3aUW4. Quantum chaos in a stochastic model.

Session: Wednesday Morning, June 18

Author: F. D. Tappert
Location: Appl. Marine Phys., Univ. of Miami, RSMAS, 4600 Rickenbacker Cswy., Miami, FL 33149
Author: M. A. Wolfson
Location: Penn State Univ., University Park, PA 16802


An idealized problem in the theory of long-range ocean acoustic propagation through randomly fluctuating mesoscale structure is reformulated as a time-dependent quantum mechanics problem with a time-varying Markovian potential function. The effective Planck constant is (Dirac h)=(delta)(cursive beta)[sup -2/3], where (delta)=(k[inf 0]l)[sup -1]<<1 measures diffraction, and (cursive beta)<<1 is the strength of the mesoscale structure. Also k[inf 0] is the acoustic wave number and l is the scale length of the mesoscale structure. In the formal classical limit ((Dirac h)->0), the parabolic ray equations are nonintegrable and the relative motion of two particles for small initial separations exhibits chaotic behavior characterized by a positive Lyapunov exponent, (nu)~(cursive beta)[sup 2/3]/l. It is shown that this physically relates to the exponential proliferation of caustics. A generalized wave kinetic equation (GWKE) is derived for the evolution in phase space of the two-particle Wigner function. The GWKE is analytically examined for (Dirac h)<<1 by a novel boundary layer method, called the ``extended quantum notch method,'' which yields several important results: First, the ``log breakdown time (range)'' t[inf b]~(nu)[sup -1]log((Dirac h)[sup -1]) is obtained where semi-classical theory breaks down due to the saturation of caustics, and it is shown that this is the range where the scintillation index saturates to unity; and, second, a wave (quantum) manifestation of classical chaos is found to be the exponential decay of the scintillation index beyond its peak value. [Work supported by ONR.]

ASA 133rd meeting - Penn State, June 1997