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Entropy

Entropy

Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...

Entropy

Entropy

The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...

Entropy Change in Reversible Processes

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

Wald-Wolfowitz Runs Test I

Wald-Wolfowitz Runs Test I

The Wald-Wolfowitz test, also known as the runs test, is a nonparametric statistical test used to assess the randomness of a sequence of two different types of elements (e.g., positive/negative values, successes/failures). It examines whether the order of the elements in a sequence is random or if there is a pattern or trend present. This nonparametric test applies to any ordered data despite the population and sample data distribution, even if a higher sample size is available.
The test works...

The Entropy as a State Function

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...

Entropy and the Second Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...

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Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

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Entropy of finite random binary sequences with weak long-range correlations.

S S Melnik¹, O V Usatenko¹

¹A. Ya. Usikov Institute for Radiophysics and Electronics Ukrainian Academy of Science, 12 Proskura Street, 61805 Kharkov, Ukraine.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 11, 2014

Summary

This study analyzes the entropy of Markov chains using pair correlation functions, enabling entropy calculations for longer sequences. A fluctuation contribution to entropy is examined, revealing self-similar structures in DNA sequences.

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Area of Science:

Information Theory
Statistical Mechanics
Computational Biology

Background:

Markov chains are fundamental in modeling sequential data.
Calculating entropy for long sequences is computationally challenging with standard methods.

Purpose of the Study:

To develop a novel method for calculating the differential entropy of N-step binary stationary ergodic Markov chains.
To analyze the impact of correlations and chain finiteness on entropy.

Main Methods:

Expressing conditional probability via pair correlation functions for weak correlations.
Representing entropy as a functional of the pair correlator.
Examining fluctuation contributions due to finite chain length.

Main Results:

The method allows entropy calculation for significantly longer sequences compared to traditional approaches.
A fluctuation contribution to entropy, comparable to the regular part, was identified even for short subsequences.
A self-similar entropy structure was observed for specific pair correlation functions.

Conclusions:

The developed theory provides an efficient way to compute entropy in Markov chains.
The findings have implications for analyzing complex biological sequences, such as DNA.
The study highlights the importance of correlation functions and finite-size effects in entropy estimation.