diffraction experiments created by the diffractometer software are shown. The reason for these disadvantages is that the traditionally used target function has a limited, local meaning, for example, to obtain the best coverage of the reciprocal space. An approach is proposed that implements the principle of statistical randomization of the experiment and makes it possible to achieve the strategic goal of structural analysis – obtaining a model capable of reflecting the subtle details of the atomic structure. A scan-list balanced by most factors has been found, which in less time leads to obtaining experimental data of significantly higher quality than traditional lists. The use of a reference crystal, previously measured dozens of times on diffractometers around the world, has shown the advantage of experimental data obtained by a new method. Increasing the balance and accuracy of the data resulted in a maximum improvement in the values of the refinement criteria to R1/wR2 = 0.53/0.59% and Δρ = –0.47/+0.30 e/Å. The achievement of the strategic goal of the research of the reference crystal (confirmation of the anharmonic model of atomic displacement parameters) could be confirmed not only by the “purification” of difference Fourier syntheses of electron density, which is sometimes visual and subjective, but also by a statistically flawless decrease in the R-factors of refinement by 30–40 relative %. Data of such high quality are needed to study the dynamics of structural models under external conditions, for the detection and modeling of phase transitions, critical points, bio- and chemical activity of compounds, verification of computational methods of structures.