Thermodynamic Behavior of Statistical Event Counting in Time: Independent and Correlated Measurements

Abstract

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: tdη − ndµ = 0. For simple Poisson counting with rate r, η = r(e µ − 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 √ πiqij/ √ 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.

Legendre-Fenchel transforms capture layering transitions in porous media

“…We have seen that the Legendre-Fenchel transforms apply to two widely different cases, so we may pose the question: Does the transform apply in general to energy conversion in small systems? From the mathematics point of view, this seems likely [28,29]. More data is needed before we may conclude, but this study brings out an interesting perspective.…”

Legendre-Fenchel transforms capture layering transitions in porous media

“…From the mathematics point of view, this seems likely. 28,29 More data is needed before we may conclude, but this study brings out an interesting perspective. If the answer is yes, we may have a new tool for energy conversion in small systems.…”

Quantifying information of intracellular signaling: progress with machine learning

“…Fourth, Shore and Johnson proved the principle of entropy maximization as requirements to be satisfied by any distribution function [44]. We refer readers to [56,57] for more detailed discussions.…”