Thermodynamics and Energy

   

On a Certain Misunderstanding in the Interpretation of the Canonical Distribution of the Symbol Sequences

Authors: Ilya Shesterikov

The canonical (Gibbs) distribution is widely used in statistical physics to describe the probabilities of microscopic states characterized by an energy value. In symbolic dynamics and the study of symbolic sequences generated by nonlinear dynamical systems, an analogous construction is frequently applied: the probability of observing a particular symbol sequence is assumed to depend exponentially on an associated "energy", often defined through a cylinder length or a Jacobian-based quantity. While this analogy is technically appealing and mathematically consistent, it has led to a persistent conceptual misunderstanding. The confusion arises when the discrete cylinder lengths are mistakenly interpreted as samples from a continuous distribution, leading to the use of probability density functions where only discrete probabilities are appropriate. In this paper, we analyze the origin of this misunderstanding, clarify the correct interpretation of the canonical distribution in symbolic dynamics, and provide practical guidance for avoiding associated pitfalls. We further illustrate the issue with examples, graphical explanations, and a discussion of implications for numerical studies of chaotic systems.

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[v1] 2025-12-11 00:45:43

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