Statistical Thermodynamics and Data ad Infinitum

Generalizing ideas of Szilard (1925), Mandelbrot (1964) and Hill (1964), we show that a statistical thermodynamic structure can emerge purely from the infinitely large data limit under a probabilistic framework independent upon their underlying mechanistic details. Systems with distinct values of a set of observables are identified as different thermodynamic states, which are parameterized by the entropic forces conjugated to the observables. The ground state with zero entropic forces corresponds to the prior probability equipped with a symmetry of interest. The entropic forces lead to symmetry breaking for each particular system that produces the data, emph{c.f.} emerging of time correlation and breakdown of detailed balance. Probabilistic models for the excited states are predicted by the Maximum Entropy Principle for sequences of i.i.d. and correlated Markov samples.

Asymptotically-equivalent models are also found by the Maximum Caliber Principle. With a novel derivation of Maximum Caliber, conceptual differences between the two principles are clarified. The emergent thermodynamics in the data infinitus limit has a mesoscopic origin from the Maximum Caliber. In the canonical probabilistic models of Maximum Caliber, the variances of the observables and their conjugated forces satisfy the asymptotic thermodynamic uncertainty principle, which stems from the reciprocal-curvature relation between ``entropy'' and ``free energy'' functions in the theory of large deviations. The mesoscopic origin of the reciprocality is identified. As a consequence of limit theorems in probability theory, the phenomenological statistical thermodynamics is universal without the need of mechanics.