{"title":"Singularity properties of the entropy in an enclosed system characterized by accumulated noise stochastic processes","authors":"","doi":"10.1016/j.cnsns.2024.108256","DOIUrl":null,"url":null,"abstract":"<div><p>Due to the irreversibility of processes in an enclosed system, the randomness in the system increases in time and can be represented by an accumulated noise stochastic process. In contrast to the thermodynamics theory, the analysis of the system is based on the information theory point of view. The system is defined by two states: a time state and a timeless state. Based on the central limit theorem, due to the additivity of the noise process, the time state is characterized by the time-dependent Gaussian stochastic process. In contrast to Shannon's theory, precise expressions for the probability density, information, and entropy functions of the random variables defining the time-dependent process are derived and analyzed for a finite, infinite, and zero value of the related time-dependent variance. It has been proven that the entropy rate is not necessarily inversely proportional to time, as presented in previous works. Furthermore, mathematical proofs are presented, showing that the system entropy is a singularity function that increases towards infinity in the time state and then drops to zero value when the system enters the timeless state. The timeless state is characterized by the undefined pdf function, infinite values of information, and zero entropy.</p></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4000,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1007570424004416/pdfft?md5=48e3c53a32d4594436e281f7eeb84951&pid=1-s2.0-S1007570424004416-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Nonlinear Science and Numerical Simulation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1007570424004416","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Due to the irreversibility of processes in an enclosed system, the randomness in the system increases in time and can be represented by an accumulated noise stochastic process. In contrast to the thermodynamics theory, the analysis of the system is based on the information theory point of view. The system is defined by two states: a time state and a timeless state. Based on the central limit theorem, due to the additivity of the noise process, the time state is characterized by the time-dependent Gaussian stochastic process. In contrast to Shannon's theory, precise expressions for the probability density, information, and entropy functions of the random variables defining the time-dependent process are derived and analyzed for a finite, infinite, and zero value of the related time-dependent variance. It has been proven that the entropy rate is not necessarily inversely proportional to time, as presented in previous works. Furthermore, mathematical proofs are presented, showing that the system entropy is a singularity function that increases towards infinity in the time state and then drops to zero value when the system enters the timeless state. The timeless state is characterized by the undefined pdf function, infinite values of information, and zero entropy.
由于封闭系统中过程的不可逆性,系统中的随机性随时间而增加,可以用累积噪声随机过程来表示。与热力学理论不同,对系统的分析是基于信息论的观点。系统由两种状态定义:时间状态和永恒状态。根据中心极限定理,由于噪声过程的可加性,时间状态的特征是随时间变化的高斯随机过程。与香农理论不同的是,在相关时变方差为有限、无限和零值时,推导并分析了定义时变过程的随机变量的概率密度、信息和熵函数的精确表达式。研究证明,熵率并不一定与时间成反比,这在以前的著作中已有论述。此外,数学证明还表明,系统熵是一个奇异函数,在时间状态下会向无穷大方向增加,当系统进入永恒状态时,熵值会降为零。永恒状态的特征是 pdf 函数未定义、信息值无限大和熵为零。
期刊介绍:
The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity.
The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged.
Topics of interest:
Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity.
No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.