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 and/or entropy:

    “The same things change their names at such a rate;
    For instance—passion in a lover’s glorious,
    But in a husband is pronounced uxorious.”
    —George Gordon Noel Byron (1788–1824)

    “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.”
    —Václav Havel (b. 1936)