Unification of the standard and gradient theories of phase transition (original) (raw)

Model of phase transitions with coupling of the order parameter and its gradient

A model of phase transitions with coupling between the order parameter and its gradient is proposed. This nonlinear model is shown to be suitable for the description of phase transitions accompanied by the formation of spatially inhomogeneous distributions of the order parameter. Exact solutions of the proposed model are obtained for the special cases which can be related to the spinodal decomposition or cosmological scenario. Such solutions manifest crucial dependence on the coupling constant.

A continuum theory for first-order phase transitions based on the balance of structure order

Mathematical Methods in the Applied Sciences, 2008

First-order phase transitions are modelled by a non-homogeneous, time-dependent scalar-valued order parameter or phase field. The time dependence of the order parameter is viewed as arising from a balance law of the structure order. The gross motion is disregarded and hence the body is regarded merely as a heat conductor. Compatibility of the constitutive functions with thermodynamics is exploited by expressing the second law through the classical Clausius-Duhem inequality. First, a model for conductors without memory is set up and the order parameter is shown to satisfy a maximum theorem. Next, heat conductors with memory are considered. Different evolution problems are established through a system of differential equations whose form is related to the manner in which the memory property is represented.

Universality in the Dynamics of Second-Order Phase Transitions

Physical Review Letters, 2016

When traversing a symmetry breaking second order phase transition at a finite rate, topological defects form whose number dependence on the quench rate is given by simple power laws. We propose a general approach for the derivation of such scaling laws that is based on the analytical transformation of the associated equations of motion to a universal form rather than employing plausible physical arguments. We demonstrate the power of this approach by deriving the scaling of the number of topological defects in both homogeneous and non-homogeneous settings. The general nature and extensions of this approach are discussed.

Phase transitions with several order parameters

Physica A: Statistical Mechanics and its Applications, 1977

The predictions of catastrophe theory for phase transitions involving more than one order parameter are given. These predictions are compared with those of other theories. For the simplest transition involving two order parameters it is found that there is a parameter which does not affect the topology of the phase diagram, which does affect certain angles in the diagram, and whose measured value will not depend on the scale of external physical variables. Comparison with renormalization theory predictions for this parameter leads to general observations on the relation of catastrophe theory and renormalization theory.

Four variants of theory of the second order phase transitions

Because of one-valued connection between the configurational entropy and the order parameter it is possible to present the theory of the second order phase transitions in terms of the configurational entropy. It is offered a variant of theory, in which the Nernst theorem is obeyed. Within the framework of heterogeneous model the phenomena of growth of level of fluctuations and their correlations are analyzed at transition of critical point as competitions of kinetic and relaxation processes in the conditions of proximity of two critical points.

Unifying Model for Several Classes of Two-Dimensional Phase Transition

Physical Review Letters, 2005

A relatively simple and physically transparent model based on quantum percolation and dephasing is employed to construct a global phase diagram which encodes and unifies the critical physics of the quantum Hall, "two-dimensional metal-insulator", classical percolation and, to some extent, superconductor-insulator transitions. Using real space renormalization group techniques, crossover functions between critical points are calculated. The critical behavior around each fixed point is analyzed and some experimentally relevant puzzles are addressed. PACS numbers: 71.30.+h,73.43.Nq,74.20.Mn Two-dimensional phase transitions have been a focus of interest for many years, as they may be the paradigms of second order quantum phase transitions (QPTs). However, in spite of the abundance of experimental and theoretical information, there are still unresolved issues concerning their behavior at and near criticality, most simply exposed by the value of the critical exponent ν (describing the divergence of the correlation length at the transition). Below we discuss three examples which underscore these problems. First, in the integer quantum Hall (QH) effect, which is usually described within the single particle framework, various numerical studies yielded a critical exponent ν ≈ 2.35 [1], in agreement with heuristic arguments [2]. Some experiments indeed reported values around 2.4 [3], while other experiments reported exponents around 1.3 [4], close to the classical percolation exponent ν p = 4/3. Even more perplexing, some experiments claim that the width of the transition does not shrink to zero at zero temperature [5], in contradiction with the concept of a QPT. Additionally, the observation of a QH insulator [6] is inconsistent with the QPT scenario [7]. Consider secondly the superconductor (SC) -insulator transition (SIT), for which theoretical studies suggest several scenarios. Similar to the QH situation, some experiments yield a value ν ≈ 1.3 [8], not far from the value ν ≃ 1, predicted by numerical simulations within the random boson model, but closer to the classical percolation value. Other experiments, however, yield ν ≃ 2.8 [9], while some experiments claim an intermediate metallic phase [10]. As a third example, consider the recently claimed metal-insulator transition (MIT) [11]. The critical exponent is again close to 1.3 [12, 13] but the occurrence of such phase transition is in clear contrast with the scaling theory of localization.

Pattern formation in the models with coupling between order parameter and its gradient

The European Physical Journal B, 2013

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.

Phase transitions induced by microscopic disorder: A study based on the order parameter expansion

Physica D: Nonlinear Phenomena, 2010

Based on the order parameter expansion, we present an approximate method which allows us to reduce large systems of coupled differential equations with diverse parameters to three equations: one for the global, mean field, variable and two which describe the fluctuations around this mean value. With this tool we analyze phase-transitions induced by microscopic disorder in three prototypical models of phase-transitions which have been studied previously in the presence of thermal noise. We study how macroscopic order is induced or destroyed by time independent local disorder and analyze the limits of the approximation by comparing the results with the numerical solutions of the self-consistency equation which arises from the property of self-averaging. Finally, we carry on a finite-size analysis of the numerical results and calculate the corresponding critical exponents.

Properties of higher-order phase transitions

Nuclear Physics B, 2006

Experimental evidence for the existence of strictly higher-order phase transitions (of order three or above in the Ehrenfest sense) is tenuous at best. However, there is no known physical reason why such transitions should not exist in nature. Here, higher-order transitions characterized by both discontinuities and divergences are analysed through the medium of partition function zeros. Properties of the distributions of zeros are derived, certain scaling relations are recovered, and new ones are presented.

Condensation vs phase ordering in the dynamics of first-order transitions

Physical Review E, 1997

The origin of the non commutativity of the limits t → ∞ and N → ∞ in the dynamics of first order transitions is investigated. In the large-N model, i.e. N → ∞ taken first, the low temperature phase is characterized by condensation of the large wave length fluctuations rather than by genuine phase-ordering as when t → ∞ is taken first. A detailed study of the scaling properties of the structure factor in the large-N model is carried out for quenches above, at and below Tc. Preasymptotic scaling is found and crossover phenomena are related to the existence of components in the order parameter with different scaling properties. Implications for phase-ordering in realistic systems are discussed.