Differential invariants of measurements, and their connection to central moments

TL;DR

This paper explores how central moments of probability distributions relate to differential invariants of measurements in affine spaces, revealing connections to thermodynamics and invariance under affine transformations.

Contribution

It introduces a geometric framework linking central moments to differential invariants of Legendrian submanifolds under affine group actions.

Findings

Central moments generate scalar differential invariants.

Heat capacity emerges as a differential invariant in thermodynamics.

The algebra of invariants is explicitly characterized.

Abstract

Due to the principle of minimal information gain, the measurement of points in an affine space $V$ determines a Legendrian submanifold of $V \times V^* \times \mathbb R$. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on $V$. We investigate the action of this group of affine transformations on Legendrian submanifolds of $V \times V^* \times \mathbb R$ by giving a detailed overview of the structure of the algebra of scalar differential invariants. We show how the central moments can be used to construct the scalar differential invariants. In the end, we view the results in the context of equilibrium thermodynamics of gases, where we notice that the heat capacity is one of the differential invariants.