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\documentclass[10pt]{beamer}
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\title{Nonequilibrium systems and computation of transport coefficients\\[.3cm]
\small \textcolor{yellow}{SINEQ Summer school}%
}
\author{%
Urbain Vaes \texorpdfstring{\\\texttt{urbain.vaes@inria.fr}}{}
}
\institute{%
MATHERIALS -- Inria Paris
\textcolor{blue}{\&} CERMICS --
École des Ponts ParisTech
}
\date{\today}
\begin{document}
\begin{frame}[plain]
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\centering
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\titlepage
\end{frame}
\begin{frame}
{Some references}
\begin{itemize}
\item \fullcite{MR3509213}
\item \fullcite{pavliotis2011applied}
\item Lecture notes by Gabriel Stoltz on computational statistical physics:
\url{http://cermics.enpc.fr/~stoltz/Cours/intro_phys_stat.pdf}
\end{itemize}
\end{frame}
\begin{frame}
{Outline}
\tableofcontents
\end{frame}
\section{Introduction}
\begin{frame}
{Transport coefficients}
At the \alert{macroscopic} level,
transport coefficients relate an external forcing to an average response expressed through some steady-state flux.
\textbf{Examples:}
\begin{itemize}
\item The \emph{mobility} relates an external force to a velocity;
\item The \emph{heat conductivity} relates a temperature difference to a heat flux;
\item The \emph{shear viscosity} relates a shear velocity to a shear stress;
\end{itemize}
\vspace{.3cm}
\textbf{Challenges we do not address:}
\begin{itemize}
\item Choose thermodynamical ensemble;
\item Prescribe microscopic dynamics;
\end{itemize}
\end{frame}
\begin{frame}
{Computation of transport coefficients}
Three main classes of methods:
\begin{itemize}
\itemsep.2cm
\item
Non-equilibrium techniques
\begin{itemize}
\item Calculations from the steady state of a system out of equilibrium.
\item Comprises bulk-driven and boundary-driven approaches.
\end{itemize}
\item
Equilibrium techniques based on the Green--Kubo formula
\[
\rho = \int_{0}^{\infty} \expect_{\mu} \bigl[\varphi(x_t) \phi(x_0)\bigr] \, \d t.
\]
\item
Transient techniques:
\end{itemize}
\end{frame}
\section{Equilibrium and nonequilibrium dynamics}
\begin{frame}
{Equilibrium and nonequilibrium dynamics}
Consider a general diffusion process of the form
\[
\d x_t = b(x_t) \, \d t + \sigma(x_t) \, \d W_t,
\]
and assume that it admits an invariant distribution $\mu$.
\vspace{.2cm}
\begin{definition}
[Time-reversibility]
A stationary ($x_0 \sim \mu$) stochastic process $(x_t)$ is time-reversible if its law is invariant under time reversal:
the law of the \emph{forward paths} $(x_s)_{0 \leq s \leq t}$
coincides with the law of the \emph{backward paths} $(x_{t-s})_{0 \leq s \leq t}$.
\end{definition}
\vspace{.2cm}
\begin{theorem}
A stationary diffusion processes $x_t$ in $\real^d$ with generator $\mathcal L$ and invariant measure~$\mu$ is reversible if and only if $\mathcal L$ is self-adjoint in~$L^2(\mu)$.
\end{theorem}
\end{frame}
\end{document}
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