Methods and Applications of White Noise Analysis in Interdisciplinary Sciences - Bernido, Christopher C. and Carpio-Bernido, M Victoria (Singapore : World Scientific Pub. Co., 2015)
Analysis, modeling, and simulation for better understanding of diverse complex natural and social phenomena often require powerful tools and analytical methods. Tractable approaches, however, can be developed with mathematics beyond the common toolbox. This book presents the white noise stochastic calculus, originated by T. Hida, as a novel and powerful tool in investigating physical and social systems. The calculus, when combined with Feynman's summation-over-all-histories, has opened new avenues for resolving cross-disciplinary problems. Applications to real-world complex phenomena are further enhanced by parametrizing non-Markovian evolution of a system with various types of memory functions. This book presents general methods and applications to problems encountered in complex systems, scaling in industry, neuroscience, polymer physics, biophysics, time series analysis, relativistic and nonrelativistic quantum systems.
1. Introduction. 1.1. Some properties of white noise. 1.2. Preview of applications. 1.3. A calculus based on white noise -- 2. White noise analysis: some basic notions and terminology. 2.1. T- and S-transforms. 2.2. Simple examples. 2.3. The Gauss kernel. 2.4. Donsker Delta function. 2.5. Levy's stochastic area -- 3. Fluctuations with memory. 3.1. Gibrat's law. 3.2. Parametrizing the effects of memory. 3.3. Memory functions and probability densities. 3.4. Fractional Brownian motion. 3.5. Bessel-modified Brownian motion. 3.6. Exponentially-modified Brownian motion. 3.7. Moments of the probability density function. 3.8. Standard deviation with memory function. 3.9. Modified diffusion equation. 3.10. Example: periodic boundary condition. 3.11. Example: the wedge boundary for diffusion with memory -- 4. Complex systems. 4.1. Scaling property. 4.2. Metabolic rate fluctuation. 4.3. Fluctuations in word use. 4.4. Growth of companies. 4.5. Sensitivity to changes in Hurst index -- 5. Time series analysis. 5.1. Time series fluctuation as modified Brownian motion. 5.2. Mean square displacement with memory. 5.3. Typhoon track fluctuations. 5.4. Particle-tracking in microrheology -- 6. Fluctuations without memory. 6.1. Correspondence between a stochastic differential equation and the Fokker-Planck equation. 6.2. Short-time solution for the Fokker-Planck equation. 6.3. Path integral for the Fokker-Planck equation. 6.4. One-dimensional random walk -- 7. Neurophysics. 7.1. Modelling single neuron activities. 7.2. Neuronal firing rate. 7.3. Interneuronal connections with memory. 7.4. Scaling property for neuronal clusters. 7.5. Synchronous burst rates -- 8. Biopolymers. 8.1. Modelling diffusive polypeptides. 8.2. Helical conformations. 8.3. Path integral for helical conformations. 8.4. Drift coefficient as modulating function. 8.5. Helix-turn-helix motif. 8.6. Matching with real proteins. 8.7. Overwinding of DNA. 8.8. Biomolecular chirality and entropy -- 9. White noise functional integrals in quantum mechanics. 9.1. Quantum propagator as sum-over-all-histories. 9.2. Feynman path integral and the Schrodinger equation. 9.3. The path integral in time-sliced form. 9.4. The free particle. 9.5. Feynman integrand as a white noise functional. 9.6. Uniform magnetic field. 9.7. Outlook -- 10. Quantum particles with boundary conditions. 10.1. Quantum particle with periodic boundaries. 10.2. The Aharonov-Bohm setup. 10.3. Infinite wall potential. 10.4. Particle in a box. 10.5. Free particle on the half-line with general boundary conditions -- 11. Relativistic quantum mechanics. 11.1. Green function for the Dirac equation. 11.2. Dirac particle in a uniform magnetic field.
Includes bibliographical references (p. 177-188) and index.
Electronic reproduction.
Keywords: White noise theory; Electronic books