Entropy2014, 16(9), 4974-4991; doi:10.3390/e16094974 (registering DOI) - published 17 September 2014 Show/Hide Abstract
Abstract: In this paper, we continue our efforts to show how maximum relative entropy (MrE) can be used as a universal updating algorithm. Here, our purpose is to tackle a joint state and parameter estimation problem where our system is nonlinear and in a non-equilibrium state, i.e., perturbed by varying external forces. Traditional parameter estimation can be performed by using filters, such as the extended Kalman filter (EKF). However, as shown with a toy example of a system with first order non-homogeneous ordinary differential equations, assumptions made by the EKF algorithm (such as the Markov assumption) may not be valid. The problem can be solved with exponential smoothing, e.g., exponentially weighted moving average (EWMA). Although this has been shown to produce acceptable filtering results in real exponential systems, it still cannot simultaneously estimate both the state and its parameters and has its own assumptions that are not always valid, for example when jump discontinuities exist. We show that by applying MrE as a filter, we can not only develop the closed form solutions, but we can also infer the parameters of the differential equation simultaneously with the means. This is useful in real, physical systems, where we want to not only filter the noise from our measurements, but we also want to simultaneously infer the parameters of the dynamics of a nonlinear and non-equilibrium system. Although there were many assumptions made throughout the paper to illustrate that EKF and exponential smoothing are special cases ofMrE, we are not “constrained”, by these assumptions. In other words, MrE is completely general and can be used in broader ways.
Entropy2014, 16(9), 4960-4973; doi:10.3390/e16094960 (registering DOI) - published 17 September 2014 Show/Hide Abstract
Abstract: In this study; the Rayleigh–Bénard convection model was established; and a great number of Bénard cells with different numbered vortexes were acquired by numerical simulation. Additionally; the Bénard cell with two vortexes; which appeared in the steady Bénard fluid with a different Rayleigh number (abbreviated Ra); was found to display the primary characteristics of the system’s entropy production. It was found that two entropy productions; which are calculated using either linear theory or classical thermodynamic theory; are all basically consistent when the system can form a steady Bénard flow in the proper range of the Rayleigh number’s parameters. Furthermore; in a steady Bénard flow; the entropy productions of the system increase alongside the Ra parameters. It was also found that the difference between the two entropy productions is the driving force to drive the system to a steady state. Otherwise; through the distribution of the local entropy production of the Bénard cell; two vortexes are clearly located where there is minimum local entropy production and in the borders around the cell’s areas of larger local entropy production.
Entropy2014, 16(9), 4937-4959; doi:10.3390/e16094937 - published 15 September 2014 Show/Hide Abstract
Abstract: In a previous study we provided analytical and experimental evidence that some materials are able to store entropy-flow, of which the heat-conduction behaves as standing waves in a bounded region small enough in practice. In this paper we continue to develop distributed control of heat conduction in these thermal-inductive materials. The control objective is to achieve subtle temperature distribution in space and simultaneously to suppress its transient overshoots in time. This technology concerns safe and accurate heating/cooling treatments in medical operations, polymer processing, and other prevailing modern day practices. Serving for distributed feedback, spatiotemporal H ∞ /μ control is developed by expansion of the conventional 1D-H ∞ /μ control to a 2D version. Therein 2D geometrical isomorphism is constructed with the Laplace-Galerkin transform, which extends the small-gain theorem into the mode-frequency domain, wherein 2D transfer-function controllers are synthesized with graphical methods. Finally, 2D digital-signal processing is programmed to implement 2D transfer-function controllers, possibly of spatial fraction-orders, into DSP-engine embedded microcontrollers.
Entropy2014, 16(9), 4923-4936; doi:10.3390/e16094923 - published 15 September 2014 Show/Hide Abstract
Abstract: An entropy-controlled bending mechanism is presented to study the nanomechanics of microcantilever-based single-stranded DNA (ssDNA) sensors. First; the conformational free energy of the ssDNA layer is given with an improved scaling theory of thermal blobs considering the curvature effect; and the mechanical energy of the non-biological layer is described by Zhang’s two-variable method for laminated beams. Then; an analytical model for static deflections of ssDNA microcantilevers is formulated by the principle of minimum energy. The comparisons of deflections predicted by the proposed model; Utz–Begley’s model and Hagan’s model are also examined. Numerical results show that the conformational entropy effect on microcantilever deflections cannot be ignored; especially at the conditions of high packing density or long chain systems; and the variation of deflection predicted by the proposed analytical model not only accords with that observed in the related experiments qualitatively; but also appears quantitatively closer to the experimental values than that by the preexisting models. In order to improve the sensitivity of static-mode biosensors; it should be as small as possible to reduce the substrate stiffness.
Entropy2014, 16(9), 4911-4922; doi:10.3390/e16094911 - published 12 September 2014 Show/Hide Abstract
Abstract: The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L∞ . The solutions attained are unique and exclusive. Performance is established by utilizing the maximum principle for certain generalized time and space-fractional diffusion equations. The fractional differential operator is inspected based on the interpretation of the Riemann–Liouville differential operator. Fractional entropy inequalities are imposed.
Entropy2014, 16(9), 4892-4910; doi:10.3390/e16094892 - published 10 September 2014 Show/Hide Abstract
Abstract: In the information theory community, the following “historical” statements are generally well accepted: (1) Hartley did put forth his rule twenty years before Shannon; (2) Shannon’s formula as a fundamental tradeoff between transmission rate, bandwidth, and signal-to-noise ratio came out unexpected in 1948; (3) Hartley’s rule is inexact while Shannon’s formula is characteristic of the additive white Gaussian noise channel; (4) Hartley’s rule is an imprecise relation that is not an appropriate formula for the capacity of a communication channel. We show that all these four statements are somewhat wrong. In fact, a careful calculation shows that “Hartley’s rule” in fact coincides with Shannon’s formula. We explain this mathematical coincidence by deriving the necessary and sufficient conditions on an additive noise channel such that its capacity is given by Shannon’s formula and construct a sequence of such channels that makes the link between the uniform (Hartley) and Gaussian (Shannon) channels.