Statistical Thermodynamics and Data ad Infinitum: Conjugate Variables as Entropic Forces, and their Statistical Variations

Maximum Entropy principle identifies notions of forces conjugated to observables and the relations between them, independent upon their underlying mechanistic details. For data about state distributions or transition rates, the principle can be derived purely from limit theorems of the idealized infinite data sampling. This derivation from data reveals the empirical origin of the principle and clarify the meaning of applying it to finite but large data. The identified forces lead to symmetry breaking for each particular system that produces the data, e.g. the emergence of time correlation and time irreversibility. We show that the leading-order statistical variations of the observables and the inferred forces satisfy an asymptotic thermodynamic uncertainty principle.