Fisher Information Metric - Change in Entropy

Change in Entropy

The action of a curve on a Riemannian manifold is given by

A=\frac{1}{2}\int_a^b \frac{\partial\theta^j}{\partial t} g_{jk}(\theta)\frac{\partial\theta^k}{\partial t} dt

The path parameter here is time t; this action can be understood to give the change in entropy of a system as it is moved from time a to time b. Specifically, one has

as the change in entropy. This observation has resulted in practical applications in chemical and processing industry: in order to minimize the change in entropy of a system, one should follow the minimum geodesic path between the desired endpoints of the process. The geodesic minimizes the entropy, due to the Cauchy–Schwarz inequality, which states that the action is bounded below by the length of the curve, squared.

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