From Information Geometry to Newtonian Dynamics

Newtonian dynamics is derived from prior information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the state of a particle is defined by a probability distribution. The corresponding configuration space is a statistical manifold the geometry of which is defined by the information metric. The trajectory follows from a principle of inference, the method of Maximum Entropy. No additional “physical” postulates such as an equation of motion, or an action principle, nor the concepts of momentum and of phase space, not even the notion of time, need to be postulated. The resulting entropic dynamics reproduces the Newtonian dynamics of any number of particles interacting among themselves and with external fields. Both the mass of the particles and their interactions are explained as a consequence of the underlying statistical manifold.

💡 Research Summary

The paper presents a radical reformulation of Newtonian mechanics using the mathematical framework of information geometry. The authors begin by postulating that the position of any particle cannot be known with absolute precision; instead, each particle’s state is represented by a probability distribution, typically a Gaussian with a finite variance that encodes an irreducible uncertainty. The collection of all such distributions forms a statistical manifold, and the Fisher information metric endows this manifold with a Riemannian geometry. In this geometry, the distance between two points measures the distinguishability of the corresponding probability distributions, which the authors identify with the physical notion of spatial separation.

Crucially, the curvature of the statistical manifold encodes interactions. By computing the curvature tensor of the information metric, the authors show that it reproduces the second derivatives of a conventional potential function; thus, forces such as gravity, electrostatics, or any pairwise interaction appear as geometric curvature. Mass, on the other hand, emerges from the width of the individual particle’s distribution: a narrower distribution (smaller variance) corresponds to a larger mass, reflecting a stronger suppression of positional uncertainty. Consequently, both mass and interaction potentials are not introduced as external postulates but arise naturally from the underlying statistical structure.

Dynamics are derived via the principle of Maximum Entropy (ME). Given an initial and a final probability distribution, the most unbiased inference about the intermediate evolution is the path that maximizes the entropy subject to the constraints imposed by the information metric. In information geometry, this entropy‑maximizing path coincides with a geodesic— the shortest‑distance curve on the manifold. The geodesic equation derived from the Fisher metric is mathematically equivalent to the Euler‑Lagrange equations obtained from a conventional action principle, and it reduces precisely to Newton’s second law, (F = ma), when the appropriate identifications of mass and potential are made. Notably, time does not appear as an externally prescribed parameter; instead, it is introduced as a monotonic parameter that tracks the progression of entropy along the geodesic, effectively making “time” an emergent quantity tied to the flow of information.

The authors extend the formalism to systems of many particles. Each particle contributes its own set of coordinates on the manifold, and inter‑particle interactions generate off‑diagonal components in the information metric. The resulting collective metric simultaneously contains the mass matrix and the interaction potential matrix. Solving the geodesic equations on this high‑dimensional manifold reproduces the full Newtonian dynamics of a multi‑particle system, including both internal forces and external fields.

By showing that the familiar Newtonian framework—mass, forces, equations of motion, and even the notion of time—can be derived from purely inferential principles, the paper argues that classical mechanics is fundamentally an instance of entropic dynamics. This perspective aligns classical mechanics with Bayesian inference: physical quantities are interpreted as parameters of probability distributions, and their evolution follows from optimal updating of knowledge. The authors suggest that such an information‑theoretic foundation may provide a natural bridge to quantum mechanics and general relativity, where uncertainty and geometry already play central roles.

In summary, the work demonstrates that a statistical manifold equipped with the Fisher information metric, together with the Maximum Entropy method, suffices to reconstruct Newtonian mechanics without invoking any traditional physical axioms. Mass and interactions are geometric manifestations of underlying statistical uncertainty, and dynamics emerge as the most probable inference path— a geodesic— on this manifold. This entropic dynamics framework offers a unified, inference‑based viewpoint that could reshape our conceptual understanding of the foundations of physics.