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Mass Transfer in MOFs
Published in T. Grant Glover, Bin Mu, Gas Adsorption in Metal-Organic Frameworks, 2018
Coincidence between ensemble and time averages is the key message of the Ergodic Theorem.96 It is expected to hold, quite in general, under macroscopic equilibrium.97,98 Its experimental proof with respect to molecular diffusion, however, has to struggle with the controversial conditions of measurement. While PFG NMR diffusion measurements attain their highest accuracies with highly mobile molecules at high concentrations, for single-particle tracking just the opposite is true—since only then can one be sure to follow a particular molecule, that is, to determine a genuine trajectory “by connecting the dots” (i.e., subsequent fluorescing points).99 With a guest molecule (Atto532—ATTO532-COOH, ATTO-TEC, Siegen, Germany) offering suitable measuring conditions for both single-particle tracking and PFG NMR and using, as a host system, a nanoporous glass with a particularly homogeneous internal surface, the gap in the measuring conditions was bridged, yielding the expected coincidence in the measured diffusivities.100
Laser Basics
Published in Mohammad E. Khosroshahi, Applications of Biophotonics and Nanobiomaterials in Biomedical Engineering, 2017
The fluctuations, Δm, that occur for a beam of light with constant intensity are referred to as particle fluctuations and Δm∝I $ {{\iDelta m}}\, \propto \sqrt {{I}} $ . As we know, the photon rate equations can be used to investigate the effect of the atomic absorption and emission processes on the statistical properties of an incident laser photon distribution. If the beam during the interaction with a material is attenuated and the number of atoms in ground state is higher than the number in the upper level (N1 > N2), then the beam is said to be weakened. If, however, N2 > N1, then the beam is said to be amplified. The occurrence of photon absorption and emission causes the number of photons in each mode of the radiation field in the cavity to fluctuate. According to the ergodic theory of statistical mechanics, which is concerned with the behaviour of a dynamical system when it is allowed to run for a long time, for the certain systems the time average of their properties is equal to the average over the entire space. In other words, time averages are equivalent to averages taken over a large number of similar systems. Each cavity in the ensemble (i.e., the supposed collection of similar systems) has a certain defined number of photons. The fraction of cavity modes that contains n photons is determined by the probability function or thermal distribution Pn,
The algebra of bounded operators on a Banach space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
In the study of discrete dynamical systems, one considers the action of some map T on a space X. Ergodic theory is the part of dynamical systems theory in which one is interested in the action of a measure preserving transformation T on a measure space X. Perhaps surprisingly, the origins of ergodic theory are in mathematical physics - statistical mechanics, to be precise1.
Measure-theoretic pressure and topological pressure in mean metrics
Published in Dynamical Systems, 2019
In classical ergodic theory, measure-theoretic entropy and topological entropy are important determinants of complexity in dynamical system. The relationship between these two quantities is the well-known variational principle. Topological pressure is a generalization of topological entropy. The theories of topological pressure, variational principle and equilibrium states play a prominent role in statistical mechanics, ergodic theory and dynamical systems (see, e.g. the book [2,11,12]). Since the works of Bowen [3] and Ruelle [15], the topological pressure turned into a basic tool of the dimension theory in dynamical systems. From a viewpoint of dimension theory, Pesin and Pitskel [13] defined the topological pressure of additive potentials for non-compact subsets of compact metric spaces and proved the variational principle under some supplementary conditions. This work extended Bowen's results in [1] on topological entropy for non-compact sets.
An optical channel modeling of a single mode fiber
Published in Journal of Modern Optics, 2018
Neda Nabavi, Peng Liu, Trevor James Hall
The basic focus of the ergodic theory is the development of conditions under which sample or time averages consisting of arithmetic means of a sequence of measurements on a random process converged to a probabilistic or ensemble average of the measurement. This analysis then can be extended to the channel matrix. Considering: