Stochastic Numerical Methods An Introduction for Students and 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.

Raul Toral is head of the department of Complex Systems at IFISC (Palma de Mallorca, Spain). He obtained his academic degrees from Barcelona university (Spain), spent two years at THE Physics Department of Edinburgh University (UK), and two more years at Lehigh University (Pennsylvania, USA), before joining the University of Balearic Islands where he has been a full professor since 1994. Prof. Toral has authored 200 scientific publications. He was the director of the Physbio International Summer School on Stochastic Processes in Biology held in St. Etienne de Tinn?e (France) in 2006, and a member of the editorial board of Fluctuations and Noise Letters (2005-2007), as well as the organizer of several conferences devoted to basic issues and applications of Nonlinear and Statistical Physics.

Pere Colet is Research Professor at IFISC (CSIC-UIB). He obtained his M.Sc. degree in physics from Universitat de Barcelona (1987) and his Ph. D. also in Physics from Universitat de les Illes Balears (1991), Spain. He was a postdoctoral Fulbright fellow at the School of Physics of the Georgia Institute of Tecnology. In May 1995, he joined the Spanish Consejo Superior de Investigaciones Cientificas. He has co-authored over 100 papers in ISI journals as well as 35 other scientific publications. His research interests include fluctuations and nonlinear dynamics of semiconductor lasers, synchronization of chaotic lasers and encoded communications, synchronization of coupled nonlinear oscillators, pattern formation, and quantum fluctuations in nonlinear optical cavities and dynamics of dissipative solitons.

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.