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

We introduce an entropy analysis of time series, repeated measurements of statistical observables, based on an Eulerian homogeneous degree-one entropy function $\Phi(t,n)$ of time $t$ and number of events $n$. The duality of $\Phi$, in terms of conjugate variables $\eta=-\Phi'_t$ and $\mu=\Phi'_n$, yields an ``equation of state'' (EoS) in differential form that resembles the Gibbs-Duhem relation in classical thermodynamics: $t d\eta-n d\mu = 0$. For simple Poisson counting with rate $r$, $\eta=r(e^{\mu}-1)$. The conjugate variable $\eta$ is then identified as being equal to the Hamiltonian function in a Hamilton-Jacobi equation for $\Phi(t,n)$. 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 $\sqrt{\pi_i}q_{ij}/\sqrt{\pi_j}$ and the conjugate variables $\mu_i\delta_{ij}$. The eigenvector, as a posterior to the naive counting measure used as the prior, suggests a set of intrinsic characteristics of Markov states.