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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Famous quotes containing the words change in, change and/or entropy:

    “No one can doubt the purpose for which the Nation now seeks to use the Democratic Party. It seeks to use it to interpret a change in its own plans and point of view.”
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    “In a time of drastic change it is the learners who inherit the future. The learned usually find themselves equipped to live in a world that no longer exists.”
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    “Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
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