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Every ergodic measure is uniquely maximizing
 Discr. & Cont. Dyn. Sys
"... Abstract. Let Mφ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f: X → R, a measure µ ∈ Mφ is called fmaximizing if � f dµ = sup { � f dm: m ∈ Mφ}. It is shown that if µ is any ergodic measure in Mφ, th ..."
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Abstract. Let Mφ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f: X → R, a measure µ ∈ Mφ is called fmaximizing if � f dµ = sup { � f dm: m ∈ Mφ}. It is shown that if µ is any ergodic measure in Mφ, then there exists a continuous function whose unique maximizing measure is µ. More generally, if E is a nonempty collection of ergodic measures which is weak ∗ closed as a subset of Mφ, then there exists a continuous function whose set of maximizing measures is precisely the closed convex hull of E. If moreover φ has the property that its entropy map is upper semicontinuous, then there exists a continuous function whose set of equilibrium states is precisely the closed convex hull of E. 1. Introduction. Let X be a compact metrizable space, and Γ a topological group or semigroup. Let φ be a topological action of Γ on X, i.e., a continuous map φ: Γ × X → X, (γ, x) ↦ → φγ(x) such that φ1 = idX and φγ ′ ◦ φγ = φγ ′ γ for all γ, γ ′ ∈ Γ. Let B denote the σalgebra of Borel subsets of X. A Borel probability
Integral Representations in Conuclear Cones
, 1994
"... this article. Wellcapped cones are conuclear. The main tool is Choquet's notion of conical measure, of which we present the necessary properties here. ..."
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this article. Wellcapped cones are conuclear. The main tool is Choquet's notion of conical measure, of which we present the necessary properties here.
Why the Quantum Must Yield to Gravity
"... After providing an extensive overview of the conceptual elements – such as Einstein’s ‘hole argument ’ – that underpin Penrose’s proposal for gravitationally induced quantum state reduction, the proposal is constructively criticised. Penrose has suggested a mechanism for objective reduction of quan ..."
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After providing an extensive overview of the conceptual elements – such as Einstein’s ‘hole argument ’ – that underpin Penrose’s proposal for gravitationally induced quantum state reduction, the proposal is constructively criticised. Penrose has suggested a mechanism for objective reduction of quantum states with postulated collapse time τ = ¯h/∆E, where ∆E is an illdefinedness in the gravitational selfenergy stemming from the profound conflict between the principles of superposition and general covariance. Here it is argued that, even if Penrose’s overall conceptual scheme for the breakdown of quantum mechanics is unreservedly accepted, his formula for the collapse time of superpositions reduces to τ → ∞ (∆E → 0) in the strictly Newtonian regime, which is the domain of his proposed experiment to corroborate the effect. A suggestion is made to rectify this situation. In particular, recognising the cogency of Penrose’s reasoning in the domain of full ‘quantum gravity’, it is demonstrated that an appropriate experiment which could in principle corroborate his argued ‘macroscopic ’ breakdown of superpositions is not the one involving nonrotating mass distributions as he has suggested, but a Leggetttype SQUID or BEC
Stochastic measures and modular evolution in nonequilibrium thermodynamics
, 908
"... We present an application of the theory of stochastic processes to model and categorize nonequilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical structure for (physical) correlation functions and nonequilib ..."
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We present an application of the theory of stochastic processes to model and categorize nonequilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical structure for (physical) correlation functions and nonequilibrium thermodynamical potentials. It is proposed that macroscopic evolution equations (such as dynamic correlation functions) may be obtained from a nonequilibrium thermodynamical description, by using the fact that extended thermodynamical potentials belongs to a certain class of statistical systems whose probability distribution functions are defined by a stationary measure; although a measure which is, in general, different from the equilibrium Gibbs measure. These probability measures obey a certain hierarchy on its stochastic evolution towards the most probable (stationary) measure. This in turns defines a convergence sequence. We propose a formalism which considers the mesoscopic stage (typical of nonlocal dissipative processes such as the ones described by extended irreversible thermodynamics) as being governed by stochastic dynamics due to the effect of nonequilibrium fluctuations. Some applications of the formalism are described. Scope The paper is outlined as follows: Section 1 is a brief introduction to the problem of applying measure theoretical tools to the study of manyparticle physical systems, also some recent developments in the field are mentioned. We sketch how the probability measures approach has been applied to equilibrium systems (states). In section 2 we present an extension of such method for the case of nonequilibrium systems (processes) by means of the ChoquetMeyer Theorem on continuous measures. In this section two propositions (2.1 and 2.3) are made in terms of nonequilibrium stochastic measures as how to deal with systems out of
Electronic Journal of Theoretical Physics Stochastic Measures and Modular Evolution in Nonequilibrium Thermodynamics
, 2007
"... Abstract: We present an application of the theory of stochastic processes to model and categorize nonequilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical structure for (physical) correlation functions and n ..."
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Abstract: We present an application of the theory of stochastic processes to model and categorize nonequilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical structure for (physical) correlation functions and nonequilibrium thermodynamical potentials. It is proposed that macroscopic evolution equations (such as dynamic correlation functions) may be obtained from a nonequilibrium thermodynamical description, by using the fact that extended thermodynamical potentials belong to a certain class of statistical systems whose probability distribution functions are defined by a stationary measure; although a measure which is, in general, different from the equilibrium Gibbs measure. These probability measures obey a certain hierarchy on its stochastic evolution towards the most probable (stationary) measure. This in turns defines a convergence sequence. We propose a formalism which considers the mesoscopic stage (typical of nonlocal dissipative processes such as the ones described by extended irreversible thermodynamics) as being governed by stochastic dynamics due to the effect of nonequilibrium fluctuations. Some applications of the formalism are described.