Unique Equilibrium State Research Articles

Unique equilibrium states for flows and homeomorphisms with non-uniform structure

Using an approach due to Bowen, Franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the Bowen property. We show that this conclusion remains true using weaker non-uniform versions of specification, expansivity, and the Bowen property. We also establish a corresponding result for homeomorphisms. In the homeomorphism case, we obtain the upper bound from the level-2 large deviations principle for the unique equilibrium state. The theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic formalism in partially hyperbolic and non-uniformly hyperbolic settings.

Criteria for the density of the graph of the entropy map restricted to ergodic states

We consider a non-uniquely ergodic dynamical system given by a$\mathbb{Z}^{l}$-action (or$(\mathbb{N}\cup \{0\})^{l}$-action)$\unicode[STIX]{x1D70F}$on a non-empty compact metrisable space$\unicode[STIX]{x1D6FA}$, for some$l\in \mathbb{N}$. Let (D) denote the following property: the graph of the restriction of the entropy map$h^{\unicode[STIX]{x1D70F}}$to the set of ergodic states is dense in the graph of$h^{\unicode[STIX]{x1D70F}}$. We assume that$h^{\unicode[STIX]{x1D70F}}$is finite and upper semi-continuous. We give several criteria in order that (D) holds, each of which is stated in terms of a basic notion: Gateaux differentiability of the pressure map$P^{\unicode[STIX]{x1D70F}}$on some sets dense in the space$C(\unicode[STIX]{x1D6FA})$of real-valued continuous functions on$\unicode[STIX]{x1D6FA}$, level-two large deviation principle, level-one large deviation principle, convexity properties of some maps on$\mathbb{R}^{n}$for all$n\in \mathbb{N}$. The one involving the Gateaux differentiability of$P^{\unicode[STIX]{x1D70F}}$is of particular relevance in the context of large deviations since it establishes a clear comparison with another well-known sufficient condition: we show that for each non-empty$\unicode[STIX]{x1D70E}$-compact subset$\unicode[STIX]{x1D6F4}$of$C(\unicode[STIX]{x1D6FA})$, (D) is equivalent to the existence of an infinite dimensional vector space$V$dense in$C(\unicode[STIX]{x1D6FA})$such that$f+g$has a unique equilibrium state for all$(f,g)\in \unicode[STIX]{x1D6F4}\times V\setminus \{0\}$; any Schauder basis$(f_{n})$of$C(\unicode[STIX]{x1D6FA})$whose linear span contains$\unicode[STIX]{x1D6F4}$admits an arbitrary small perturbation$(h_{n})$so that one can take$V=\text{span}(\{f_{n}+h_{n}:n\in \mathbb{N}\})$. Taking$\unicode[STIX]{x1D6F4}=\{0\}$, the existence of an infinite dimensional vector space dense in$C(\unicode[STIX]{x1D6FA})$constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of the measure of maximum entropy.

Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if$f$has an expanding repeller and$\unicode[STIX]{x1D719}$is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval$I\subset \mathbb{R}$can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems

It has previously been suggested that small subsystems of closed quantum systems thermalize under some assumptions; however, this has been rigorously shown so far only for systems with very weak interaction between subsystems. In this work, we give rigorous analytic results on thermalization for translation-invariant quantum lattice systems with finite-range interaction of arbitrary strength, in all cases where there is a unique equilibrium state at the corresponding temperature. We clarify the physical picture by showing that subsystems relax towards the reduction of the global Gibbs state, not the local Gibbs state, if the initial state has close to maximal population entropy and certain non-degeneracy conditions on the spectrum are satisfied. Moreover, we show that almost all pure states with support on a small energy window are locally thermal in the sense of canonical typicality. We derive our results from a statement on equivalence of ensembles generalizing earlier results by Lima, and give numerical and analytic finite-size bounds, relating the Ising model to the finite de Finetti theorem. Furthermore, we prove that global energy eigenstates are locally close to diagonal in the local energy eigenbasis, which constitutes a part of the eigenstate thermalization hypothesis that is valid regardless of the integrability of the model.

Weak expansion properties and large deviation principles for expanding Thurston maps

In this paper, we prove that an expanding Thurston map f:S2→S2 is asymptotically h-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is h-expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on S2, there exists at least one equilibrium state. For such maps, we also establish large deviation principles for iterated preimages and periodic points. It follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding Thurston map without periodic critical points and a potential that is Hölder continuous with respect to a visual metric on S2.

Wild attractors and thermodynamic formalism

Fibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family $$f_\lambda $$ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential $$\phi _t = -t\log |f'_\lambda |$$ , there is a unique phase transition at some $$t_1 \leqslant 1$$ , and the pressure $$P(\phi _t)$$ is analytic (with unique equilibrium state) elsewhere. The pressure is majorised by a non-analytic $$C^\infty $$ curve (with all derivatives equal to $$0$$ at $$t_1 < 1$$ ) at the emergence of a wild attractor, whereas the phase transition at $$t_1 = 1$$ can be of any finite order for those $$\lambda $$ for which $$f_\lambda $$ is Lebesgue conservative. We also obtain results on the existence of conformal measures and equilibrium states, as well as the hyperbolic dimension and the dimension of the basin of $$\omega (c)$$ .

Hypocoercivity for linear kinetic equations conserving mass

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted L 2 L^2 norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.

Emergence of phase-locked states for the Winfree model in a large coupling regime

We study the large-time behavior of the globally coupled Winfree model in a large coupling regime. The Winfree model is the first mathematical model for the synchronization phenomenon in an ensemble of weakly coupled limit-cycle oscillators. For the dynamic formation of phase-locked states, we provide a sufficient framework in terms of geometric conditions on the coupling functions and coupling strength. We show that in the proposed framework, the emergent phase-locked state is the unique equilibrium state and it is asymptotically stable in an $l^1$-norm; further, we investigate its configurational structure. We also provide several numerical simulations, and compare them with our analytical results.

Predicting financial time series data using artificial immune system-inspired neural networks

This paper investigates a set of approaches for the prediction of noisy time series data; specifically, the prediction of financial signals. A novel dynamic self-organised multilayer neural network based on the immune algorithm for financial time series prediction is presented, combining the properties of both recurrent and self-organised neural networks. In an attempt to overcome inherent stability and convergence problems, the network is derived to ensure that it reaches a unique equilibrium state. The accuracy of the comparative evaluation is enhanced in terms of profit earning; empirical testing used in this work includes normalised mean square error NMSE to evaluate forecast fitness and also evaluates predictions against financial metrics to assess profit generation. Extensive simulations for multi-step prediction in stationary and non-stationary time series were performed. The resulting forecast made by the proposed network shows substantial profits on financial historical signals when compared to various solely neural network approaches. These simulations suggest that dynamic immunology-based self-organised neural networks have a better ability to capture the chaotic movement in financial signals.

The thermodynamic approach to multifractal analysis

AbstractMost results in multifractal analysis are obtained using either a thermodynamic approach based on the existence and uniqueness of equilibrium states or an orbit-gluing approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the orbit-gluing approach. We compare the thermodynamic and orbit-gluing approaches and give a survey of many of the multifractal results in the literature.

Complex dynamics of monopolies with gradient adjustment

This paper aims to show that delay matters in continuous- and discrete-time framework. It constructs a simple dynamic model of a boundedly rational monopoly. First the existence of the unique equilibrium state is proved under general price and cost function forms. Conditions are derived for its local asymptotical stability with both continuous and discrete time scales. The global dynamic behavior of the systems is then numerically examined, demonstrating that the continuous system is globally asymptotically stable without delay and in the presence of delay if the delay is sufficiently small. Then stability of the continuous system is lost via Hopf bifurcation. In the discrete case without delay, the steady state is locally asymptotically stable if the speed of adjustment is small enough, then stability is lost via period-doubling bifurcation. If the delay is one or two steps, then stability loss occurs via Neimark–Sacker bifurcation.

Equilibrium states of interval maps for hyperbolic potentials

We study the thermodynamic formalism of sufficiently regular interval maps for Hölder continuous potentials. We show that for a hyperbolic potential there is a unique equilibrium state, and that this measure is exponentially mixing. Moreover, we show the absence of phase transitions: the pressure function is real analytic at such a potential.

The Kac Model Coupled to a Thermostat

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $\beta$. The system admits the canonical distribution at inverse temperature $\beta$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.

Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, “hyperbolic”. We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Hölder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Hölder continuous functions. Another consequence is that every Hölder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.

On the work of Sarig on countable Markov chains and thermodynamic formalism

The paper is a nontechnical survey and is aimed to illustrate Sarig'sprofound contributions to statistical physics and in particular,thermodynamic formalism for countable Markov shifts. I will discusssome of Sarig's work on characterization of existence of Gibbsmeasures, existence and uniqueness of equilibrium states as well asphase transitions for Markov shifts on a countable set of states.

Network congestion control with Markovian multipath routing

In this paper we consider an integrated model for TCP/IP protocols with multipath routing. The model combines a Network Utility Maximization for rate control based on end-to-end queueing delays, with a Markovian Traffic Equilibrium for routing based on total expected delays. We prove the existence of a unique equilibrium state which is characterized as the solution of an unconstrained strictly convex program. A distributed algorithm for solving this optimization problem is proposed, with a brief discussion of how it could be implemented by adapting the current Internet protocols.

Diffusion in inhomogeneous flows: Unique equilibrium state in an internal flow

The role of diffusion in creating rotationality (enstrophy) is studied here and a transport equation for enstrophy is derived to explain this connection. As an illustration, flow instabilities and pattern formation are investigated here for an inhomogeneous internal flow with definitive boundary conditions. Results obtained by direct numerical simulation (DNS) of flow inside a two-dimensional rectangular lid driven cavity (RLDC) show that diffusion is responsible in forming patterns at a post-critical Reynolds numbers. The transport equation for enstrophy derived from the Navier–Stokes equation in Eulerian framework helps to explain the enstrophy spectrum in flows, specially in 2D flows, where vortex stretching is absent as the dominant energy cascade mechanism to small scales. For the 2D flow in RLDC, diffusion and convection provide a unique equilibrium state in an intermediate post-critical range of Reynolds number around 6000. This is independent of the geometric aspect ratio (height to width of the cavity) of the cavity greater than or equal to two. Such equilibrium can be observed in numerical simulations, only when special care is exercised for diffusion discretization at high wavenumbers. Another motivation in this work is to show that diffusion and dissipation are not identical for inhomogeneous flows, as opposed to equating these in studies of homogeneous turbulent flows. Organized enstrophy is shown as a consequence of over-riding action of diffusion in creating rotationality in this flow.

Markov approximations of Gibbs measures for long-range interactions on 1D lattices

We study one-dimensional lattice systems with pair-wise interactions of infinite range. We show projective convergence of Markov measures to the unique equilibrium state. For this purpose we impose a slightly stronger condition than summability of variations on the regularity of the interaction. With our condition we are able to explicitly obtain stretched exponential bounds for the rate of mixing of the equilibrium state. Finally we show convergence for the entropy of the Markov measures to that of the equilibrium state via the convergence of their topological pressure (free energy).

Intelligent control for long-term ecological systems

An intelligent control methodology for a long-term ecological systems is developed in this paper. This intelligent control methodology is called as robust recurrent fuzzy neural network control RRFNNC. This control methodology is used to deal with multi-biomass ecological system which is an uncertain nonlinear system subject to unpredictable but bounded disturbances. This RRFNNC system is comprised of a recurrent fuzzy neural network RFNN controller and a robust controller. The RFNN controller is used to approximate an ideal controller; and the robust controller is designed to compensate for the approximation error between the RFNN controller and the ideal controller. The proposed RRFNNC system is applied to keep the multi-biomasses of ecological system within a stay small neighborhood of the unique nontrivial optimal equilibrium state of the undisturbed exploited ecosystem. For the simulation results of accumulative yield of harvest, more harvest can be obtained by applying the proposed RRFNNC system when compared with state feedback control.

Modeling the Epidemiology of Malaria and Control with Estimate of the Basic Repro-duction Number

Strategies for controlling the epidemiology of many infectious diseases such as malaria include a rapid reduc-tion in both the infected and susceptible population via treatment and vaccination. In this paper, we have modified the Tumwiine et al. (2007) mathematical model for the transmission of malaria by including a vaccination parameter. We have shown that the model has a unique disease-free equilibrium state which is locally and globally asymptotically stable, if 1, and that the endemic equilibrium exist provided > 1, where is a parameter which depends on the given model parameters. Numerical simulations of the modified model clearly show that, with a proper combination of treatment and vaccination, offered at about 65% each on the susceptible and infected population, malaria can be eradicated from the community.