The equilibrium states for semigroups of rational maps

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)-conformal measure, where P(ψ) denotes the topological pressure of ψ. The research of M. Urbański was supported in part by the NSF Grant DMS 0400481. H. Sumi thanks University of North Texas for kind hospitality, during his stay there.     

The stability of constant equilibrium states in relaxation models

We consider various first-order systems of PDEs with partial dissipation, as well as partial conservation. This class includes relaxation models, for instance the one designed by S. Jin and Z. Xin, as well as discrete velocity models for gases, as the Broadwell system. As we showed in a recent paper, the Jin-Xin model admits a convex compact positively invariant region, whenever the equilibrium system does. As a by-product, we obtained the existence of global weak solutions for the Cauchy problem with large data. For more general systems, the global existence of a uniformly bounded entropy solution will be a basic assumption in this work. We consider one-dimensional data which are either space periodic or square integrable. We prove that the (expected globally bounded) entropy solution relaxes to the equilibrium state; the latter is either zero or is determined by the mean value of the conserved components. We emphasize that we do not need any assumption about the nonlinearity of the underlying equilibrium system. We give two different proofs of the stabilization, which apply in different contexts. The first one uses compensated compactness and has a rather broad efficiency. For instance, it applies to several quasi-linear models. But the convergence result does not provide any decay rate in the periodic setting. The other one uses a dispersion estimate for the principal part of the model. It applies to periodic data and needs the strong assumption of semi-linearity, but yields an exponential decay in theL ²-norm. We expect that it could extend to multi-dimensional contexts.     

Rotation, entropy, and equilibrium states

For a dynamical system and function we consider the corresponding generalised rotation set. This is the convex subset of consisting of all integrals of with respect to -invariant probability measures. We study the entropy of rotation vectors , and relate this to the directional entropy of Geller & Misiurewicz. For a mixing subshift of finite type, and of summable variation, we prove that if the rotation set is strictly convex then the functions and are in fact one and the same. For those rotation sets which are not strictly convex we prove that and can differ only at non-exposed boundary points .

     

Nonuniqueness of equilibrium states for bars in tension

This paper presents an infinite linear program with the optimal value of the maximizing problem strictly greater than the optimal value of the minimizing problem.     

The influence of original imperfections on the equilibrium states of physically nonlinear panels

The influence of original imperfections, curvature of the panels, and the level of growth of plastic deformations on the equilibrium states of longitudinally compressed cylindrical panels have been studied. The physical relationships were established on the basis of the theory of localized deformations. It has been found that original imperfections substantially lower the upper critical stresses in the zone of elastic and slowly growing plastic deformations, depending on the curvature parametera/R. In a zone of growing plastic deformations, the solution according to nonlinear shell theory approximates the solution according to linear shell theory, and change of curvature of the panels does not cause a sharp increase of the upper critical stress as occurs in a zone of plastic deformations.Institute for Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 345–351, March–April, 1970.     

Isoperimetric inequality and equilibrium states of a two-phase medium

A two-sided estimate for the volume of each phase of a two-phase equilibrium state is derived provided that the equilibrium displacement field of the one-phase problem is smooth. The obtained estimate is uniform with respect to the temperature and surface tension coefficient. Bibliography: 8 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 81–88.     

Pressure and equilibrium states for countable state markov shifts

We give a general definition of the topological pressureP _top (f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for functions satisfying a mild distortion property. We introduce a new notion of Z-recurrent functions. Given any such functionf, we show a general method how to obtain tight sequences of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state forf if and only if εf ⁻ dμ<∞. We discuss some conditions that ensure this integrability. As an application we obtain the Gauss measure as a weak limit of measures supported on periodic points.     

A method for determining the equilibrium states of dynamic systems

A heuristic method is presented for determining the equilibrium states of motion of dynamic systems, in particular, spacecraft. The method can also be applied to the solution of sets of linear or nonlinear algebraic equations. A positive-semidefinite functional is formed to convert the problem to that of finding those minimum points where the functional vanishes. The process is initiated within a selecteddomain of interest by random search; convergence to a minimum is obtained by a modified Davidon's deflected gradient technique. To render this approach feasible in the presence of constraints, the functional is modified to include penalty terms which cause the functional to approach infinity at the constraint boundaries. Close approximations to solutions near the constraint boundaries are found by applying Carroll's approach in successively reducing the weighting factors of the penalty terms. After finding a minimum, the local domain around this point is eliminated by adding to the functional an interior constraint term, representing the surface under a hypersphere centered at the minimum point. The domain of consideration now becomes the subdomain formed by subtracting the space contained within this hypersphere from the previous domain of interest. Minima are now sought within the remaining space, as before.This paper is derived from research performed by the author while employed by TRW Systems Group, Redondo Beach, California.The author acknowledges the helpful suggestions of Dr. G. Bekey, University of Southern California, and those of Mr. E. A. Quast, Dr. M. P. Scher, and Dr. R. J. Wiley, Dynamics Department, TRW Systems Group, Redondo Beach, California.     

Application of optimization methods for finding equilibrium states of two-dimensional crystals

A two-dimensional model of a multilayer material and a procedure for simulating its properties based on global optimization methods are proposed. This model is applied for the case of a two-dimensional crystal. Global minima of the interaction energy of the material’s atoms are found, and geometric characteristics of its corresponding equilibrium states are described. The resulting lattices, in particular graphene’s lattices, agree with experimental data, which confirms the validity of the proposed approach. This approach can be extended to a wider class of layered structures, and it can be used for determining the mechanical properties of materials.     

Effect of initial imperfections on the equilibrium states of elastoplastic sheets

The local strains theory provides the basis for an analysis of the equilibrium states of initially imperfect compressed sheets of elastoplastic strain-hardening material. In the presence of initial imperfections of the sheet geometry there is no clearly expressed limit at which the deflection increases sharply. Such a limit, corresponding to the critical stress in the linear theory of plates and shells, emerges as the initial imperfection parameter tends to zero. The difference between the total transverse deflections of the initially imperfect and ideal sheets reaches a maximum at the stress level corresponding to this limit. Beyond this limit the initial imperfection effect is important at leastic and weakly developed plastic strains but diminishes considerably as the plastic strain level develops.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 897–905, September–October, 1968.     

Instability of equilibrium states for coupled heat reservoirs at different temperatures

We consider quantum systems consisting of a “small” system coupled to two reservoirs (called open systems). We show that such systems have no equilibrium states normal with respect to any state of the decoupled system in which the reservoirs are at different temperatures, provided that either the temperatures or the temperature difference divided by the product of the temperatures are not too small. Our proof involves an elaborate spectral analysis of a general class of generators of the dynamics of open quantum systems, including quantum Liouville operators (“positive temperature Hamiltonians”) which generate the dynamics of the systems under consideration.     

Principal eigenvalues,topological pressure,and stochastic stability of equilibrium states

Suppose thatL is a second order elliptic differential operator on a manifoldM, B is a vector field, andV is a continuous function. The paper studies by probabilistic and dynamical systems means the behavior asɛ → 0 of the principal eigenvalueλ ^ε (V) for the operatorL ^ε = ɛL + (B, ∇) +V considered on a compact manifold or in a bounded domain with zero boundary conditions. Under certain hyperbolicity conditions on invariant sets of the dynamical system generated by the vector fieldB the limit asɛ → 0 of this principal eigenvalue turns out to be the topological pressure for some function. This gives a natural transition asɛ → 0 from Donsker-Varadhan’s variational formula for principal eigenvalues to the variational principle for the topological pressure and unifies previously separate results on random perturbations of dynamical systems. This work was supported by US-Israel Binational Science Foundation.     

Existence,uniqueness and stability of equilibrium states for non-uniformly expanding maps

We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact metric spaces and Hölder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation.     

Convergence to equilibrium states for fluid models of many-server queues with abandonment

Fluid models, in particular their equilibrium states, have become an important tool for the study of many-server queues with general service and patience time distributions. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under some conditions. Our method builds on the framework of measure-valued processes, which keeps track of the remaining patience and service times.     

A local variational principle of pressure and its applications to equilibrium states

We prove a local variational principle of pressure for any given open cover. More precisely, for a given dynamical system (X, T), an open cover of X, and a continuous, real-valued function f on X, we show that the corresponding local pressure P(T, f; ) satisfies

Solution of transient markovian state probabilities and rate of approach to equilibrium states

The differential equations for transient state probabilities for Markovian processes are examined to derive the rate of convergence of transient states to equilibrium states. There is an acute need to solve the balance equations for large states, particularly for handling computer per- formance modeling with a network of queues that do not satisfy product form solutions or can- not be cast into the forms convenient for mean value analysis. The rate of convergence to equilibrium states is derived for irreducible aperiodic homogeneous Markov chains on the basis of a geometrical interpretation. A numerical integration method with dynamic step-size adjustments is applied and compared against the power method of Wallace and Rosenberg.