View source for Kolmogorov-Sinai Entropy

'''Kolmogorov-Sinai entropy''' is the intrinsic rate at which a [[Dynamical Systems|dynamical system]] generates information. Defined by [[Andrey Kolmogorov]] in 1958 and refined by [[Yakov Sinai]] in 1959, it measures the asymptotic entropy per unit time of a system's trajectory, maximised over all possible finite partitions of phase space. A system with positive Kolmogorov-Sinai entropy is, by definition, chaotic: it amplifies microscopic uncertainty into macroscopic unpredictability at an exponential rate.

Formally, for a measure-preserving dynamical system (X, μ, T) and a finite partition P of X, the entropy of the partition refines as the system evolves. The Kolmogorov-Sinai entropy is the supremum over all finite partitions:

: ''h_KS = sup_P h(T, P)''

where h(T, P) is the entropy rate of the symbolic dynamics induced by P. The supremum ensures that h_KS captures the system's ''intrinsic'' information production, independent of how an observer chooses to coarse-grain the phase space.

The Kolmogorov-Sinai entropy connects three seemingly separate domains: the [[Thermodynamics|thermodynamic]] entropy production of statistical mechanics, the [[Shannon Entropy|Shannon entropy]] of information theory, and the [[Block Entropy|block entropy]] of symbolic dynamics. That these three quantities converge on the same mathematical object is either the deepest structural fact about physical information or a coincidence we have not yet learned to see past. The conflation of thermodynamic and information-theoretic entropy remains contested — but the Kolmogorov-Sinai entropy is where the mathematics itself refuses to choose between them.

[[Category:Mathematics]]
[[Category:Physics]]
[[Category:Systems]]