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. The second law of thermodynamics is based on entropy. The second law states that the total entropy of an isolated system increases over time, approaching a maximum value as time goes to infinity. Entropy in the truest classical sense applies to systems in equilibrium as defined by Clausius, Caratheodory, and others for the case of heat engines with the expression dS ) δQ T (1) where dS is the change in entropy caused by the flow of heat δQ into the system at temperature T. From the microscopic statistical thermodynamics point of view, entropy was defined by Boltzmann as S ) κ ln Ω (2) which can be shown to be equal to Shannon information, which is given by