scholarly journals Extendability of Equilibria of Nematic Polymers

2008 ◽  
Vol 2008 ◽  
pp. 1-10
Author(s):  
Hongyun Wang ◽  
Hong Zhou
Keyword(s):  
Nonlinear System ◽  
Existence And Uniqueness ◽  
Jacobian Matrix ◽  
Intersection Point ◽  
Equilibrium States ◽  
Intermolecular Potential ◽  
Small Perturbations ◽  
Nematic Polymers ◽  
Folding Point ◽  
Uniqueness Of Equilibrium

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.

scholarly journals Equilibrium states for a class of skew products

2019 ◽  
Vol 40 (11) ◽  
pp. 3030-3050
Author(s):  
MARIA CARVALHO ◽  
SEBASTIÁN A. PÉREZ
Keyword(s):  
Existence And Uniqueness ◽  
Global Analysis ◽  
American Mathematical Society ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Axiom A ◽  
Skew Products ◽  
Pure Mathematics ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We consider skew products on $M\times \mathbb{T}^{2}$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\unicode[STIX]{x1D6FA}$-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in$T^{4}$ (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.

scholarly journals Equilibrium states for Mañé diffeomorphisms

2018 ◽  
Vol 39 (9) ◽  
pp. 2433-2455 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
TODD FISHER ◽  
DANIEL J. THOMPSON
Keyword(s):  
Large Deviations ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Potential Functions ◽  
Equilibrium States ◽  
Unique Equilibrium ◽  
The Family ◽  
Natural Classes ◽  
Uniqueness Of Equilibrium

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the Sinaĭ–Ruelle–Bowen measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.

scholarly journals On equilibrium states for partially hyperbolic horseshoes

2016 ◽  
Vol 38 (1) ◽  
pp. 301-335 ◽  
Author(s):  
I. RIOS ◽  
J. SIQUEIRA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Systems ◽  
Three Dimensional ◽  
Small Variation ◽  
Equilibrium States ◽  
Dimensional System ◽  
Hölder Continuous ◽  
The Family ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

scholarly journals Existence and uniqueness of equilibrium states for some spin and continuum systems

1973 ◽  
Vol 32 (2) ◽  
pp. 153-165 ◽  
Author(s):  
M. Cassandro ◽  
G. Gallavotti ◽  
J. L. Lebowitz ◽  
J. L. Monroe
Keyword(s):  
Existence And Uniqueness ◽  
Equilibrium States ◽  
Uniqueness Of Equilibrium ◽  
Continuum Systems

Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian

2010 ◽  
Vol 31 (2) ◽  
pp. 423-447 ◽  
Author(s):  
RENAUD LEPLAIDEUR
Keyword(s):  
Lyapunov Exponent ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Fractal Dimensions ◽  
Equilibrium States ◽  
Real Analytic ◽  
Homoclinic Tangencies ◽  
Uniqueness Of Equilibrium

AbstractIn this article we prove the existence and uniqueness of equilibrium states for the potential $\phi _{t}= -t\logju \ (t\in \R )$ and the class of non-uniformly hyperbolic horseshoes which was introduced in Rios [Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity14 (2001), 431–462]. We show that the pressure t↦𝒫(t) for −tlog Ju is real-analytic on $\R $. We give the exact equations of the two asymptotes to the graph of 𝒫(t) at ±∞ and we prove that these non-uniformly hyperbolic horseshoes do not have measures which minimize the unstable Lyapunov exponent.

scholarly journals Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

2021 ◽  
pp. 1-43
Author(s):  
DOMINIC VECONI
Keyword(s):  
Thermodynamic Formalism ◽  
Equilibrium States ◽  
Maximal Entropy ◽  
Srb Measure ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Surface Homeomorphisms ◽  
Young Towers ◽  
Uniqueness Of Equilibrium ◽  
Measure Of Maximal Entropy

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

2010 ◽  
Vol 22 (10) ◽  
pp. 1147-1179 ◽  
Author(s):  
LUIS BARREIRA
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Present Stage ◽  
The Past ◽  
Recent Developments ◽  
General Sequences ◽  
Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

A Full-System Approach of the Elastohydrodynamic Line/Point Contact Problem

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
W. Habchi ◽  
D. Eyheramendy ◽  
P. Vergne ◽  
G. Morales-Espejel
Keyword(s):  
Finite Element ◽  
Nonlinear System ◽  
System Approach ◽  
Jacobian Matrix ◽  
Point Contact ◽  
System Of Equations ◽  
Nonlinear System Of Equations ◽  
Full System ◽  
Element Discretization ◽  
Fully Coupled

The solution of the elastohydrodynamic lubrication (EHL) problem involves the simultaneous resolution of the hydrodynamic (Reynolds equation) and elastic problems (elastic deformation of the contacting surfaces). Up to now, most of the numerical works dealing with the modeling of the isothermal EHL problem were based on a weak coupling resolution of the Reynolds and elasticity equations (semi-system approach). The latter were solved separately using iterative schemes and a finite difference discretization. Very few authors attempted to solve the problem in a fully coupled way, thus solving both equations simultaneously (full-system approach). These attempts suffered from a major drawback which is the almost full Jacobian matrix of the nonlinear system of equations. This work presents a new approach for solving the fully coupled isothermal elastohydrodynamic problem using a finite element discretization of the corresponding equations. The use of the finite element method allows the use of variable unstructured meshing and different types of elements within the same model which leads to a reduced size of the problem. The nonlinear system of equations is solved using a Newton procedure which provides faster convergence rates. Suitable stabilization techniques are used to extend the solution to the case of highly loaded contacts. The complexity is the same as for classical algorithms, but an improved convergence rate, a reduced size of the problem and a sparse Jacobian matrix are obtained. Thus, the computational effort, time and memory usage are considerably reduced.

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