On the Statistical Origin of Physical Laws

Abstract

We consider an embedded observer with no prior knowledge of the existence or nature of the dynamical process generating her observations and ask how such an observer should infer the underlying dynamics from a finite observational history. Rather than taking fixed dynamical laws as the starting point and observations as their consequence, we begin with observations and ask how stable effective laws emerge from them. To this end, we develop an inferential framework in which dynamical laws are treated as statistical objects. The observer, modeled as a quantum reference frame interacting with an environment, explains past observations and predicts future observations by averaging over an ensemble of computable dynamical models, each represented by a CPTP map. Within the shortest-program approximation, we show that the posterior distribution over dynamical models assumes a Gibbs-Boltzmann form in which the minimum description length (MDL) plays the role of an effective energy. The posterior arises as the minimizer of a variational free-MDL functional balancing algorithmic simplicity against epistemic entropy. Sequential updating statistically tends to reinforce compressible regularities, thereby stabilizing the corresponding effective dynamical laws. These emerge as the optimal explanatory and predictive structures for past and future observations, respectively. The results suggest that the lawfulness experienced by embedded observers is a statistical property of inductive inference and can arise and persist without assuming fixed fundamental laws. More specifically, effective laws remain stable under typical continuations of the observational history while still allowing, in principle, transitions to competing laws when sufficiently atypical observations accumulate. The resulting picture is that of a statistical mechanics on the space of computable dynamical models, connecting universal induction, algorithmic information theory, and equilibrium statistical mechanics within an observer-centered theory of dynamical inference.