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Dear colleague,
Next follows the index of a tutorial review on the topic of Stochastic
processes in Physics, corresponding to a series of lectures given at
the meeting Instabilities and Nonequilibrium Structures VI, to be
published shortly by Kluwer Academic. If you wish, you might find the
complete version in COND-MAT/9707147 or at
http://formentor.uib.es/PhysDept/publicationsDB/0091.html

Maxi San Miguel and Raul Toral
Instituto Mediterraneo de Estudios Avanzados IMEDEA (CSIC-UIB)
Campus Universitat de les Illes Balears
07071-Palma de Mallorca
Spain
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{\baselineskip=11truept
\noindent{\bf STOCHASTIC EFFECTS IN PHYSICAL SYSTEMS\footnote{
COND-MAT/9707147\\
This paper can be found in\\
http://formentor.uib.es/PhysDept/publicationsDB/0091.html\\
To be published
in {\sl Instabilities and Nonequilibrium Structures},
VI, E. Tirapegui and W.Zeller,
eds. Kluwer Academic Pub. (1997).}
}
\vskip13truept
\vskip13truept
{\noindent \bf MAXI SAN MIGUEL and RA{\'UL} TORAL}\\
Departamento de F\'{\i}sica Interdisciplinar \\
Instituto Mediterr\'a neo de Estudios Avanzados IMEDEA (CSIC-UIB)\\
Campus Universitat de les Illes Balears\\
E-07071 Palma de Mallorca, Spain\\
http://www.imedea.uib.es/PhysDept/
}

\contentsline {section}{\numberline {1}Introduction}{2}
\contentsline {section}{\numberline {2}Stochastic Processes}{3}
\contentsline {subsection}{\numberline {2.1}Basic Concepts}{3}
\contentsline {subsection}{\numberline {2.2}Stochastic Differential Equations}{9}
\contentsline {subsection}{\numberline {2.3}The Fokker--Planck Equation}{14}
\contentsline {subsection}{\numberline {2.4}Numerical generation of trajectories}{16}
\contentsline {subsubsection}{\numberline {2.4.1}The white noise case: Basic algorithms}{16}
\contentsline {subsubsection}{\numberline {2.4.2}The Ornstein--Uhlenbeck noise}{21}
\contentsline {subsubsection}{\numberline {2.4.3} Runge--Kutta type methods}{24}
\contentsline {subsubsection}{\numberline {2.4.4}Numerical solution of Partial Stochastic Differential Equations}{25}
\contentsline {subsection}{\numberline {2.5}A trivial (?) example: The linear equation with multiplicative noise}{27}
\contentsline {section}{\numberline {3}Transient stochastic dynamics}{30}
\contentsline {subsection}{\numberline {3.1}Relaxational dynamics at a pitchfork bifurcation}{33}
\contentsline {subsection}{\numberline {3.2}Statistics of laser switch-on}{41}
\contentsline {section}{\numberline {4}Noise in Dynamical Systems}{45}
\contentsline {subsection}{\numberline {4.1}Classification of Dynamical Systems: Potentials and Lyapunov Functions}{45}
\contentsline {subsection}{\numberline {4.2}Potentials and Stationary Distributions}{53}
\contentsline {subsection}{\numberline {4.3}The K\"uppers-Lortz instability}{55}
\contentsline {section}{\numberline {5}Noise effects in spatially extended systems}{62}
\contentsline {subsection}{\numberline {5.1}Symmetry restoring by noise in pattern formation}{62}
\contentsline {subsection}{\numberline {5.2}The stochastic Swift-Hohenberg equation: Symmetry restoring, pattern selection and noise modified Eckhaus instability.}{65}
\contentsline {subsection}{\numberline {5.3}Noise amplification in convective instabilities: Noise sustained structures}{72}
\contentsline {section}{\numberline {6}Fluctuations, phase transitions and noise--induced transitions}{75}
\contentsline {subsection}{\numberline {6.1}Noise--induced transitions}{75}
\contentsline {subsection}{\numberline {6.2}Noise--induced phase transitions}{77}
\end{document}

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