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.

Read more about this topic:  Fisher Information Metric

Famous quotes containing the words change and/or entropy:

    “If the children and youth of a nation are afforded opportunity to develop their capacities to the fullest, if they are given the knowledge to understand the world and the wisdom to change it, then the prospects for the future are bright. In contrast, a society which neglects its children, however well it may function in other respects, risks eventual disorganization and demise.”
    —Urie Bronfenbrenner (b. 1917)

    “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)