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.

中文翻译：

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以累积噪声随机过程为特征的封闭系统中熵的奇异性


由于封闭系统中过程的不可逆性，系统中的随机性随时间增加，可以用累积噪声随机过程来表示。与热力学理论相反，系统的分析是基于信息论的观点。该系统由两种状态定义：时间状态和永恒状态。基于中心极限定理，由于噪声过程的可加性，时间状态用依赖于时间的高斯随机过程来表征。与香农的理论相反，针对相关时间相关方差的有限、无限和零值，导出并分析了定义时间相关过程的随机变量的概率密度、信息和熵函数的精确表达式。已经证明，熵率不一定与时间成反比，如先前的工作中所提出的。此外，还提出了数学证明，表明系统熵是一个奇点函数，在时间状态下向无穷大增加，然后在系统进入无时间状态时降至零值。永恒状态的特点是未定义的 pdf 函数、无限的信息值和零熵。