Extreme physical information and the nonlinear wave equation

B. Roy Frieden

doi:10.1117/12.219553

15 September 1995 Extreme physical information and the nonlinear wave equation

B. Roy Frieden

Proceedings Volume 2528, Optical and Photonic Applications of Electroactive and Conducting Polymers; (1995) https://doi.org/10.1117/12.219553
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

The nonlinear wave equation an be derived from a principle of extreme physical information (EPI) K. This is for a scenario where a probe electron moves through a medium in a weak magnetic field. The field is caused by a probabilistic line current source. Assume that the probability current density S of the electron is approximately constant, and directed parallel to the current source. Both the source probability amplitudes (rho) and the electron probability amplitudes (phi) are unknowns (called 'modes') of the problem. The net physical information K here consists of two components: functional K₁[(phi) ] due to modes (phi) and K₂[(rho) ] due to modes (rho) , respectively. To form K₁[(phi) ], the Fisher information functional I₁[(phi) ] for the electron modes is first constructed. This is of a fixed mathematical form. Then, a unitary transformation on (phi) to a physical space is sought that leaves I₁ invariant, as form J₁. This is, of course, the Fourier transformation, where the transform coordinates are momenta and I₁ is essentially the mean-square electron momentum. Information K₁[(phi) ] is then defined as (I₁ - J₁). Information K₂ is formed similarly. The total information K is formed as the sum of the two components K₁[(phi) ] and K₂[(rho) ], by the additivity of Fisher information, and is then extremized in both (phi) and (rho) . Extremizing first in (rho) gives a Taylor series in powers of (phi) _n*(phi) _n, which is cut off at the quadratic term. Back-substituting this into the total Lagrangian gives one that is quadratic in (phi) _n*(phi) _n. Now varying (phi) * gives the required cubic wave equation in (phi) .

Citation Download Citation

B. Roy Frieden "Extreme physical information and the nonlinear wave equation", Proc. SPIE 2528, Optical and Photonic Applications of Electroactive and Conducting Polymers, (15 September 1995); https://doi.org/10.1117/12.219553

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