Maps of intervals with indifferent fixed points: Thermodynamic formalism and phase transitions

We develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points. For such systems the formalism yields onedimensional systems with many-body infinite-range interactions for which the thermodynamics is well defined but Gibbs states are not. (Piecewise linear systems of this kind yield the soluble, in a sense, Fisher models.) We prove that such systems exhibit phase transitions, the order of which depends on the behavior at the indifferent fixed points. We obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures. Our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable (expanding) induced version. The technique allows one also to obtain a version of the Bowen-Ruelle formula for the Hausdorff dimension of repellers for maps with indifferent fixed points, and to generalize Fisher results to some nonsoluble models.Meyerhoff Visiting Professor, on leave from the Center for Transport Theory and Mathematical Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061.     

Unusual spin transitions

New heterospin complexes of Cu(hfac)₂ (hfac, hexafluoroacetylacetonate) with pyrazole-substituted nitronyl nitroxides have been found that in the solid state exhibit thermally induced spin transitions analogous to spin crossover. For the first complex, [Cu(hfac)₂L^i-Pr], at room temperature, the Cu—O_L distances, where O_L is the oxygen atom of the nitroxyl group, are very short (2.143 Å). This leads to a strong antiferromagnetic exchange (～-120cm^?1) in the > N—^·O—Cu²⁺—O^·—N < exchange clusters. The CuO₆ coordination units formed by four O atoms of the two hfac anions and by the nitroxyl O atoms of the two bridging nitroxides have a rare form of flattened octahedra, transformed at low temperatures into elongated octahedra with shorter Cu—O_L distances (2.143 Å→2.002Å) and two longer Cu—O_hfac distances (2.130 Å→2.293 Å). For the second complex, [Cu(hfac)₂L^Bu·0.5C₆H₁₄], unusual low temperature structural dynamics of heterospin systems have been found. It is characterized by the formation of two types of CuO₆ unit. The axial Cu—O_L distances are lengthened in one unit (2.250 Å→2.347 Å) and shortened in the other (2.250 Å → 2.006 Å). This leads to a sophisticated μ_eff(T) dependence with μ_eff drastically decreased at 163 K as a result of full coupling of two spins in half of all >N—^·O—Cu²⁺—O^·—N < exchange clusters and to a shift from antiferromagnetic to ferromagnetic exchange in the other half.     

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

The dynamics near a Hopf saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.     

Boundary conditions for phase transitions and interfacial reactions

Abstract

Si et al. [1 Si, X., Oh, E.S. and Slattery, J.C. 2010. Phil. Mag., 90: 655[Taylor & Francis Online] , [Google Scholar]] pointed out that it was inappropriate to use continuity of displacement at interfaces during phase transitions or in the case of reactions at interfaces as in the case of oxidation, since appropriate reference configurations cannot be identified. They instead derived a new compatibility constraint, when atleast one of the adjoining phases is crystalline. The test of these ideas offered by Slattery et al. [2 Slattery, J.C., Si, X., Fu, K.B. and Oh, E.S. 2010. Phil. Mag., 90: 665[Taylor & Francis Online] , [Google Scholar]] was successful, but it likely was too simple, since the deformations were so small. A more stringent and successful test has recently been offered by [3 Slattery, J.C., Fu, K-B and Philos, E-SOh. 2012. Mag., 92: 1788 [Google Scholar]]. Here, we analyze oxidation on the surface of a cylinder both using an extension of the compatibility constraint and using continuity of displacement, comparing the results with the experimental observations of Imbrie and Lagoudas [4 Imbrie, P.K. and Lagoudas, D.C. 2001. Oxid. Met., 55: 359[Crossref], [Web of Science ®] , [Google Scholar]].     

On the fixed points for circle maps

Renormalization group transformations have been developed to study the critical behavior of circle maps. When the winding number equals the golden mean, the fixed point functions must satisfy two functional equations. However, it turns out that one of these equations already determines the fixed point solutions. It is shown that under certain conditions the second functional equation is automatically satisfied.     

Undecidable hopf bifurcation with undecidable fixed point

We exhibit a polynomial dynamical system where one cannot decide whether a Hopf bifurcation occurs. Therefore one cannot decide whether there will be parameter values such that a stable fixed point becomes an unstable one. Related incompleteness results for previously described axiomatized versions of dynamical systems theory are also discussed.Dedicated to the memory of Leopoldo Nachbin (1922–1993), mathematician, mentor, and friend.     

Infra-red fixed points in supersymmetry

Model independent constraints on supersymmetric models emerge when certain couplings are drawn towards their infra-red (quasi) fixed points in the course of their renormalization group evolution. The general principles are first reviewed and the conclusions for some recent studies of theories with R-parity and baryon and lepton number violations are summarized.     

Theory of the dynamics of first-order phase transitions: unstable fixed points, exponents, and dynamical scaling

Phase transitions are of great importance in a diversity of fields. They are usually classified into continuous phase transitions and first-order phase transitions (FOPTs). Whereas the former has a well-developed theoretical framework of the renormalization-group (RG) theory, no general theory has yet been developed for the latter that appear far more frequently. Focusing on the dynamics of a generic FOPT in the phi4 model below its critical point, we show by a field-theoretic RG method that it is governed by an unexpected unstable fixed point of the corresponding phi3 model. Accordingly, it exhibits a distinct scaling and universality behavior with unstable exponents different from the critical ones.     

Noise-sensitive hysteresis loops around period-doubling bifurcation points

Summary The noise-sensitive hysteresis loops observed in the bouncing-ball model are described. The phenomenon is analysed within the formalisms of the square map and the dissipative standard mapping. The notion of steady-state paths is introduced. A linear approximation of the simplest steady-state path is found.

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Riassunto Si descrivono gli anelli d’isteresi sensibili al rumore osservati nel modello della palla che rimbalza. Il fenomeno è analizzato nell’ambito dei formalismi della mappa quadrata e della funzione dissipativa standard. S’introduce la nozione di percorsi dello stato stazionario. Si trova un’approssimazione lineare del percorso piú semplice dello stato stazionario.

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Резюме Описываются петли гистерезиса, чувствительные к шуму, которые наблюдаются в модели подскакивающего мяча. Это явление анализируется в рамках формализма отображения квадрата и формализма диссипативного стандартного отображения. Вводится понятие установившихся траекторий. Получается линейная аппроксимация для простейших установившихся траекторий.

     

Universal slowing-down exponent near period-doubling bifurcation points

The critical exponent Δ, describing the slowing of convergence near period-doubling bifurcations, should be a universal constant with “mean-field” value Δ = 1.     

Theory of interfacial phase transitions in surfactant systems

The spin-1 Ising model, which is equivalent to the three-component lattice gas model, is used to study wetting transitions in three-component surfactant systems consisting of an oil, water, and a nonionic surfactant. Phase equilibria, interfacial profiles, and interfacial tensions for three-phase equilibrium are determined in mean field approximation, for a wide range of temperature and interaction parameters. Surfactant interaction parameters are found to strongly influence interfacial tensions, reducing them in some cases to ultralow values. Interfacial tensions are used to determine whether the middle phase, rich in surfactant, wets or does not wet the interface between the oil-rich and water-rich phases. By varying temperature and interaction parameters, a wetting transition is located and found to be of the first order. Comparison is made with recent experimental results on wetting transitions in ternary surfactant systems.This paper is dedicated to J. K. Percus in honor of his 65th birthday.     

Symplectic fixed points and holomorphic spheres

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.     

Critical fixed points in discrete hydrodynamics

A discrete-cell formulation of hydrodynamics was recently introduced, which is exactly renormalizable in a certain sense: if one knows the discrete equations of motion for a certain cell size W and discrete time interval τ, one can accurately numerically calculate the equations of motion on the coarser scales 2W or 2τ. These coarsening transformations have previously been investigated for the one-dimensional diffusive system. A line of fixed points was found, parameterized by the (positive) diffusivity D'. In this paper we examine the behavior of the coarsening transformation on the D' = 0 manifold in the space of equations of motion for one-dimensional systems. We find another line of fixed points, this one parameterized by the super-Burnett coefficient D'₃. This corresponds to a Gaussian critical point. The possibility of generalizing this to non-Gaussian (Ising-like) critical points is discussed.