Related papers: Some basic facts on the system \Delta u - W_u (u) …
The main objective of this article are two-fold. First, we introduce some general principles on phase transition dynamics, including a new dynamic transition classification scheme, and a Ginzburg-Landau theory for modeling equilibrium phase…
In the present paper we consider the system {\Delta}u - W_u (u) = 0, where u: R^n to R^n, for a class of potentials W: R^n to R that possess several global minima and are invariant under a general finite reflection group G. We establish…
Recently, Giorgio Fusco and the author studied the system {\Delta}u - W_u (u) = 0 for a class of potentials that possess several global minima and are invariant under a general finite reflection group, and established existence of…
We consider the phase transition in a model which consists of a Ginzburg-Landau free energy for superconductors including a Chern-Simons term. The mean field theory of Halperin, Lubensky and Ma [Phys. Rev. Lett. 32, 292 (1974)] is applied…
Systems kept out of equilibrium in stationary states by an external source of energy store an energy $\Delta U=U-U_0$. $U_0$ is the internal energy at equilibrium state, obtained after the shutdown of energy input. We determine $\Delta U$…
In this paper, we revisit the classical linear turning point problem for the second order differential equation $\epsilon^2 x'' +\mu(t)x=0$ with $\mu(0)=0,\,\mu'(0)\ne 0$ for $0<\epsilon\ll 1$. Written as a first order system, $t=0$…
We investigate the Allen-Cahn system \begin{equation*} \Delta u-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the…
This paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the Ginzburg-Landau theory as a unified language. In the first part, mean-field theory is…
Paying attention to the difference of density of states, \Delta ln g(E) = ln g(E+\Delta E) - ln g(E), we study the convergence of the Wang-Landau method. We show that this quantity is a good estimator to discuss the errors of convergence,…
Using a physically motivated stress energy tensor, we prove weak and strong monotonicity formulas for solutions to the semilinear elliptic system $\Delta u=\nabla W(u)$ with $W$ nonnegative. In particular, we extend a recent two dimensional…
We consider a Ginzburg-Landau type equation in $\R^2$ of the form $-\Delta u = u J'(1-|u|^{2})$ with a potential function $J$ satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context…
Geometrical approach to the phenomenological theory of phase transitions of the second kind at constant pressure $P$ and variable temperature $T$ is proposed. Equilibrium states of a system at zero external field and fixed $P$ and $T$ are…
In this study, we present theoretical investigations of phase transitions and critical phenomena in materials through the lens of second-order Ginzburg-Landau theory, in conjunction with considerations of symmetry groups and thermal…
A simple solid-on-solid model of adsorbate-induced faceting is studied by using a modified Wang-Landau method. The phase diagram for this system is constructed by computing the density of states in a special two-dimensional energy space. A…
We study the effective geometric motions of an anisotropic Ginzburg--Landau equation with a small parameter $\varepsilon>0$ which characterizes the width of the transition layer. For well-prepared initial datum, we show that as…
In this Letter, the dynamic phase transitions of the time-dependent Ginzburg-Landau equations are analyzed using a newly developed dynamic transition theory and a new classification scheme of dynamics phase transitions. First, we…
We develop a Landau like theory to characterize the phase transitions in resetting systems. Restart can either accelerate or hinder the completion of a first passage process. The transition between these two phases is characterized by the…
We investigate the Allen-Cahn system \begin{equation*} \Delta u-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the…
We consider the following elliptic system \Delta u =\nabla H (u) \ \ \text{in}\ \ \mathbf{R}^N, where $u:\mathbf{R}^N\to \mathbf{R}^m$ and $H\in C^2(\mathbf{R}^m)$, and prove, under various conditions on the nonlinearity $H$ that, at least…
The nonlinear $\sigma$-model for disordered interacting electrons is studied in spatial dimensions $d>4$. The critical behavior at the metal-insulator transition is determined exactly, and found to be that of a standard…