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Entropic Approach to Critical Materials Assessment

Alan J. Hurd

TL;DR

The paper tackles the problem of materials criticality by linking market price to crustal abundance through a geologic entropy framework. It develops a variational method that maximizes the entropy S_m = -k \sum_{i=1}^{N} n_i \log n_i under constraints, yielding a price distribution p(n) = \frac{c}{n} + \eta (\log n - \lambda) and explaining p(n) \sim n^{-1} for small n with empirical support across ~39 elements. By connecting price, abundance, and mining production within a statistically grounded entropy model, the work identifies how a highly peaked abundance distribution can influence criticality metrics, including a power-law with a kink over several decades. The approach offers a flexible tool for critical materials assessment that can incorporate economic, regulatory, or national-security constraints, with potential implications for policy and supply-chain resilience.

Abstract

Most methodologies for materials criticality assessment score supply risk and societal importance. Market-based criteria offer quantitative measures for assessment. Here we develop a statistical approach based on a geologic entropy function in which flexible constraints, such as economic, national security related, or regulatory, can be applied. As an example, the formulation describes the relation between elemental price and crustal abundance for selected elements, both important to supply risk. The method may be applicable to parameters resulting from collective decisions exhibiting a highly peaked probability distribution.

Entropic Approach to Critical Materials Assessment

TL;DR

The paper tackles the problem of materials criticality by linking market price to crustal abundance through a geologic entropy framework. It develops a variational method that maximizes the entropy S_m = -k \sum_{i=1}^{N} n_i \log n_i under constraints, yielding a price distribution p(n) = \frac{c}{n} + \eta (\log n - \lambda) and explaining p(n) \sim n^{-1} for small n with empirical support across ~39 elements. By connecting price, abundance, and mining production within a statistically grounded entropy model, the work identifies how a highly peaked abundance distribution can influence criticality metrics, including a power-law with a kink over several decades. The approach offers a flexible tool for critical materials assessment that can incorporate economic, regulatory, or national-security constraints, with potential implications for policy and supply-chain resilience.

Abstract

Most methodologies for materials criticality assessment score supply risk and societal importance. Market-based criteria offer quantitative measures for assessment. Here we develop a statistical approach based on a geologic entropy function in which flexible constraints, such as economic, national security related, or regulatory, can be applied. As an example, the formulation describes the relation between elemental price and crustal abundance for selected elements, both important to supply risk. The method may be applicable to parameters resulting from collective decisions exhibiting a highly peaked probability distribution.
Paper Structure (6 sections, 11 equations, 2 figures, 1 table)

This paper contains 6 sections, 11 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Entropic Approach to Criticality Assessment
  3. Acknowledgments and Funding
  4. Contributions by Authors
  5. Conflicts of Interest
  6. Data Availability

Figures (2)

  • Figure 1: Data of price (US$/kg) vs mined mass (metric ton) for 39 chemical elements showing a power-law relationship. A similar power law relates price to crustal abundance. [Jonathan Price, personal communication]
  • Figure 2: Expected distribution of element masses $m_i$ in an ensemble of regional geologies