A Continuous Description of Different Means with Application To Mixing Rules

Area of Science:

• Mathematical analysis
• Computational chemistry
• Physical chemistry

Background:

• Traditional means (arithmetic, geometric, harmonic) present discrete choices for averaging two numbers.
• Selecting the appropriate mean can be ambiguous, leading to discrete problems in calculations.

Purpose of the Study:

• To introduce and analyze one-parameter functions that provide a continuous connection between harmonic mean (HM), geometric mean (GM), and arithmetic mean (AM).
• To identify functions suitable for a unified description of mixing rules in computational chemistry and thermophysical property calculations.

Main Methods:

• Exploration of three one-parameter functions to bridge HM, GM, and AM.
• Detailed analysis of the Lehmer mean and Hölder mean functions.

Main Results:

• The proposed functions establish a continuous relationship between different mean values.
• The Hölder mean is identified as particularly effective for creating a uniform, one-parameter description.
• Demonstrated applicability in quantum chemical force field and thermophysical property calculations.

Conclusions:

• One-parameter functions offer a continuous alternative to discrete mean selection.
• The Hölder mean provides a powerful and versatile tool for unifying various mixing rules in scientific computations.