100 Years of dimensional analysis: New steps toward empirical law deduction

TL;DR

This paper extends dimensional analysis by introducing a matrix generalization of the Buckingham Theorem, enabling the deduction of empirical laws from data, exemplified by deriving the Planck Radiation Law without initial assumptions.

Contribution

It presents a novel matrix-based approach to dimensional analysis, generalizing the Buckingham Theorem and demonstrating its application in discovering physical laws from observational data.

Findings

Matrix generalization of Buckingham Theorem developed.

Inverse transform S' is non-unique, affecting scaling relations.

Planck Radiation Law derived using the matrix method without first principles.

Abstract

On the verge of the centenary of dimensional analysis (DA), we present a generalisation of the theory and a methodology for the discovery of empirical laws from observational data. It is well known that DA: a) reduces the number of free parameters, b) guarantees scale invariance through dimensional homogeneity and c) extracts functional information encoded in the dimensionless grouping of variables. Less known are the results of Rudolph and co-workers that DA also gives rise to a new pair of transforms - the similarity transform (S) that converts physical dimensional data into dimensionless space and its inverse (S'). Here, we present a new matrix generalisation of the Buckingham Theorem, made possible by recent developments in the theory of inverse non-square matrices, and show how the transform pair arises naturally. We demonstrate that the inverse transform S' is non-unique and how this casts doubt on scaling relations obtained in cases where observational data has not been referred to in order to break the degeneracy inherent in transforming back to dimensional (physical) space. As an example, we show how the underlying functional form of the Planck Radiation Law can be deduced in only a few lines using the matrix method and without appealing to first principles; thus demonstrating the possibility of a priori knowledge discovery; but that subsequent data analysis is still required in order to identify the exact causal law. It is hoped that the proof presented here will give theoreticians confidence to pursue inverse problems in physics using DA.