Entropy metric

What is complexity There is no good general definition of complexity, though there are many. Intuitively, complexity lies somewhere between order and disorder, between regularity and randomness, between perfect crystal and gas. Complexity has been measured by logical depth, metric entropy, information content (Shannon s entropy), fluctuation complexity, and many other techniques some of them are discussed below. These measures are well suited to specific physical or chemical applications, but none describe the general features of complexity. Obviously, the lack of a definition of complexity does not prevent researchers from using the term. 

It turns out convenient to define the metrical entropy S as that function for which dS(s) - (s)ds then, provided only... 

In other words, T - 0 is attainable unless, for all z, (dS/dz)T approaches zero more rapidly than does Cz/T in the limit of low temperatures. The attainability problem is thus intricately linked to the behavior of the metrical entropy at low temperatures. 

We develop instead a shortcut for the Second Law along the following lines Just as we were able to link the performance of work under adiabatic conditions to the existence of a function of state, the energy, so we will postulate that the transfer of heat under reversible conditions is related to another functions of state, termed the empirical entropy, symbolized by s (or, later, the metrical entropy, S), which will then found to be useful to characterize various processes. In short, it is claimed as part of the Second Law ofThermodynamics that... 

The idea here was to examine which pair of techniques and individual columns could lead to the best separations in 2DLC. This is achievable by using ID separations and then comparing how the retention of each component varies across the separation space. Another innovation here was the use of IT-derived metrics such as information entropy, informational similarity, and the synentropy. As stated in this paper, The informational similarity of 2D chromatographic systems, H is a measure of global... 

Continuing with the mini-theme of computational materials chemistry is Chapter 3 by Professor Thomas M. Truskett and coworkers. As in the previous chapters, the authors quickly frame the problem in terms of mapping atomic (chemical) to macroscopic (physical) properties. The authors then focus our attention on condensed media phenomena, specifically those in glasses and liquids. In this chapter, three properties receive attention—structural order, free volume, and entropy. Order, whether it is in a man-made material or found in nature, may be considered by many as something that is easy to spot, but difficult to quantify yet quantifying order is indeed what Professor Truskett and his coauthors describe. Different types of order are presented, as are various metrics used for their quantification, all the while maintaining theoretical rigor but not at the expense of readability. The authors follow this section of their... 

In contrast to the unit dependence of the thermodynamic vector lengths and metric eigenvalues, the thermodynamic angles are pure dimensionless numbers. Figure 11.9 exhibits the angle 6Sy that each entropy vector S) makes with respect to the volume (abscissa) and temperature (ordinate) axes. 

In essence, we wish to follow the Gibbs entropy maximization procedure of Section 5.2 backward in time. Specifically, we seek to characterize the final stage of equilibration when effective local equilibrium has been achieved in each small cell n, but cell intensities are not yet equalized throughout the system. To make direct contact with Section 5.2, we shall temporarily revert to the entropy representation (Section 5.3), which generates a scalar product and metric geometry that is conformally equivalent to the (/-based metric... 

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

To capture the relaxation times change from molecular vibration times to values prototypical near the glass transition temperature, other metrics of fragility have been used that exploits the shape of the normalized excess entropy data in a reduced-7), form. This includes the ratio F of VFT temperature, where the relaxation time diverges, to the glass transition temperature, To/Tg [18,107] or... 

By a synthesis of the partition function for a supercooled liquid at some hctive temperature in the inherent structure formalism [ 1,4] with the configurational entropy obtained by restricting mrepiica replicas to be in the same state [161], Mohanty has uncovered a relationship between Parisi s replica symmetry-breaking parameter mrepiica(T), and the Narayanaswamy-Moynihan non-linear parameter x, a parameter that provides a metric on the deviation of the glassforming system from equilibrium [162] ... 

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