Stochastic Numerical Methods: An Introduction for Scientists

The book introduces at a master?s level the numerical methods that use probability or stochastic concepts to analyze random processes. The book aims at being rather general and is addressed at students of natural sciences (Physics, Chemistry, Mathematics, Biology, etc.) and Engineering, but also social sciences (Economy, Sociology, etc.) where some of the techniques have been used recently to numerically simulate different agent–based models. The authors develop in detail examples from the phase–transitions field to explain the whole process from the numerical simulation (design of the convenient algorithm) to the data analysis (extraction of critical exponents, finite–size effects, etc). The core of the book covers Monte Carlo type methods with applications to statistical physics and phase transitions, numerical methods for stochastic differential equations – both ordinary and partial (including advanced pseudo–spectral methods–, Gillespie?s method to simulate the dynamics of systems described by master equations (e.g. birth and death processes, and applications to Biology, such as protein expression and transcription). Finally, and in order to explain modern hybrid algorithms (combining Monte Carlo and stochastic differential equations), the authors explain the basics of molecular dynamics. Appendices with supplementary material for more advanced topics, end–of–chapter practical exercises, and useful codes for the core methods are included. INDICE: 1–Review of probability and statistics: random variables, probability density, joint and conditional probabilities, moments, correlations, law of large numbers, statistical description of data. 2–Basic Monte Carlo integration: one–dimensional problems. 3–Generation of random numbers with arbitrary distribution. 4–Multi–dimensional Monte Carlo integration: Metropolis and heat bath. 5–Applications to statistical mechanics: Ising and Potts models, hard spheres, Landau–Wilson Hamiltonian. 6–Applications to phase transitions: critical phenomena, finite–size scaling. 7–Introduction to Markov processes: master equations, birth and death processes, Poisson processes, stationary solutions, detailed balance. 8–Numerical simulation of master equations: Gillespie?s algorithm. 9–Introduction to stochastic differential equations. Brownian motion: Einstein and Langevin descriptions. Wiener process. Ito and Stratonovich interpretations. Ornstein–Uhlenbeck process. 10–Main algorithms for the numerical integration of stochastic differential equations: Euler, Heun and Runge–Kutta stochastic methods. 11–Molecular dynamics: numerical integration of equations of motion. Time reversal and simplectic algorithms. Hybrid Montecarlo. 12–Numerical integration of stochastic partial differential equations: finite differences and pseudospectral methods. Appendixes: –Generation of uniform random numbers. –Collective algorithms for Ising and Potts models: Wang–Swendsen and Wolff. –Extrapolation techniques: Ferrenberg–Swendsen algorithm, multicanonical ensemble, partition function. –Montecarlo renormalization group. –First passage time problems. Absorbing barriers. –Constructive role of noise: noise–induced phase transitions, stochastic resonance, coherence resonance, noisy precursors, etc. –Fokker–Planck equations. Non–equilibrium potentials. –Data ordering: index and ranking.

• ISBN: 978-3-527-41149-8
• Editorial: Wiley VCH
• Encuadernacion: Rústica
• Páginas: 416
• Fecha Publicación: 09/07/2014
• Nº Volúmenes: 1
• Idioma: Inglés