28,090 research outputs found
Invariance entropy, quasi-stationary measures and control sets
For control systems in discrete time, this paper discusses measure-theoretic
invariance entropy for a subset Q of the state space with respect to a
quasi-stationary measure obtained by endowing the control range with a
probability measure. The main results show that this entropy is invariant under
measurable transformations and that it is already determined by certain subsets
of Q which are characterized by controllability properties.Comment: 30 page
Gibbs entropy and irreversible thermodynamics
Recently a number of approaches has been developed to connect the microscopic
dynamics of particle systems to the macroscopic properties of systems in
nonequilibrium stationary states, via the theory of dynamical systems. This way
a direct connection between dynamics and Irreversible Thermodynamics has been
claimed to have been found. However, the main quantity used in these studies is
a (coarse-grained) Gibbs entropy, which to us does not seem suitable, in its
present form, to characterize nonequilibrium states. Various simplified models
have also been devised to give explicit examples of how the coarse-grained
approach may succeed in giving a full description of the Irreversible
Thermodynamics. We analyze some of these models pointing out a number of
difficulties which, in our opinion, need to be overcome in order to establish a
physically relevant connection between these models and Irreversible
Thermodynamics.Comment: 19 pages, 4 eps figures, LaTeX2
Propagation of Memory Parameter from Durations to Counts
We establish sufficient conditions on durations that are stationary with
finite variance and memory parameter to ensure that the
corresponding counting process satisfies () as , with the same memory parameter that was assumed for the durations. Thus, these conditions ensure that
the memory in durations propagates to the same memory parameter in counts and
therefore in realized volatility. We then show that any utoregressive
Conditional Duration ACD(1,1) model with a sufficient number of finite moments
yields short memory in counts, while any Long Memory Stochastic Duration model
with and all finite moments yields long memory in counts, with the same
Gibbs entropy and irreversible thermodynamics
Recently a number of approaches has been developed to connect the microscopic
dynamics of particle systems to the macroscopic properties of systems in
nonequilibrium stationary states, via the theory of dynamical systems. This way
a direct connection between dynamics and Irreversible Thermodynamics has been
claimed to have been found. However, the main quantity used in these studies is
a (coarse-grained) Gibbs entropy, which to us does not seem suitable, in its
present form, to characterize nonequilibrium states. Various simplified models
have also been devised to give explicit examples of how the coarse-grained
approach may succeed in giving a full description of the Irreversible
Thermodynamics. We analyze some of these models pointing out a number of
difficulties which, in our opinion, need to be overcome in order to establish a
physically relevant connection between these models and Irreversible
Thermodynamics.Comment: 19 pages, 4 eps figures, LaTeX2
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