Four variants of theory of the second order phase transitions (original) (raw)
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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.
STATISTICAL MECHANICS OF EQUILIBRIUM AND NONEQUILIBRIUM PHASE TRANSITIONS: THE YANG–LEE FORMALISM
International Journal of Modern Physics B, 2005
Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang-Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions. :43 WSPC/INSTRUCTION FILE Yang-Lee YANG-LEE FORMALISM FOR PHASE TRANSITIONS 5 particle separations, i.e., in terms of the interaction potential, |u(r)| C 1 /r d+ε as r → ∞ (with C 1 > 0 and ε positive constants), where d is the dimension of the system. Such systems are currently called short-range interaction systems. (ii) u(r) has to have a repulsive part increasing sufficiently rapid at small interparticle distances (preventing the system from collapse at high particle number densities). Throughout this paper, we shall consider systems with a hard-core interparticle repulsion at small distances, u(r) = ∞ for r < r 0 ∼ b 1/d (where b is the single-particle excluded volume), which means that a system of volume V can accommodate a finite maximum number of particles M = V /b. c (iii) Finally, the interaction potential has to be everywhere bounded from below, u(r) −u 0 whatever r (with u 0 a positive constant).
Thermodynamics of a higher-order phase transition: scaling exponents and scaling laws
2002
The well known scaling laws relating critical exponents in a second order phase transition have been generalized to the case of an arbitrarily higher order phase transition. In a higher order transition, such as one suggested for the superconducting transition in Ba 0.6 K 0.4 BiO 3 and in Bi 2 Sr 2 CaCu 2 O 8 , there are singularities in higher order derivatives of the free energy. A relation between exponents of different observables has been found, regardless of whether the exponents are classical (mean-field theory, no fluctuations, integer order of a transition) or not (fluctuation effects included). We also comment on the phase transition in a thin film.
Dynamical fluctuations in critical regime and across the 1st order phase transition
Nuclear Physics A, 2017
In this proceeding, we study the dynamical evolution of the sigma field within the framework of Langevin dynamics. We find that, as the system evolves in the critical regime, the magnitudes and signs of the cumulants of sigma field, C 3 and C 4 , can be dramatically different from the equilibrated ones due to the memory effects near T c. For the dynamical evolution across the 1st order phase transition boundary, the supercooling effect leads the sigma field to be widely distributed in the thermodynamical potential, which largely enhances the cumulants C 3 , C 4 , correspondingly.
Statistical physics and phase transitions
Phase Transitions in Machine Learning, 2009
The short introduction to thermodynamics is given in minimalist approach. Basic knowledge of classical physics is preassumed for the reader of this manuscript. First lecture covers statistical physics with reference to technical sciences.
Journal of Statistical Physics, 1987
Two-dimensional lattice-gas models with attractive interactions and particleconserving happing dynamics under the influence of a very large external electric field along a principal axis are studied in the case of a critical density. A finite-size scaling analysis allows the evaluation of critical indexes for the infinite system asβ=0.230±0.003,v=0.55±0.2, and α 0. We also describe some qualitative features of the system evolution and the existence of certain anisotropic order even well above the critical temperature in the case of finite lattices.
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.
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.
Unification of the standard and gradient theories of phase transition
We show, that the standard model of phase transition can be unified with the gradient model of phase transitions using the description in terms of the gradient of order parameter. The generalization of the gradient theory of phase transitions with regard to the fourth power of the order parameter and its gradient is proposed. Such generalization makes it possible to described wide class of phase transitions within a unified approach. In particular it is consistent with the nonlinear models, that can be used to describe a phase transition with the formation of spatially inhomogeneous distribution of the order parameter. Typical examples of such structures (with or without defects) are considered. We show that formation of spatially inhomogeneous distributions of the order parameter in the course of a phase transitions is a characteristic feature of many nonlinear models of phase transitions.
Physical Review E
Development of thermodynamic induction up to second order gives a dynamical bifurcation for thermodynamic variables and allows for the prediction and detailed explanation of nonequilibrium phase transitions with associated spontaneous symmetry breaking. By taking into account nonequilibrium fluctuations, longrange order is analyzed for possible pattern formation. Consolidation of results up to second order produces thermodynamic potentials that are maximized by stationary states of the system of interest. These potentials differ from the traditional thermodynamic potentials. In particular a generalized entropy is formulated for the system of interest which becomes the traditional entropy when thermodynamic equilibrium is restored. This generalized entropy is maximized by stationary states under nonequilibrium conditions where the standard entropy for the system of interest is not maximized. These nonequilibrium concepts are incorporated into traditional thermodynamics, such as a revised thermodynamic identity and a revised canonical distribution. Detailed analysis shows that the second law of thermodynamics is never violated even during any pattern formation, thus solving the entropic-coupling problem. Examples discussed include pattern formation during phase front propagation under nonequilibrium conditions and the formation of Turing patterns. The predictions of second-order thermodynamic induction are consistent with both observational data in the literature as well as the modeling of this data.