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This study introduces a new dynamics to understand typical sets for finite sequences of random variables. This method simplifies analysis for small systems and connects finite properties to asymptotic results.

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Area of Science:

  • Information Theory
  • Statistical Mechanics
  • Dynamical Systems

Background:

  • The asymptotic equipartition property describes how long random sequences converge to a typical set.
  • Estimating typical set properties is challenging for finite sequences due to exponential growth.
  • Current methods often require asymptotic limits, limiting applications to small or transient systems.

Purpose of the Study:

  • To derive a time-inhomogeneous dynamics for finite sequences of independent and identically distributed random variables.
  • To bridge the gap between finite sequence properties and asymptotic results in information theory.
  • To enable the application of typical set concepts to small and transient systems.

Main Methods:

  • Derivation of a time-inhomogeneous dynamics.
  • Development of a geometric mapping, the 'triangle map', relating sequences of length n to n+1.
  • Application of the framework to the Bernoulli process and the Schlögl model.

Main Results:

  • The dynamics successfully construct typical set properties for all finite length sequences.
  • The number of points required by the triangle map grows linearly with sequence length, not exponentially.
  • Demonstrated convergence to asymptotic limits and reproduction of exact calculations for specific models.

Conclusions:

  • The derived dynamics provide a novel framework for analyzing typical sets in finite systems.
  • The linear growth in complexity allows for practical application to smaller systems.
  • This work extends the applicability of information theory and statistical mechanics principles to transient phenomena.