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Finite Time Dynamics and Finite Time Predictions for Stochastic and Deterministic Chaotic Systems.

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Title: Finite Time Dynamics and Finite Time Predictions for Stochastic and Deterministic Chaotic Systems.
Authors: Bunimovich, L. A. (AUTHOR)
Source: Theory of Probability & Its Applications. 2026, Vol. 70 Issue 4, p650-658. 9p.
Subjects: Stochastic systems, Deterministic processes, Transient analysis, Probability theory, Phase transitions, Nonequilibrium statistical mechanics, Limit theorems
Abstract: Probability theory deals with limit theorems, which consider limits (when time tends to infinity) of some functions (observables) on a sample space, or averages of these observables over an infinite time interval. But what is happening in a finite time or over finite time intervals? Such questions, important for virtually all applications, seem to be intractable mathematically (and generally sound unreasonable). For instance, equilibrium statistical mechanics deals with phase transitions (a number of equilibrium probability distributions/states) rather than time evolution, while nonequilibrium statistical mechanics is concerned with convergence of nonequilibrium states to equilibrium ones. Again, such processes occur on infinite time intervals. It turns out, however, that there are natural and reasonable questions about finite time dynamics of random and deterministic chaotic systems, which can be answered and, moreover, rigorously answered. This allows one to make predictions about a finite time evolution of such systems. [ABSTRACT FROM AUTHOR]
Copyright of Theory of Probability & Its Applications is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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PubType: Academic Journal
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  Data: <searchLink fieldCode="DE" term="%22Stochastic+systems%22">Stochastic systems</searchLink><br /><searchLink fieldCode="DE" term="%22Deterministic+processes%22">Deterministic processes</searchLink><br /><searchLink fieldCode="DE" term="%22Transient+analysis%22">Transient analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Probability+theory%22">Probability theory</searchLink><br /><searchLink fieldCode="DE" term="%22Phase+transitions%22">Phase transitions</searchLink><br /><searchLink fieldCode="DE" term="%22Nonequilibrium+statistical+mechanics%22">Nonequilibrium statistical mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Limit+theorems%22">Limit theorems</searchLink>
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  Data: Probability theory deals with limit theorems, which consider limits (when time tends to infinity) of some functions (observables) on a sample space, or averages of these observables over an infinite time interval. But what is happening in a finite time or over finite time intervals? Such questions, important for virtually all applications, seem to be intractable mathematically (and generally sound unreasonable). For instance, equilibrium statistical mechanics deals with phase transitions (a number of equilibrium probability distributions/states) rather than time evolution, while nonequilibrium statistical mechanics is concerned with convergence of nonequilibrium states to equilibrium ones. Again, such processes occur on infinite time intervals. It turns out, however, that there are natural and reasonable questions about finite time dynamics of random and deterministic chaotic systems, which can be answered and, moreover, rigorously answered. This allows one to make predictions about a finite time evolution of such systems. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Theory of Probability & Its Applications is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/S0040585X97T992689
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      – Code: eng
        Text: English
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        PageCount: 9
        StartPage: 650
    Subjects:
      – SubjectFull: Stochastic systems
        Type: general
      – SubjectFull: Deterministic processes
        Type: general
      – SubjectFull: Transient analysis
        Type: general
      – SubjectFull: Probability theory
        Type: general
      – SubjectFull: Phase transitions
        Type: general
      – SubjectFull: Nonequilibrium statistical mechanics
        Type: general
      – SubjectFull: Limit theorems
        Type: general
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      – TitleFull: Finite Time Dynamics and Finite Time Predictions for Stochastic and Deterministic Chaotic Systems.
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              Text: 2026
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              Y: 2026
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            – TitleFull: Theory of Probability & Its Applications
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