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Electrochemistry and Electroanalytical Techniques
Published in Ashutosh Kumar Dubey, Amartya Mukhopadhyay, Bikramjit Basu, Interdisciplinary Engineering Sciences, 2020
Ashutosh Kumar Dubey, Amartya Mukhopadhyay, Bikramjit Basu
In very generic terms, the third law of thermodynamics states that the entropy of a closed system approaches a constant value as the temperature approaches absolute zero. As mentioned in the preceding paragraph, entropy is proportional to the number of microstates, and, in case there is only a unique ground state with minimum energy at absolute zero, then the entropy will not only be constant, but also will be equal to zero. Nevertheless, if the concerned system lacks a well-defined order and the minimum energy state is not well-defined (as is typically for glassy systems), then there is a possibility of some remnant finite entropy at temperatures close to absolute zero, which is then called the system's residual entropy.
Enhancing the formation of ionic defects to study the ice Ih/XI transition with molecular dynamics simulations
Published in Molecular Physics, 2021
We also showed the usefulness of this approach by studying the ice XI/Ih transition. For this purpose, we constructed an ad hoc order parameter for ice XI based on the local atomic environments around oxygen atoms. An appropriate quasi-static bias potential as a function of this order parameter sufficed to drive reversible transitions between ice XI and ice Ih. From these simulations we calculated the free energy difference between the two phases and the residual entropy of ice Ih. We obtained a residual entropy of that is in agreement with the experiment within the error bar of our calculation. We argue that it represents an upper bound to the true residual entropy since finite size effects reduce the residual entropy. As far as we know, this is the first calculation of the residual entropy that fully takes into account the subtle variation in energy between different proton configurations. The approximate transition temperature obtained here from a rough extrapolation of high temperature simulations is around 50 K. A simpler estimate of the transition temperature based on the enthalpy at 0 K for this model, results in a transition temperature of 68 K in quite good agreement with the experimental result of 72 K, even if we take into account that nuclear quantum effect will likely change our result [30].
Bayesian Estimation of Dynamic Cumulative Residual Entropy for Classical Pareto Distribution
Published in American Journal of Mathematical and Management Sciences, 2018
K. R. Renjini, E. I. Abdul Sathar, G. Rajesh
A fundamental uncertainty measure of a random variable is known as entropy and was introduced by Shannon (1948). Let X be a non-negative absolutely continuous random variable with probability density function f, then the Shannon entropy is defined as follows. An alternative to Shannon’s entropy, namely, cumulative residual entropy (CRE), has been defined by Rao et al. (2004). Asadi and Zohrevand (2007) have introduced a dynamic version of CRE, namely, dynamic cumulative residual entropy (DCRE), and is defined as follows. where is the, survival function of X. Given that an item has survived up to time t, measures the uncertainty in its remaining life. For a discussion of the properties and applications of DCRE we refer to Asadi and Zohrevand (2007), Crescenzo and Longobardi (2009), Navarro, Aguila, and Asadi (2010), Rajesh et al. (2014), and Renjini, Sathar, and Rajesh (2016a, 2016b).
Spectral segmentation based dimension reduction for hyperspectral image classification
Published in Journal of Spatial Science, 2022
Ayasha Siddiqa, Rashedul Islam, Masud Ibn Afjal
Cross Cumulative Residual Entropy (CCRE) is a popular similarity measurement tool (Wang and Vemuri 2007). CCRE can be used to measure the similarity of two images in which cumulative residual distribution is used instead of probabilistic distribution (Rao et al. 2004). The CCRE of two images I and J is given by