Fisher Information, Entropy, and the Second and Third Laws of Thermodynamics

Abstract

We propose Fisher information as a new calculable thermodynamic property that can be shown to follow the second and third laws of thermodynamics. However, Fisher information is qualitatively different from entropy and potentially possesses much more structure. Hence, a mathematical expression is derived for computing the Fisher information of a system of many molecules from the canonical partition function. This development is further illustrated through the derivation of Fisher information expressions for a pure ideal gas and an ideal gas mixture. Some of the unique properties of Fisher information are then explored through the classic experiment of the isochoric mixing of two ideal gases. Note that, although the entropy of isochorically mixing two ideal gases is always positive and is dependent only on the respective mole fractions of the two gases, the Fisher information of mixing has far more structure, involving the mole numbers, molecular masses, temperature, and volume. Although the application of Fisher information to molecular systems is clearly in its infancy, it is hoped that the present work will catalyze further investigation into a new and truly unique line of thought on thermodynamics.

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