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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}

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\begin{frame}
  {Some references}
  \begin{itemize}
    \itemsep.2cm
    \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
      \[
        \rho = \int_{0}^{\infty} \expect_{\mu} \bigl[\varphi(x_t) \phi(x_0)\bigr] \, \d t.
      \]
      We will derive this formula from linear response.
    \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}

\begin{frame}
  {Existence of an invariant measure for noneq.\ dynamics}
  Consider the paradigmatic dynamics
  \begin{align*}
    \d q_t &= M^{-1} p_t \, \d t, \\
    \d p_t &= - \bigl(\grad V(q_t) + \eta F\bigr) \, \d t - \gamma M^{-1} p_t \, \d t + \sqrt{\frac{2 \gamma}{\beta}} \, \d W_t,
  \end{align*}
  where $(q_t, p_t) = \torus^d \times \real^d$ and $F \in \real^d$ with $\abs{F} = 1$ is a given direction.

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