Extreme physical information and the nonlinear wave equation

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 K<SUB>1</SUB>[(phi) ] due to modes (phi) and K<SUB>2</SUB>[(rho) ] due to modes (rho) , respectively. To form K<SUB>1</SUB>[(phi) ], the Fisher information functional I<SUB>1</SUB>[(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 I<SUB>1</SUB> invariant, as form J<SUB>1</SUB>. This is, of course, the Fourier transformation, where the transform coordinates are momenta and I<SUB>1</SUB> is essentially the mean-square electron momentum. Information K<SUB>1</SUB>[(phi) ] is then defined as (I<SUB>1</SUB> - J<SUB>1</SUB>). Information K<SUB>2</SUB> is formed similarly. The total information K is formed as the sum of the two components K<SUB>1</SUB>[(phi) ] and K<SUB>2</SUB>[(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) <SUB>n</SUB>*(phi) <SUB>n</SUB>, which is cut off at the quadratic term. Back-substituting this into the total Lagrangian gives one that is quadratic in (phi) <SUB>n</SUB>*(phi) <SUB>n</SUB>. Now varying (phi) * gives the required cubic wave equation in (phi) .