2021/09/27 by Hong Qian, Qian, Hong
Economics, Econometrics and Finance · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Complex Systems and Time Series Analysis #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2109.12806
openalex publication_date 2021/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce an entropy analysis of time series, repeated measurements of statistical observables, based on an Eulerian homogeneous degree-one entropy function Φ(t,n) of time t and number of events n. The duality of Φ, in terms of conjugate variables η=-Φ't and μ=Φ'n, yields an ``equation of state'' (EoS) in differential form that resembles the Gibbs-Duhem relation in classical thermodynamics: t dη-n dμ= 0. For simple Poisson counting with rate r, η=r(eμ-1). The conjugate variable η is then identified as being equal to the Hamiltonian function in a Hamilton-Jacobi equation for Φ(t,n). Applying the same logic to the entropy function of time correlated events yields a Hamiltonian as the principal eigenvalue of a matrix. For time reversible case it is the sum of a symmetric Markovian part √(πi)qij/√(πj) and the conjugate variables μiδij. The eigenvector, as a posterior to the naive counting measure used as the prior, suggests a set of intrinsic characteristics of Markov states.