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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]
  \begin{figure}[ht]
    \centering
    % \includegraphics[height=1.5cm]{figures/logo_matherials.png}
    % \hspace{.5cm}
    \includegraphics[height=1.2cm]{figures/logo_inria.png}
    \hspace{.5cm}
    \includegraphics[height=1.5cm]{figures/logo_ponts.png}
    \hspace{.5cm}
    \includegraphics[height=1.5cm]{figures/logo_ERC.jpg}
    \hspace{.5cm}
    \includegraphics[height=1.5cm]{figures/logo_EMC2.png}
  \end{figure}
  \titlepage
\end{frame}

\begin{frame}
  {Some references}
  \begin{itemize}
    \itemsep.2cm
    \item \fullcite{MR3509213}
    \item \fullcite{pavliotis2011applied}
    \item \fullcite{MR2723222}
    \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}

\section{Introduction}

\begin{frame}
  \frametitle{Introduction}
  {\bf Aims of computational statistical physics}
  \begin{itemize}
    \item {\red numerical microscope}
    \item computation of {\blue average properties}, static or dynamic
  \end{itemize}
  \begin{center}
    \begin{minipage}[t]{.6\textwidth}
      \begin{figure}[ht]
        \centering
        \resizebox{\textwidth}{!}{%
          \begin{tikzpicture}
            \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=\textwidth]{figures/nanotube.png}};
            \node [draw, color=red!60, fill=red!5, very thick, rectangle, minimum height=1cm] (warm) at (0,1) {$T_+$};
            \node [draw, color=blue!60, fill=blue!5, very thick, rectangle, minimum height=1cm] (cold) at (7.3,4.1) {$T_-$};
            % \node [draw=none] (flux) at (5,2) {$J$};
            \draw[->, line width=1mm] (3.5,0.9) to node[below]{$J$} ++(2.5,2);
            \node [draw=none, minimum height=1cm] (unknown) at (0,4) {%
                \begin{minipage}{4cm}
                  \centering
                  \textbf{Fourier's law:}%
                  \[
                    J = - \alert{\kappa} \grad T
                  \]
                \end{minipage}
              };
        \end{tikzpicture}}
      \end{figure}
    \end{minipage}
  \end{center}

``Given the structure and the {\red laws of interaction} of the particles, what are the {\blue macroscopic properties} of the systems composed of these particles?''

\end{frame}

\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}
  They can be estimated from molecular simulation at the \blue{microscopic level}.
  \begin{itemize}
      \item Defined from \emph{nonequilibrium} dynamics;
      \item Three main classes of methods to calculate them.
  \end{itemize}

  \vspace{.3cm}
  \textbf{\blue Outline of this talk}
  \begin{itemize}
    \item Equilibrium vs nonequilibrium dynamics;
    \item Definition and computation of the mobility;
    \item Computation of other transport coefficients;
    \item Error analysis.
  \end{itemize}
\end{frame}


\begin{frame}
    \begin{center}
      \Large
      \color{blue}
      Part I: Definition and examples of nonequilibrium systems
    \end{center}

    \centering
    \begin{minipage}{.8\textwidth}
    \begin{itemize}
        \item Equilibrium vs nonequilibrium dynamics
        \item Existence of an invariant measure for nonequilibrium dynamics
        \item Convergence to the invariant measure
        \item Perturbation expansion of the invariant measure
    \end{itemize}
    \end{minipage}
\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}

  In this course, equilibrium = reversible,
  possibly up to a one-to-one transformation preserving the invariant measure.
\end{frame}

\begin{frame}
    {Paradigmatic examples of nonequilibrium dynamics}
    \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term}
        \begin{equation}
            \label{eq:overdamped_Langevin_F}
            \tag{NO}
            \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F} +  \sqrt{2} \, \d W_t
        \end{equation}
    \end{block}

    \begin{block}{Langevin dynamics perturbed by a constant force term}
        \begin{equation}
            \label{eq:Langevin_F}
            \tag{NL}
            \left\{
                \begin{aligned}
                    \d q_t & = M^{-1} p_t  \D t, \\*
                    \d p_t & =  \bigl( -\nabla V(q_t) + {\red \eta F}  \bigr) \D t - \gamma M^{-1} p_t  \D t
                    + \sqrt{2\gamma}  \D W_t,
                \end{aligned}
            \right.
        \end{equation}
        In the rest of this section, we take {\blue $ M = \I$} for simplicity.
    \end{block}
    where
    \begin{itemize}
        \item $F \in \real^d$ with $\abs{F} = 1$ is a given direction
        \item $\eta \in \real$ is the strength of the external forcing.
    \end{itemize}

    Is there an invariant probability measure?
\end{frame}

\begin{frame}
    {When {\yellow $\eta = 0$}, these dynamics are reversible}
    \begin{itemize}
        \item For overdamped Langevin dynamics
            \[
                \mathcal L_{\rm ovd} \Big\vert_{\red \eta = 0} = - \grad V \cdot \grad + \laplacian
                = - \grad^* \grad,
                \qquad
                \mu(\d q) = \frac{1}{Z} \e^{- V(q)} \, \d q.
            \]
            where $\grad^* := (\grad V - \grad) \cdot $.
            For any $f, g \in C^{\infty}_{\rm c}(\mathcal E)$, we have
            \[
                \int_{\mathcal E} (\mathcal L_{\rm ovd} f ) g \, \d \mu
                = - \int_{\mathcal E} \nabla f \cdot \nabla g \, \d \mu
                = \int_{\mathcal E} (\mathcal L_{\rm ovd} g ) f \, \d \mu.
            \]
        \item For Langevin dynamics, $\mu(\d q \, \d p) = \frac{1}{Z} \exp \left( - V(q) - \frac{\abs{p}^2}{2} \right) \, \d q \, \d p$.
            \begin{align*}
                \mathcal L\Big\vert_{\red \eta = 0}
                = p \cdot \grad_q - \grad V \cdot \grad_p + \gamma \left( - p \cdot \grad_p + \laplacian_p \right)
                = \grad_p^* \grad_q - \grad_q^* \grad_p  - \gamma \grad_p^* \grad_p,
            \end{align*}
            where  $\grad_q^* := (\grad V - \grad_q) \cdot $ and $\grad_p^* = (p -\grad_p) \cdot$ are the formal~$L^2(\mu)$ adjoints.
            \begin{align*}
                \int_{\mathcal E} (\mathcal Lf ) g \, \d \mu
                &= \int_{\mathcal E} g \left(\grad_p^* \grad_q - \grad_q^* \grad_p\right) f -  \gamma \grad_p f \cdot \grad_p g \, \d \mu  \\
                &= \int_{\mathcal E} {\red -} f \left(\grad_p^* \grad_q - \grad_q^* \grad_p\right) g -  \gamma \grad_p f \cdot \grad_p g \, \d \mu \\
                &= \int_{\mathcal E} (f \circ S) \bigl(\mathcal L (g \circ S)\bigr) \, \d \mu
                \qquad S f(q, p) := f(q, -p).
            \end{align*}
    \end{itemize}
\end{frame}

\begin{frame}
    {Another example useful for thermal transport}
    \begin{block}{Langevin dynamics with modified fluctuation}
        \[
            \left\{
                \begin{aligned}
                    \d q_t & = p_t  \, \d t, \\*
                    \d p_t & =  -\nabla V(q_t) \, \d t - \gamma p_t  \, \d t
                    + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t,
                \end{aligned}
            \right.
        \]
    \end{block}
    with non-negative temperature
    \[
        T_\eta(q) = T_{\rm ref} + \eta \widetilde{T}(q)
    \]
    Typically, $\widetilde{T}$ constant and positive on $\mathcal D_+ \subset \mathcal C$,
    and constant and negative on $\mathcal D_- \subset \mathcal D$.
    \begin{itemize}
        \item
            Non-zero energy flux from $\mathcal D_+$ to $\mathcal D_-$ expected in the steady-state

        \item

            Simplified model of thermal transport (in 3D materials or atom chains)
    \end{itemize}
\end{frame}

\begin{frame}
  {Worked example in dimension one}
  Consider the perturbed overdamped Langevin dynamics with~$q_t \in \torus$
  \[
      \d q_t = - V'(q_t) \, \d t + {\red \eta} \, \d t + \sqrt{2} \, \d W_t,
  \]
  The associated Fokker--Planck equation reads
  \[
      \frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0.
  \]
  \begin{minipage}[t]{.45\textwidth}
      \vspace{.5cm}
      The solution is unique and given by
      \[
          \rho_{\eta}(q) \propto \e^{-V(q)} \int_{\torus} \e^{V(q+y) - \eta y} \, \d y.
      \]

      \textbf{Example:} $\rho_{\eta}$ with $V(q) = \frac{1}{2} (1 - \cos q)$.
  \end{minipage}
  \begin{minipage}[t]{.5\textwidth}
  \end{minipage}
  \begin{minipage}[t]{.45\textwidth}
      \begin{figure}[ht]
          \centering
          \includegraphics[width=\linewidth]{figures/invariant_perturbed_ol.pdf}
      \end{figure}
  \end{minipage}
\end{frame}

\begin{frame}
  {Nonequilibrium overdamped Langevin dynamics}
  In general, how can we prove existence of an invariant measure for
  \[
      \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F} +  \sqrt{2} \, \d W_t \, ?
  \]

  \begin{itemize}
      \item
          If the state space is compact (e.g. $\torus^d$),
          apply Doeblin's theorem.

      \item
          If not, use its generization, Harris' theorem.
  \end{itemize}

  \medskip
  Fix ${\blue t = 1}$ and denote by $p\colon \mathcal E \times \mathcal B(\mathcal E)$ the Markov transition kernel
  \[
      p(x, A) := \proba \left[ q_t \in A \, \middle| \, q_0 = x \right].
  \]
  For an observable $\phi \colon \mathcal E \to \real$ and a probability measure $\mu$,
  we let
  \[
      (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, p(x, \d y),
      \qquad
      (\mathcal P^{\dagger} \mu)(A)  := \int_{A} p(x, A) \, \mu(\d x).
  \]
  Note that $\mathcal P$ and $\mathcal P^{\dagger}$ are formally $L^2$ adjoints:
  \[
      \int_{\mathcal E} (\mathcal P \phi) \, \d \mu = \int_{\mathcal E} \phi \, \d (\mathcal P^\dagger \mu).
  \]
\end{frame}

\begin{frame}
  {Existence of an invariant measure for compact state space (1/2)}
  Let $d(\placeholder, \placeholder)$ denote the total variation metric.
  \begin{theorem}
      [Doeblin's theorem]
      If there exists $\alpha \in (0, 1)$ and a probability measure $\pi$ such that
      \[
          \forall \mu, \qquad
          \mathcal P^{\dagger} \mu \geq \alpha \pi,
          \qquad \text{ (Minorization condition) }
      \]
      then there exists $\mu_*$ such that $\mathcal P^{\dagger} \mu_* = \mu_*$,
      and $d(\mathcal P^{\dagger^n}  \mu, \mu_*) \leq (1-\alpha)^n d(\mu, \mu_*)$.
  \end{theorem}

  \emph{Sketch of proof.} Define the Markov transition kernel
  \[
      \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \pi(\placeholder),
  \]
  Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$.
  We have
  \begin{align*}
      d(\mathcal P^\dagger \mu, \mathcal P^\dagger \nu)
      &= \sup_{\phi \in \mathcal F} \int_{\mathcal E} \phi(q) (\mathcal P^{\dagger} \mu - \mathcal P^{\dagger} \nu) (\d q)
      = \sup_{\phi \in \mathcal F} \int_{\mathcal E} \mathcal P \phi(q) \bigl(\mu - \nu\bigr) (\d q) \\
      &= (1 - \alpha) \sup_{\phi \in \mathcal F} \int_{\mathcal E} \widetilde {\mathcal P} \phi(q) (\mu - \nu) (\d q)
      \leq (1 - \alpha) \, d(\mu, \nu).
  \end{align*}
  Conclude using Banach's fixed point theorem.
\end{frame}

\begin{frame}
  {Existence of an invariant measure for compact state space (2/2)}
  Two simple corollaries:
  \begin{itemize}
      \item
          Suppose that $\phi$ is uniformly bounded. Then
          \begin{align*}
              \left\lvert \mathcal P^n \phi(x) - \overline \phi \right\rvert
              &= \int_{\mathcal E} \mathcal P^n (\phi - \overline \phi) \, \d(\delta_x - \mu_{*})
              = \int_{\mathcal E} (\phi - \overline \phi) \, (\mathcal P^{\dagger n} \delta_x -  \mathcal P^{\dagger n} \mu_{*}) (\d q) \\
              &\leq \norm{\phi - \overline \phi}_{L^{\infty}} (1-\alpha)^n d(\delta_x, \mu_{*})
              \leq 2 \norm{\phi - \overline \phi}_{L^{\infty}} (1 - \alpha)^n.
          \end{align*}
          This shows that
          \[
              \left\lVert \mathcal P^n \phi(x) - \overline \phi \right\rVert_{L^{\infty}}
              \leq 2 (1 - \alpha)^n \norm{\phi - \overline \phi}_{L^{\infty}}.
          \]

      \item
          The Neumann series $\I + \mathcal P + \mathcal P^2 + \dotsb$ is convergent as a bounded operator on
          \[
              L^{\infty}_{*} := \left\{ \phi \in L^{\infty}(\mathcal E) : \int_{\mathcal E} \phi \, \d \mu_{*} = 0 \right\}.
          \]
          Thus $\I - \mathcal P$ is invertible on~$L^{\infty}_{*}$ and
          \[
              (\I - \mathcal P)^{-1} = \I + \mathcal P + \mathcal P^2 + \dotsb
          \]
  \end{itemize}
\end{frame}

\begin{frame}
    {Connection with the time-continuous setting}
    Consider the overdamped Langevin dynamics on~$\torus^d$:
    \[
        \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F \, \d t} +  \sqrt{2} \, \d W_t,
        \qquad q_t \in \torus^d.
    \]

    \begin{itemize}
        \itemsep.5cm
        \item
            The \textbf{minorization condition} is satisfied.
            Indeed for $t > 0$
            \begin{align*}
                \mathcal P^{\dagger}\mu (A)
                &= \expect \left[ q_t \in A \, \middle| \, q_0 \sim \mu \right]
                =  \int_{\mathcal E} \int_{A} p_t(x, y) \, \mu(\d x) \,
                && p_t = \text{transition pdf} \\
                &\geq \left( \inf_{(x,y) \in \mathcal E^2} p_t(x, y) \right) \lambda(A) && \lambda := \text{Lebesgue measure}.
            \end{align*}
            The infimum is $> 0$ by parabolic regularity and Harnack's inequality.
        \item
            \textbf{Decay of the semigroup}:
            For $t \in [0, \infty)$ and $\varphi \in L^{\infty}_*$, it holds that
            \begin{align*}
                \lVert \e^{t \mathcal L_{\rm ovd}} \varphi \rVert_{L^{\infty}}
        &= \left\lVert \e^{(t-  \lfloor t \rfloor) \mathcal L_{\rm ovd}} \left( \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi \right) \right\rVert_{L^{\infty}} \\
        &\leq  \left\lVert   \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi  \right\rVert_{L^{\infty}}
        \leq 2 \e^{\alpha} \e^{- \alpha t}  \lVert \varphi \rVert_{L^{\infty}}.
            \end{align*}

        \item
            \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{*}$,
            and
            \[
                \mathcal L_{\rm ovd}^{-1}
                = - \int_{0}^{\infty} \e^{t \mathcal L_{\rm ovd}} \, \d t.
            \]
    \end{itemize}
\end{frame}

\begin{frame}
  {Existence of an invariant measure for perturbed Langevin dynamics}
  Consider the paradigmatic dynamics
  \begin{align*}
    \d q_t &= p_t \, \d t, \\
    \d p_t &= - \grad V(q_t) \, \d t + {\red \eta F \, \d t} - \gamma p_t \, \d t + \sqrt{2 \gamma} \, \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.

  \begin{figure}[ht]
    \centering
    \includegraphics[width=0.39\linewidth]{figures/intro_position.pdf}
    \includegraphics[width=0.39\linewidth]{figures/intro_velocity.pdf}
    \caption{%
      Marginals of the steady state solution of the Langevin dynamics with forcing
    }
  \end{figure}
\end{frame}

\begin{frame}
    {Harris' theorem \footnote{M. Hairer and J. Mattingly, \emph{Progr. Probab.} (2011)}}
      Let $p(x, A)$ denote a Markov transition kernel and let
      \[
          (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, p(x, \d y),
          \qquad
          (\mathcal P^{\dagger} \mu)(A)  := \int_{A} p(x, A) \, \mu(\d x).
      \]
      \vspace{-.2cm}
  \begin{theorem}
      [Harris's theorem]
      Suppose that the following conditions are satisfied:
      \begin{itemize}
          \item
            There exists $\mathcal K\colon \mathcal E \to [1, \infty)$
            and constants~$a > 0$ and $b \geq 0$ such that
            \[
                \forall x \in \mathcal E, \qquad
                \mathcal L \mathcal K(x) \leq - a \mathcal K(x) + b,
            \]
          \item
            There exists a constant $\alpha \in (0, 1)$ and a probability measure~$\pi$ such that
            \[
                \inf_{x \in \mathcal C} p(x, \d y) \geq \, \alpha \, \pi(\d y),
            \]
            where $\mathcal C = \{x \in \real \, | \, \mathcal K(x) \leq K_{\max} \}$ for some $K_{\max} \geq 1 + 2 \, \frac{b}{a}$.
    \end{itemize}
    Then there $\exists! \, \, \mu_{*}$ such that $\mathcal P^{\dagger} \mu_{*} = \mu_{*}$.
    Furthermore  there is $\gamma \in (0, 1)$ such that
    \[
        \left\lVert \frac{\mathcal P^n \phi - \overline \phi}{\mathcal K} \right\rVert_{L^{\infty}}
        \leq C \gamma^n \norm{ \frac{\mathcal P^n \phi - \overline \phi}{\mathcal K} }_{L^{\infty}},
        \qquad \overline \phi := \int_{\mathcal E} \phi \, \d \mu_*.
    \]
  \end{theorem}
  
\end{frame}

\begin{frame}
  {Application to perturbed Langevin dynamics}
  For $\mathcal K \colon \mathcal E \to [1, \infty)$, let
  \[
      L^{\infty}_{\mathcal K}
      := \left\{ \varphi \text{~measureable } : \norm{\frac{\varphi}{\mathcal K}}_{L^{\infty}} < \infty \right\}
  \]

  \begin{theorem}
      Fix~$\eta > 0$ and $n \geq 2$,
      and let $\mathcal K_n(q, p) := 1 + \abs{p}^n$.
      There exists a unique invariant probability measure,
      with a smooth density~$\psi_{\eta}(q, p)$ with respect to the Lebesgue measure.
      Furthermore there exists $C = C(n, \eta) > 0$ and $\lambda = \lambda(n, \eta) > 0$ such that
      \[
          \forall \phi \in L^{\infty}_{\mathcal K_n}(\mathcal E), \qquad
          \left\lVert \e^{t \mathcal L_n} \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}}
          \leq C \e^{-\lambda t} \left\lVert \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}}
      \]
  \end{theorem}

  \textbf{Idea of the proof.}
  Show that the assumptions of Harris' theorem are satisfied,
  in particular that
  \begin{align*}
    \mathcal L \mathcal K_n &\leq - a \mathcal K_n(q, p) + b,
  \end{align*}
  for $a > 0$ and $b \geq 0$.
\end{frame}

\begin{frame}
{Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/3)}
    Consider the perturbed Langevin dynamics and  write
    \[
        \mathcal L_{\eta} = \mathcal L_0 + {\red \eta \widetilde {\mathcal L}},
        \qquad \widetilde {\mathcal L} = F \cdot \grad_p
    \]

    It is {\red expected} that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and
    \[
        f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathcal O(\eta^2)
    \]
    The invariance of $\psi_\eta$ can be written as
    \[
        \int_{\mathcal E} (\mathcal L_\eta \varphi) \psi_\eta = 0 = \int_{\mathcal E} (\mathcal L_\eta \varphi) f_\eta \psi_0
    \]
    \begin{block}{Fokker-Planck equation on $L^2(\psi_0)$}
        \[
            \mathcal L_\eta^* f_\eta = 0
        \]
        Observe that $\mathcal L_{\eta}^* = \mathcal L_0^* + \widetilde {\mathcal L}^*$ with
        \[
            \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p  - \gamma \grad_p^* \grad_p,
            \qquad \widetilde {\mathcal L}^* \placeholder = \grad_p^* (F  \placeholder)
        \]
    \end{block}


    {\bf Questions:} Can the expansion for $f_\eta$ be made rigorous? What is $\mathfrak{f}_1$?


\end{frame}

\begin{frame}
    {Perturbation expansion for {\yellow $\eta$ sufficiently small} (2/3)}
    \begin{block}
        {Formal asymptotics}
        Write $f_\eta = \mathfrak f_0 + \eta \mathfrak{f}_1 + \eta^2 \mathfrak{f}_2 + \dotsb$ and expand
        \begin{align*}
            \mathcal L_{\eta}^* f_{\eta}
            &= \mathcal L_0^* \mathfrak f_0 \\
            &\quad + \eta \left(\widetilde {\mathcal L}^* \mathfrak f_0 + \mathcal L_0^* \mathfrak f_1\right) \\
            &\quad + \eta^2 \left(\widetilde {\mathcal L}^* \mathfrak f_1 + \mathcal L_0^* \mathfrak f_2\right) \\
            &\quad + \eta^3 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_3\right) + \dotsb
        \end{align*}
        This suggests that $\mathfrak f_{i+1} = -(\mathcal L_0^*)^{-1} (\widetilde {\mathcal L}^* \mathfrak f_i)$ and so
        \[
            f_\eta = \sum_{i=0}^{\infty} (-\eta)^i \Bigl((\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^*\Bigr)^i \mathbf 1
            = \left(\I + \eta(\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right)^{-1} \mathbf 1.
        \]
    \end{block}
\end{frame}


\begin{frame}
    {Elements of proof}
    Let $\Pi_0$ denote the following projection operator
    \[
        \Pi_0 f := f - \int_{\mathcal E} f \, \psi_0
    \]
    \vspace{-.3cm}
    \begin{itemize}
        \itemsep.2cm
        \item
            The operator $\mathcal L_0^{-1}$ is a well defined bounded operator on $L_0^2(\psi_0)$ \\

            ({\red Hypocoercivity} + {\red hypoelliptic regularization})

        \item Since $\dps \gamma \| \nabla_p \varphi \|^2_{L^2(\psi_0)} = -\langle \mathcal L_0 \varphi,\varphi \rangle_{L^2(\psi_0)}$, it follows that
            \vspace{-0.2cm}
            \[
                \| \widetilde {\mathcal L} \varphi \|^2_{L^2(\psi_0)} \leq \| \nabla_p \varphi \|^2_{L^2(\psi_0)} \leq \frac{1}{\gamma} \| \mathcal L_0 \varphi \|_{L^2(\psi_0)} \| \varphi \|_{L^2(\psi_0)}
            \]
            Thus {\blue $\Pi_0 \widetilde {\mathcal L} \mathcal L_0^{-1}$ is bounded on $L^2_0(\psi_0)$}.
            \[
                \| \widetilde {\mathcal L} \mathcal L_0^{-1} \varphi \|^2_{L^2(\psi_0)}\leq \frac{\beta}{\gamma} \| \varphi \|_{L^2(\psi_0)} \| \mathcal L_0^{-1} \varphi \|_{L^2(\psi_0)}.
            \]


        \item It follows that $(\widetilde {\mathcal L} \mathcal L_0^{-1})^* \Pi_0 = (\widetilde {\mathcal L} \mathcal L_0^{-1})^*$ is also bounded on $L^2_0(\psi_0)$

            \medskip

        \item Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L_0^* + \eta \wcL^*$
            \vspace{-0.2cm}
            \[
                \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\wcL \mathcal L_0^{-1})^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0
            \]

        \item {\red Prove that $f_\eta \geq 0$}.
    \end{itemize}
\end{frame}

% \begin{frame}
%     {Elements of proof}
%     Let us introduce
%     \[
%         H^1_{p}(\psi_0) =
%         \Bigl\{ \varphi \in L^2(\psi_0) : \grad_p \varphi \in L^2(\psi) \Bigr\},
%         \qquad  \| \varphi \|_{H^1_{p}(\psi_0)}^2 = \| \varphi \|_{L^2(\psi_0)}^2 + \| \nabla_p \varphi \|_{L^2(\psi_0)}^2.
%     \]
%     \vspace{-.3cm}
%     \begin{itemize}
%         \itemsep.2cm
%         \item
%             The operator {\blue $\widetilde {\mathcal L}^*\colon H^1_p(\psi_0) \to L^2_0(\psi)$} is well-defined and bounded.
%             Indeed
%             \[
%                 \lVert \widetilde {\mathcal L}^* \varphi \rVert_{L^2_0(\psi_0)}^2
%                 = \ip{\nabla_p^* F \varphi}{\nabla_p^* F \varphi}_{L^2_0(\psi_0)}
%                 \leq \lVert \varphi \rVert_{H^1_p(\psi_0)}^2
%             \]
%             and
%             \[
%                 \int_{\mathcal E} \widetilde {\mathcal L}^* \phi \, \psi_0
%                 = \int_{\mathcal E} \nabla_p^* (F \phi) \, \psi_0 = 0.
%             \]
%         \item
%             The operator {\blue $(\mathcal L_0^*)^{-1} \colon L^2_0(\psi_0) \to H^1_p(\psi_0)$} is well-defined and bounded,
%             by {\red hypocoercivity} and {\red hypoelliptic regularization}.
%             % In particular, for $\phi = (\mathcal L_0^*)^{-1} \varphi$
%             % \begin{align*}
%             %     \| \phi \|_{L^2(\psi_0)}^2
%             %     +  \| \nabla_p \phi \|_{L^2(\psi_0)}^2
%             %     &= \|(\mathcal L_0^*)^{-1} \varphi \|_{L^2(\psi_0)}^2
%             %     + \frac{1}{\gamma} \ip{-\mathcal L_0^* \phi}{\phi}_{L^2(\psi_0)} \\
%             %     &\leq \frac{1}{\gamma} \norm{(\mathcal L_0^*)^{-1}}_{\mathcal B\bigl(L^2(\psi_0)\bigr)}^2
%             %     \norm{\varphi}_{L^2(\psi_0)}
%             % \end{align*}
%
%         \item Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L_0^* + \eta \wcL^*$
%             \vspace{-0.2cm}
%             \[
%                 \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0.
%             \]
%
%         \item {\red Prove that $f_\eta \geq 0$}.
%     \end{itemize}
% \end{frame}

\begin{frame}
    {Perturbation expansion for {\yellow $\eta$ sufficiently small} (3/3)}

\begin{block}{Power expansion of the invariant measure}
Spectral radius $r$ of the bounded operator
  $(\wcL \mathcal L_0^{-1})^* \in \mathcal{B}(L_0^2(\psi_0))$:
  \[
  r = \lim_{n \to +\infty}  \left\| \left[ \left(\wcL \mathcal L_0^{-1}\right)^* \right]^n  \right\|^{1/n}.
  \]
  Then, for $|\eta| < r^{-1}$, the unique invariant measure can be written as $\psi_\eta = f_\eta\psi_0$,
  where~$f_\eta \in L^2(\psi_0)$ can be expanded as
  \begin{equation}
    \label{eq:expansion_psi_xi_general}
    f_\eta =  \left( 1+\eta (\wcL \mathcal L_0^{-1})^*  \right)^{-1} \mathbf{1}
  = \biggl( 1 + \sum_{n=1}^{+\infty} (-\eta)^n
   [ (\wcL \mathcal L_0^{-1})^* ]^n \biggr) \mathbf{1}.
  \end{equation}
\end{block}

Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
\end{frame}

\section{Computation of transport coefficients}

\begin{frame}
    \begin{center}
      \Large
      \color{blue}
      Part II: Definition and calculation of the mobility
    \end{center}

    \centering
    \begin{minipage}{.6\textwidth}
    \begin{itemize}
        \item Definition through linear response
        \item Green--Kubo reformulation
        \item Link with effective diffusion
    \end{itemize}
    \end{minipage}
\end{frame}

\begin{frame}
  {Computation of transport coefficients}
  Three main classes of methods:
  \begin{itemize}
    \itemsep.2cm
    \item
      Non-equilibrium steady state 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 methods.
      \begin{itemize}
          \item System locally perturbed
          \item Relaxation of this perturbation enables to calibrate macroscopic model.
      \end{itemize}
  \end{itemize}

  We illustrate the first two for the simplest transport coefficient:
  the {\blue mobility}.
\end{frame}

\begin{frame}
    {Linear response of nonequilibrium dynamics}
    Consider the nonequilibirium dynamics with $V$ periodic:
    \begin{align*}
        \d q_t &= p_t \, \d t, \\
        \d p_t &= - \grad V(q_t) \, \d t + {\red \eta F \, \d t} - \gamma p_t \, \d t + \sqrt{2 \gamma} \, \d W_t,
    \end{align*}

    \begin{itemize}
        \item The force {\red $\eta F$} induces a non-zero velocity in the direction $F$
        \item Encoded by $\dps \expect_\eta(R) = \int_{\mathcal E} R \, \psi_\eta$ with $\dps R(q,p) = F^\t  p$
    \end{itemize}

    \begin{definition}
        [Mobility]
        The mobility in direction $F$ is defined mathematically as
        \[
            \rho_{F} =
            \lim_{\alert{\eta} \to 0} \frac{\expect_{\red \eta} [R] - \expect_{0} [R]}{\red \eta}
            = \lim_{\eta \to 0} \frac{1}{\alert{\eta}}\expect_{\red \eta} [R]
        \]
    \end{definition}

    We proved that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and
    \[
        f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathcal O(\eta^2), \qquad \mathfrak f_1 = - (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \mathbf 1.
    \]
    Therefore
    \[
        \rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0
        = -\int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L}^* \mathbf 1) \, \psi_0
    \]
\end{frame}

\begin{frame}
    {Numerical results (1)}
    \begin{figure}
        \centering
        \includegraphics[width=.75\textwidth]{figures/LR.eps}
    \end{figure}
\end{frame}

\begin{frame}
    {Numerical results (2)}
    \begin{figure}
        \centering
        \includegraphics[width=.75\textwidth]{figures/mobilityFctGamma.pdf}
        \caption{Mobility as a function of~$\gamma$~\footnote{See J.~Roussel and G.~Stoltz, \emph{ESAIM: M2AN} (2018)}}
    \end{figure}
\end{frame}

\begin{frame}
    {Reformulation as integrated correlation function}
    Define the conjugate response
    \[
        S
        = \wcL^* \mathbf{1}
        = \nabla_p^* (F \mathbf 1)
        = F^\t p.
    \]

    \begin{block}{Green--Kubo formula}
        For any $R \in L^2_0(\psi_0)$,
        \[
            \lim_{\eta \to 0} \frac{\expect_\eta(R)}{\eta} = \int_0^{+\infty} \expect_0 \Big(R(q_t,p_t)S(q_0,p_0)  \Big) d t,
        \]
        where $\expect_\eta$ is w.r.t.\ to $\psi_\eta(q,p)\, \d q\, \d p$,
        while $\expect_0$ is w.r.t.\ initial conditions~$(q_0,p_0) \sim \psi_0$ and over all realizations of the equilibrium dynamics.
    \end{block}

    For the mobility,
    it holds $S(q,p) = R(q,p) = F^\t p$ and so
    \[
        \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta \bigl(F^\t p  \bigr)}{\eta}
        = \int_0^{+\infty} \expect_0 \Big( \bigl(F^\t p_t\bigr) \bigl(F^\t p_0\bigr)  \Big) \, \d t
    \]
\end{frame}

\begin{frame}
    {Elements of proof}

\bu Proof based on the following equality on $\mathcal{B}\bigl(L_0^2(\psi_0)\bigr)$
\[
-\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, \d t.
\]

\bu Then,
\begin{align*}
\lim_{\eta \to 0} \frac{\expect_\eta(R)}{\eta} & = -\int_{\mathcal E} R \left[(\wcL \mathcal L_0^{-1})^* \mathbf{1}\right] \psi_0
= -\int_{\mathcal{E}}  [\mathcal L_0^{-1}R  ] [\wcL^* \mathbf{1} ] \, \psi_0 \\
& = \int_0^{+\infty} \left( \int_{\mathcal{E}} \left(\mathrm{e}^{t \mathcal L_0} R\right) \, S \, \psi_0\right) \d  t \\
& = \int_0^{+\infty} \expect \Big( R(q_t,p_t)S(q_0,p_0)  \Big) \, \d t
\end{align*}

\bu Note also that $S$ has average 0 w.r.t.\ invariant measure since
\[
\int_\cX S \, \d\pi = \int_\cX \wcL^* \mathbf{1} \, \d\pi = \int_\cX \wcL\mathbf{1} \, \d\pi = 0
\]

\end{frame}

\begin{frame}
  {Connection with effective diffusion}
  It is possible to show a {\blue functional central limit theorem} for the Langevin dynamics:
  \begin{equation*}
    \varepsilon  \widetilde {q}_{s/\varepsilon^2} \xrightarrow[\varepsilon \to 0]{} \sqrt{2 \mat D} \, W_s
    \qquad  \text{weakly on } C([0, \infty)), \qquad \widetilde {q}_t := q_0 + \int_{0}^{t} p_s \, \d s \in {\blue \real^{d}}.
  \end{equation*}
  In particular, $\widetilde {q}_t /\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ weakly.

  \vspace{-.25cm}
  \begin{figure}[ht]
  \centering
  \href{run:videos/gle/effective-diffusion.webm?autostart&loop}%
  {\includegraphics[width=0.75\textwidth]{videos/gle/effective-diffusion.png}}%
  \caption{Histogram of $q_t/\sqrt{t}$. The potential $V(q) = - \cos(q) / 2$ is illustrated in the background.}
  \end{figure}
\end{frame}

\begin{frame}
  {Mathematical expression for the effective diffusion (dimension 1)}
  \vspace{.2cm}
  \begin{block}{Expression of $D$ in terms of the solution to a Poisson equation}
      Effective diffusion tensor given by $D =  \ip{\phi}{p}_{L^2(\mu)}$ and $\phi$ is the solution to
  \[
    - \mathcal L \phi = p,
    \qquad \phi \in L^2_0(\mu).
  \]
  \end{block}
  \textbf{Key idea of the proof:} Apply It\^o's formula to $\phi$
  \begin{align*}
    \d \phi(q_s, p_s)
    % &= \frac{1}{\varepsilon^2} \mathcal L_{L} \phi (q_t, p_t)  + \frac{1}{\varepsilon} \, \sqrt{2 \gamma \beta^{-1}} \, \derivative{1}[\phi]{p}(q_t, p_t) \, \d W_t, \\
    &= - p_s \, \d s + \sqrt{2 \gamma} \, \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s
  \end{align*}
  and then rearrange:
  \begin{align*}
    \alert\varepsilon (\widetilde q_{t/\alert\varepsilon^2} - \widetilde q_{0}) &= \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} p_s \, \d s \\
                                                            &= \underbrace{\alert\varepsilon \bigl(\phi(q_0, p_0) - \phi(q_{t/\alert\varepsilon^2}, p_{t/\alert\varepsilon^2})\bigr)}_{\to 0
                                                            % ~\text{in $L^p(\Omega, C([0, T], \real))$}
                                                          }
    + \underbrace{\sqrt{2 \gamma} \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s}_{\to \sqrt{2 D} W_t~\text{weakly by MCLT}}.
  \end{align*}
  % where
  % \begin{align*}
  %   D &= \gamma \beta^{-1} \, \int \abs{\textstyle \derivative{1}[\phi]{p}(q, p)}^2 \, \mu(\d q \, \d p)
  %   = - \int \phi (\mathcal L \phi) \, \d \mu
  %   = \ip{\phi}{p}.
  % \end{align*}

  \textbf{In the multidimensional setting}, $D_{F} = \ip{\phi_{F}}{F^\t p}$ with $- \mathcal L \phi_{F} = F^\t p$.

  \textbf{Einstein's relation:} we just showed
  \(
      D_F = \beta^{-1} \rho_F.
  \)
\end{frame}


\begin{frame}
  {Summary: numerical approaches for calculating the mobility}
  \begin{itemize}
    \itemsep.5cm
    \item {\blue Linear response approach}:
        \begin{equation*}
            \rho_F = \lim_{\eta \to 0} \frac{1}{\alert{\eta}} \expect_{\alert{\eta}} \, \bigl[F^\t p\bigr].
        \end{equation*}
        where $\mu_{\eta}$ is the invariant distribution of the system with external forcing.

    \item {\blue Einstein's relation}:
      \[
        \rho_F = \lim_{t \to \infty}  \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl| F^\t (\widetilde {q}_t - q_0)\bigr|^2 \Bigr].
      \]

    \item Deterministic method, e.g. {\blue Fourier/Hermite Galerkin}, for the Poisson equation
      \[
        - \mathcal L_0 \phi_{F} = F^\t p, \qquad \rho_F = \ip{\phi_F}{F^\t p}.
      \]

    \item {\blue Green--Kubo formula}:
      \begin{align*}
        \rho_F &= \int_{0}^{\infty} \expect_{\blue 0}\bigl((F^\t p_0) (F^\t p_t)\bigr) \, \d t.
      \end{align*}
  \end{itemize}
\end{frame}

\begin{frame}
    \begin{center}
      \Large
      \color{blue}
      Part III: Computation of other transport coefficients
    \end{center}

    \centering
    \begin{minipage}{.6\textwidth}
    \begin{itemize}
        \item Thermal conductivity
        \item Shear viscosity
    \end{itemize}
    \end{minipage}
\end{frame}

\begin{frame}
    {Thermal transport in one-dimensional chain (1/3)}
    Consider a chain of $N$ atoms with nearest-neighbor interactions
    \begin{tikzpicture}
        \coordinate (origin) at (0,0);
        \coordinate (shift) at (1.8,0);
        \node [draw, color=red!60, fill=red!5, very thick, rectangle, minimum height=1cm] (nc) at (0,0) {$T_L$};
        \node [draw, color=blue!60,  fill=blue!5,  very thick, rectangle, minimum height=1cm] (nh) at ($ (origin) + 6*(shift) $) {$T_R$};
        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n1)  at ($ (origin) + 1*(shift) $)  {$p_1$};
        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n2)  at ($ (origin) + 2*(shift) $)  {$p_2$};
        \node [draw=none, circle, minimum size=1cm]                                    (n3)  at ($ (origin) + 3*(shift) $)  {$\dotsb$};
        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n-2) at ($ (origin) + 4*(shift) $) {$p_{N-1}$};
        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n-1) at ($ (origin) + 5*(shift) $) {$p_{N}$};
        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n1) -- node[below=.25cm]{$r_1$} (n2);
        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n2) -- node[below=.25cm]{$r_2$} (n3);
        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n3) -- node[below=.25cm]{$r_{N-2}$} (n-2);
        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n-2) -- node[below=.25cm]{$r_{N-1}$} (n-1);
        \draw[red, ->] (nc) to [out=45,in=135] node[above]{$j_0$} (n1);
        \draw[red, ->] (n1) to [out=45,in=135] node[above]{$j_1$} (n2);
        \draw[red, ->] (n2) to [out=45,in=135] node[above]{$j_2$} (n3);
        \draw[red, ->] (n3) to [out=45,in=135] node[above]{$j_{N-2}$} (n-2);
        \draw[red, ->] (n-2) to [out=45,in=135] node[above]{$j_{N-1}$} (n-1);
        \draw[red, ->] (n-2) to [out=45,in=135] node[above]{$j_{N-1}$} (n-1);
        \draw[red, ->] (n-1) to [out=45,in=135] node[above]{$j_{N}$} (nh);
    \end{tikzpicture}

    Mathematical model:
    \begin{equation*}
        \left\{ \begin{aligned}
                \d r_n &= (p_{n+1} - p_n) \, \d t, \\
                \d p_1 &= v'(r_1) \, \d t - \gamma p_1 \d t + \sqrt{2 \gamma {\color{red} (T+\Delta T)}} \, \d W_t^L, \\
                \d p_n &= \bigl(v'(r_n) - v'(r_{n-1})\bigr) \, \d t, \\
                \d p_N &= -v'(r_{N-1}) \, \d t - \gamma p_N \d t + \sqrt{2 \gamma {\color{blue} (T-\Delta T)}} \, \d W_t^R,
        \end{aligned} \right.
    \end{equation*}

The Hamiltonian of the system is the sum of the potential and kinetic energies:
\begin{equation*}
    H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2},
    \qquad V(r) = \sum_{n=1}^{N-1} v(r_n).
\end{equation*}
\end{frame}

\begin{frame}
    {Thermal transport in one-dimensional chains (2/3)}

    \begin{itemize}
        \item
            When ${\red \Delta T} = 0$,
            invariant distribution given by
            \[
                \pi(\d r \, \d p) = Z_\beta^{-1} \exp\left(- \beta \left( \frac {|p|^2} {2} + V(r) \right)\right) \, \d r \, \d p,
                \qquad \beta = T^{-1}.
            \]

        \item
            Generator of the dynamics:
            \begin{equation*}
                \begin{aligned}
                    \mathcal L
            &= \sum_{n=1}^{N-1} (p_{n+1} - p_n) \partial_{r_n}
            + \sum_{n=1}^N \Bigl(v'(r_n) - v'(r_{n-1})\Bigr) \partial_{p_n}  \\
            &\qquad - \gamma p_1 \partial_{p_1} + \gamma T \partial_{p_1}^2 - \gamma p_N \partial_{p_N} + \gamma T \partial_{p_N}^2
            + {\red \gamma \Delta T (\partial_{p_1}^2 - \partial_{p_N}^2)}.
                \end{aligned}
            \end{equation*}

            The {\red perturbation} $\widetilde {\mathcal L} = \gamma( \partial_{p_1}^2 - \partial_{p_N}^2)$
            is not bounded relatively to $\mathcal L_0$...

            \vspace{.5cm}
            $\rightarrow$ Existence/uniqueness of the invariant measure more difficult to prove\footnote{P. Carmona, \emph {Stoch. Proc. Appl.} (2007)}
    \end{itemize}
\end{frame}


\begin{frame}
    {Thermal transport in one-dimensional chains (3/3)}

    \bu Response function: {\blue total energy current}
    \begin{block}
        {Definition of the heat flux}
        \[
            J = \frac{1}{N-1}\sum_{n=1}^{N-1} j_{n},
            \qquad
            j_{n} = -v'(r_n)\frac{p_n+p_{n+1}}{2}
        \]
    \end{block}
    \smallskip

    \bu Motivation: Local conservation of the energy (in the bulk $2 \leq n \leq N-1$)
    \[
        \frac{\d\varepsilon_n}{\d t} =
        \mathcal L \varepsilon_n = j_{i-1} - j_{i},
        \qquad
        \varepsilon_n = \frac{p_n^2}{2} + \frac12 \Big( v(r_{i-1}) + v(r_n) \Big)
    \]

    \bu Definition of the {\blue thermal conductivity}: linear response
    \[
        \kappa_N = \lim_{\Delta T \to 0} \frac{(N-1)}{2\Delta T} \expect_{\Delta  t} [J].
    \]

\end{frame}

\begin{frame}
    {Shear viscosity in fluids (1/4)}

    Consider a fluid in $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$ subjected to a sinusoidal forcing
    \begin{figure}
        \centering
        \includegraphics[height=.5\textwidth]{figures/osc_shear.eps}
    \end{figure}

    Suppose that the box contains $N$ particles of mass $m$,
    each subjected to a force $F$.
\end{frame}

% \begin{frame}
%     {Shear viscosity in fluids (2/4)}
%     Macroscopic description by Navier--Stokes equation
%     \[
%         \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \nu \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x}
%     \]
%     Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives
%     \[
%         - \nu U_x''(y) = \overline{\rho} F(y), \qquad \overline \rho := \frac{\rho}{m} = \frac{N}{|\mathcal D|}
%     \]
% \end{frame}

\begin{frame}
    {Shear viscosity in fluids (2/4)}
    Assume pairwise interactions
    \[
        V(q) = \sum_{1 \leq \ell < n \leq N} \mathcal V(\abs{q_\ell - q_n}).
    \]
    \bu Add a smooth {\blue nongradient force} in the $x$ direction, depending on~$y$
    \begin{block}{Langevin dynamics under flow}
        \centerequation{\left \{ \begin{aligned}
                    \d q_{n} &= \frac{p_{n}}{m} \, \d t,\\
                    \d p_{n,x} &= - \partial_{q_{n,x}} V(q_t) \, \d t + {\red \eta F(q_{n,y}) \, \d t}
                    - \gamma \frac{p_{n,x}}{m} \, \d t + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{n,x}_t, \\
                    \d p_{n,y} &= - \partial_{q_{n,y}} V(q_t) \, \d t - \gamma \frac{p_{n,y}}{m} \, \d t
                    + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{n,y}_t.
            \end{aligned} \right.
        }
    \end{block}

    \smallskip

    \bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$

    \smallskip

    \bu The perturbation $\dps \wcL = \sum_{n=1}^N \! F(q_{n,y}) \partial_{p_{n,x}}$ is $\mathcal{L}_0$-bounded
\end{frame}


\begin{frame}
    {Shear viscosity in fluids (3/4)}

    \bu {\blue Linear response}:
    \[
        \lim_{\eta \rightarrow 0} \frac{\expect_{\eta} [\mathcal L_0 h]}{\eta}
        = - \frac{\beta}{m} \!
        \left\langle \!h, \sum_{n=1}^N p_{n,x} F(q_{n,y}) \!\right\rangle_{L^2(\psi_0)}.
    \]

\bu Average {\red longitudinal velocity}
  $u_x(Y) = \dps \lim_{\varepsilon \to 0}
  \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\placeholder) \right]}{\eta}$
  where
  \vspace{-0.3cm}
  \[
  U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{n=1}^N p_{n,x}
  \, \chi_{\varepsilon}(q_{n,y}-Y)
  \]
  \vspace{-0.5cm}

\bu Average {\red off-diagonal stress}
  $\dps \sigma_{xy}(Y) = \lim_{\varepsilon \to 0}
  \lim_{\eta \to 0} \frac{\expect_{\eta} [...]}{\eta}$,
  where
  \vspace{-0.4cm}
  \[
  \hspace{-0.1cm}
  ... =
  \frac{1}{L_x} \left( \sum_{n=1}^N \frac{p_{n,x} p_{n,y}}{m}\chi_{\varepsilon}(q_{n,y}-Y)
  - \! \! \! \! \! \! \! \!
  \sum_{1 \leq n < \ell \leq N} \! \! \! \!
  \mathcal V'(|q_n-q_\ell|)\frac{ q_{n,x}-q_{\ell,x}}{|q_n-q_\ell|}
  \!\int_{q_{\ell,y}}^{q_{n,y}} \!\chi_{\varepsilon}(s-Y) \, ds \right)
  \]

\bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$
\[
    \frac{\d\sigma_{xy}(Y)}{\d Y} + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y),
    \qquad \overline{\rho} = \frac{N}{|\mathcal{D}|}.
\]
\end{frame}

\begin{frame}
    {Shear viscosity in fluids (4/4)}

\bu {\blue Definition} $\sigma_{xy}(Y) := -\nu(Y) u_x'(Y)$,
{\red closure} assumption $\nu(Y) = \nu > 0$.

\begin{block}{Velocity profile in Langevin dynamics under flow}
\centerequation{-\nu u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)}
\end{block}

Therefore, integrating against the test function~$\e^{2i\pi \frac{y}{L_y}}$ and rearranging,
we have
\[
    \nu = \overline \rho \left( \frac{F_1}{U_1} - \gamma \right) \left(\frac{L_y}{2\pi}\right)^2,
\]
where
\[
    U_1 = \frac{1}{L_y} \int_{0}^{L_y} u_x(x) \e^{2i\pi \frac{y}{L_y}} \, \d y,
    \qquad
    F_1 = \frac{1}{L_y} \int_{0}^{L_y} F(y) \e^{2i\pi \frac{y}{L_y}} \, \d y.
\]
The coefficient $U_1$ can be rewritten as
\[
    U_1 = \lim_{\eta \to 0} \frac{1}{\eta} {\dps \expect_{\eta} \left[ \frac{1}{N}\sum_{n=1}^{N} \frac{p_{n,x}}{m} \exp \left( 2i\pi \frac{q_{n,y}}{L_y} \right) \right]}.
\]

\end{frame}

\begin{frame}
    {Numerical illustration}
    \begin{figure}[ht]
        \centering
        \includegraphics[width=\linewidth]{figures/shear1.png}
        \caption{Numerical results from~\footnote{See R.~Joubaud and G.~Stoltz, \emph{Multiscale Model. Simul.} (2012)}}
    \end{figure}
\end{frame}

\begin{frame}
    {Numerical illustration}
    \begin{figure}[ht]
        \centering
        \includegraphics[width=\linewidth]{figures/shear2.png}
        \caption{Numerical results from~\footnote{See R.~Joubaud and G.~Stoltz, \emph{Multiscale Model. Simul.} (2012)}}
    \end{figure}
\end{frame}

\begin{frame}
    \begin{center}
      \Large
      \color{blue}
      Part IV: Error estimates on the estimation of transport coefficients
    \end{center}

    \centering
    \begin{minipage}{.8\textwidth}
    \begin{itemize}
        \item Reminders: strong order, weak order
        \item Error analysis for the linear response method
        \item Error analysis for the Green--Kubo method
    \end{itemize}
    \end{minipage}
\end{frame}



\begin{frame}
    {Reminder: Error estimates in Monte Carlo simulations}

Consider the general SDE
\[
    \d x_t = b(x_t)\,\d t + \sigma(x_t) \, \d W_t
\]
with invariant measure $\pi$.

\bigskip

\bu {\red Discretization} $x^{n} \simeq x_{n\dt}$, {\blue invariant measure $\pi_\dt$}. For instance,
\[
x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \stackrel{\rm{i.i.d.}}{\sim} \mathcal N(0,{\rm Id})
\]

\medskip

\bu {\blue Ergodicity} of the numerical scheme with invariant measure~$\pi_\dt$
\[
\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(\d x)
\]

\begin{block}{Error estimates for {\red finite} trajectory averages}
\[
\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n)
= \expect_\pi(A) + \underbrace{\frac{C}{N_{\rm iter} \dt}}_{\rm bias} + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error}
\]
\end{block}

\smallskip

\bu Bias $\expect_{\pi_\dt}(A)-\expect_\pi(A) \longrightarrow$ {\bf Focus today}

\medskip

\end{frame}


\begin{frame}\frametitle{Weak type expansions}

\bu Numerical scheme = {\red Markov chain} characterized by {\blue evolution operator}
\[
P_\dt \varphi(x) = \expect\Big( \varphi\left(x^{n+1}\right)\Big| x^n = x\Big)
\]
where $(x^n)$ is an approximation of $(x_{n \dt})$

\medskip

\bu Standard notions of error: {\red fixed integration time $T < +\infty$}
\begin{itemize}
\item {\blue Strong error}:
    \[
        \dps \sup_{0 \leq n \leq T/\dt} \expect | x^n - x_{n\dt} | \leq C \dt^p
    \]
\item {\blue Weak error}: for any $\varphi$,
    \[
        \dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(x^n\right)\right] - \expect\left[\varphi\left(x_{n\dt}\right)\right] \Big| \leq C \dt^p
    \]
%\item ``mean error'' \emph{vs.} ``error of the mean''
\end{itemize}

\begin{block}{$\dt$-expansion of the evolution operator}
    \[
        P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}
    \]
\end{block}

\smallskip

{\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$.

\end{frame}

\begin{frame}
    {Elements of proof}
    \begin{itemize}
        \item
            Rewrite the weak error as a telescopic sum
            \begin{align*}
                \expect \bigl[\varphi(x^N)\bigr] - \expect \bigl[\varphi(x_{N \dt})\bigr]
                &= P_{\dt}^N \varphi (x_0) - \e^{N \dt \mathcal L} \varphi(x_0) \\
                &= \sum_{n=0}^{N-1} \left( P_{\dt}^{N-n} \e^{n \dt \mathcal L}  \varphi(x_0) - P_{\dt}^{N-(n+1)} \e^{(n+1) \dt \mathcal L} \varphi (x_0) \right) \\
                &= \sum_{n=0}^{N-1} P_{\dt}^{N-(n+1)} \left( P_{\dt} - \e^{\dt \mathcal L} \right) \e^{n \dt \mathcal L} \varphi (x_0)
            \end{align*}

        \item
            Since $u(t, x) := \e^{t \mathcal L} \varphi(x)$ solves the backward Kolmogorov equation
            \begin{align*}
                \partial_t u = \mathcal L u,
                \qquad u(0, x) = \varphi.
            \end{align*}
            we can write formally
            \[
                \e^{\dt \mathcal L} \varphi = \I + \dt \mathcal L \varphi + \frac{\dt^2}{2} \mathcal L^2\varphi + \dotsb
            \]
    \end{itemize}
\end{frame}

\begin{frame}
    {Example: Euler-Maruyama, weak order~1}
    Consider the scheme
    \[
        x^{n+1} = \Phi_\dt(x^n,G^n) = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n
    \]

    \bigskip

    \bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$

    \bigskip

    \bu Technical tool: {\blue Taylor expansion}
    \vspace{-0.2cm}
    \[
        \varphi(x + \delta) = \varphi(x) + \delta^\t \nabla \varphi(x) + \frac12 \delta^\t \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots
    \]

    \medskip

    \bu Replace $\delta$ with $\sqrt{\dt}\, \sigma(x)\,G + \dt\,b(x)$ and {\blue gather in powers of $\dt$}
    \[
        \begin{aligned}
            \varphi\big(\Phi_\dt(x,G)\big) & = \varphi(x) + \sqrt{\dt}\, \sigma(x)\,G \cdot \nabla \varphi(x) \\
                                           & \ \ \ + \dt \left(\frac{\sigma(x)^2}{2} G^\t \left[\nabla^2\varphi(x)\right]G + b(x)\cdot\nabla \varphi(x) \right) + \dots
        \end{aligned}
    \]

    \medskip

    \bu Taking {\blue expectations w.r.t. $G$} leads to
    \[
        P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathcal O(\dt^2)
    \]
\end{frame}

\begin{frame}
    {Error estimates on the invariant measure (equilibrium)}

\begin{block}{Error estimates on $\pi_\dt$}
    Suppose that
    \begin{itemize}
        \item
            For all smooth $\varphi$, the following expansion holds
            \[
                P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}
            \]
        \item The probability measure $\pi$ is invariant by $\mathcal A_k$ for $1 \leq k \leq p$, namely
            \[
                \int_\cX \mathcal A_k \varphi \, d\pi = 0
            \]
        \item
            + {\red Technical assumptions} usually satisfied
    \end{itemize}
    Then
    \[
        \int_\cX \varphi \, \d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) \d\pi + \dt^{p+1} R_{\varphi,\dt},
    \]
    where $g_{p+1} = \mathcal A_{p+1}^* \mathbf 1$ and  $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$.
\end{block}

Error on invariant measure can be {\blue (much) smaller} than the weak error

\end{frame}

\begin{frame}
    {Motivation of the result}

We verify the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$.

\medskip

\bu Idea: $\pi_\dt = \pi (1 + \dt^p f_{p+1} + \dots)$

\medskip

\bu by definition of $\pi_\dt$
\[
\int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right) \psi \right] d\pi_\dt = 0
\]
\bu compare to first order correction to the invariant measure
\[
\begin{aligned}
& \int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right)\psi\right] (1+\dt^{p}f_{p+1})\, d\pi \\
& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathcal O\left(\dt^{p+1}\right) \\
& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathcal O\left(\dt^{p+1}\right)
\end{aligned}
\]

\begin{block}{}
Suggests $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$
\end{block}
\end{frame}

\begin{frame}
    {Examples of splitting schemes for Langevin dynamics (1)}

\bu Example: Langevin dynamics, discretized using a {\blue splitting} strategy
\[
A = M^{-1} p \cdot \nabla_q,
\quad
B_\eta = \Big(-\nabla V(q) + \eta\,F\Big)\cdot \nabla_p,
\quad
C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p
\]

\bu Note that $\mathcal L_\eta = A + B_\eta + \gamma C$

\medskip

\bu Trotter splitting $\to$ weak order 1
\[
P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \, \mathcal O(\dt^2)
\]

\bu Strang splitting $\to$ {\blue weak order 2}
\[
P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \, \mathcal O(\dt^3)
\]

\bu Other category: {\red Geometric Langevin}\footnote{N.~Bou-Rabee and H.~Owhadi, {\em SIAM J. Numer. Anal.} (2010)} algorithms, \textit{e.g.} $P_\dt^{\gamma C,A,B_\eta,A}$ \\
$\to$ weak order 1 but measure preserved at order 2 in $\dt$

\end{frame}


\begin{frame}\frametitle{Examples of splitting schemes for Langevin dynamics (2)}

\small

\bu $P_\dt^{B_\eta,A,\gamma C}$ corresponds to
%\begin{equation}
%\label{eq:Langevin_splitting}
$\dps \left\{ \begin{aligned}
\widetilde{p}^{n+1} & = p^n + \Big(-\nabla V(q^{n}) + \eta F\Big)\dt, \\
q^{n+1} & = q^n + \dt \, M^{-1} \widetilde{p}^{n+1}, \\
p^{n+1} & = \alpha_\dt \widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha^2_\dt}{\beta}M} \, G^n
\end{aligned} \right.$ \\[5pt]
%\end{equation}
where $G^n$ are i.i.d. Gaussian and $\alpha_\dt = \exp(-\gamma M^{-1} \dt)$

\bigskip

\bu $P^{\gamma C,B_\eta,A,B_\eta,\gamma C}_\dt$ for
%\[
$\dps \left\{ \begin{aligned}
\widetilde{p}^{n+1/2} & = \alpha_{\dt/2} p^{n} + \sqrt{\frac{1-\alpha_{\dt}}{\beta}M} \, G^{n}, \\
p^{n+1/2} & = \widetilde{p}^{n+1/2} + \frac{\dt}{2} \Big( -\nabla V(q^{n})+\eta F\Big), \\
q^{n+1} & = q^n + \dt \, M^{-1} p^{n+1/2}, \\
\widetilde{p}^{n+1} & = p^{n+1/2} + \frac{\dt}{2} \Big(- \nabla V(q^{n+1}) +\eta F\Big), \\
p^{n+1} & = \alpha_{\dt/2} \widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha_{\dt}}{\beta}M} \, G^{n+1/2}
\end{aligned} \right.$
%\]

\end{frame}


\begin{frame}
    {Error estimates on linear response (1/3)}
    \textbf{Aim:} For observable~$R$, approximate 
    \[
        \alpha = \lim_{\eta \to 0} \frac{\expect_{\red \eta} [R]}{\eta}
    \]


    \textbf{Estimator} of linear response (up to time discretization):
    \[
        \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, \d s \xrightarrow[t\to+\infty]{\mathrm{a.s.}}
        \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, \d \mu = \alpha + \mathcal O(\eta)
    \]

    {\bf Contributions to the error}
    \begin{itemize}
        \item Statistical error with {\red asymptotic variance $\mathcal O(\eta^{-2})$}
        \item Bias $\mathcal O(\eta)$ due to $\eta \neq 0$
        \item Bias from finite integration time
        \item Timestep discretization bias
    \end{itemize}


\end{frame}

\begin{frame}
    {Error estimates on linear response (2/3)}

    \bu {\bf Statistical error} dictated by {\blue Central Limit Theorem}:
    \[
        \sqrt{t} \left(\widehat{A}_{\eta,t} - \alpha_\eta \right) \xrightarrow[t \to +\infty]{\mathrm{law}} \mathcal{N}\left(0,\frac{\sigma_{R,\eta}^2}{\eta^2}\right),
        \qquad
        \sigma_{R,\eta}^2 = \sigma_{R,0}^2 + \mathcal O(\eta)
    \]
    so $\dps \widehat{A}_{\eta,t} = \alpha_\eta + \mathcal O_{\rm P}\left(\frac{1}{\eta \sqrt{t}}\right)$ $\to$ requires {\red long simulation times} $t \sim \eta^{-2}$

    \bigskip

    \bu {\bf Finite time integration bias}: $\dps \left| \expect\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\
    Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error}

    %\bigskip
    %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$

    \bigskip

    \bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, \d\mu$
    \[
        \widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, \d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T \d W_s
    \]

\end{frame}




\begin{frame}
    {Error estimates on linear response (3/3)}

    \begin{block}
        {Finite integration time bias and timestep bias}
        There exist functions $f_{0,1}$, $f_{\alpha,0}$ and $f_{\alpha,1}$ such that
        \[
            \int_{\mathcal E} R \, \d{\mu}_{\eta,\dt} = \int_{\mathcal E} R \Big(1+ \eta f_{0,1} + \dt^\alpha f_{\alpha,0} + \eta \dt^\alpha f_{\alpha,1} \Big) \d{\mu} + r_{\psi,\eta,\dt},
        \]
        where the remainder is compatible with linear response
        \vspace{-0.1cm}
        \[
            \left|r_{\psi,\eta,\dt}\right| \leq K(\eta^2 + \dt^{\alpha+1}),
            \qquad
            \left|r_{\psi,\eta,\dt} - r_{\psi,0,\dt}\right| \leq K \eta (\eta + \dt^{\alpha+1})
        \]
    \end{block}

\medskip

\bu Corollary: error estimates on the {\blue numerically computed mobility}
\[
\begin{aligned}
\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t p \, \mu_{\eta,\dt}(\d{q}\,\d{p}) - \int_{\mathcal E}  F^\t p \, \mu_{0,\dt}(\d{q}\,\d{p}) \right) \\
& = \rho_{F} + \dt^\alpha \int_{\mathcal E}  F^\t p  \, f_{\alpha,1} \, \d{\mu} + \dt^{\alpha+1} r_{\dt}
\end{aligned}
\]

\end{frame}


\begin{frame}\frametitle{Numerical results}

\begin{figure}
\begin{center}
\includegraphics[width=.8\textwidth]{figures/mobility_Langevin.eps}
\end{center}
\end{figure}

Scaling of the mobility for the first order scheme $P_\dt^{A,B_\eta,\gamma C}$ and the second order scheme $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$.

\end{frame}

\begin{frame}
    {Error estimates on the Green--Kubo formula (1/3)}
    \textbf{Aim:} For observable~$R$, approximate
    \[
        \alpha = \int_0^{+\infty} \!\! \expect_0\Big(R(q_t,p_t)S(q_0,p_0) \Big) \, \d t
    \]
  

    \textbf{``Natural'' estimator} (up to time discretization) 
    \[ 
        \widehat{A}_{K,T} = \frac1K \sum_{k=1}^K \int_0^T R(q_t^k,p_t^k)S(q_0^k,p_0^k)\, \d t
    \]

    \bu {\bf Contributions to the error:}
    \begin{itemize}
        \item Truncature of time (exponential convergence of $\e^{t \mathcal L}$)
        \item The {\red statistical error} increases linearly with $T$.
        \item {\blue Timestep bias and quadrature formula}
    \end{itemize}
\end{frame}

\begin{frame}
    {Error estimates on the Green--Kubo formula (2/3)}

    \bu {\bf Truncation bias}: {\blue small} due to generic exponential decay of correlations
    \[
        \left|\expect\left(\widehat{A}_{K,T}\right)-\alpha\right| \leq C \e^{-\kappa T}
    \]

    \bigskip

    \bu {\bf Statistical error}: {\red large}, increases with the integration time
    \[
        \forall T \geq 1, \qquad \mathrm{Var}\left(\widehat{A}_{K,T}\right) \leq C \frac{T}{K}
    \]


    \bu {\bf Time discretization and quadrature bias}: if

    \begin{itemize}
        \item {\red uniform-in-$\Delta t$ convergence}  
        \item error on the invariant measure of order~$\dt^a$
        \item $P_\dt = \mathrm{Id} + \dt \mathcal L + \dt^2 L_2 + \dots + \dt^{a} L_a + \dots$
    \end{itemize}
    Then for $R,S$ with average~0 w.r.t.~$\mu$, 
    \[
        \hspace{-0.1cm}\int_0^{+\infty} \expect \Big( R(X_t) S(X_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{R}_{\dt}\left(X^{n}\right)S\left(X^0\right)\right) + \mathrm{O}(\dt^a) \vspace{-0.5cm}
    \]
    with 
    \[
        \widetilde{R}_{\dt} = \Big(\mathrm{Id} + \dt \,L_2 \mathcal L^{-1} + \dots + \dt^{a-1} L_a \mathcal L^{-1} \Big)R - \mu_\dt(\dots)
    \]
\end{frame}

\begin{frame}
    {Error estimates on Green-Kubo formulas (1/3)}

    \bu For methods of {\bf weak order}~1, {\red Riemman sum} ($\phi,\varphi$ average 0 w.r.t. $\pi$)
    \vspace{-0.2cm}
    \[
        \begin{aligned}
&
\int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt) \\[-7pt]
& \mathrm{where} \ \Pi_\dt \phi = \phi - \int_\cX \phi \, d\pi_\dt
        \end{aligned}
    \]

    \bu For methods of {\bf weak order} 2, {\blue trapezoidal rule}
    \vspace{-0.1cm}
    \[
        \begin{aligned}
            \int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} & = \frac{\dt}{2} \expect_\dt \left(\Pi_\dt \phi\left(x^{0}\right)\varphi\left(x^0\right)\right) \\
                                                                             & \ \ + \dt \sum_{n=1}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt^2)
        \end{aligned}
    \]

\end{frame}

% \begin{frame}\frametitle{Error estimates on Green-Kubo formulas (2)}
% 
% \bu Error of {\red order~$\alpha$ on invariant measure}: $\dps \int_\cX \psi \, d{\pi}_\dt = \int_\cX \psi \, d{\pi} + \mathrm{O}(\dt^\alpha)$
% 
% \medskip
% 
% \bu Expansion of the evolution operator ($p+1 \geq \alpha$ and $\mathcal A_1 = \mathcal L$)
% \[
% P_\dt \varphi = \varphi + \dt \, \mathcal L \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}
% \]
% 
% \begin{block}{Ergodicity of the numerical scheme}
% \centerequation{
% \forall n \in \mathbb{N}, \qquad \left\| P_\dt^n \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq C_s \e^{-\lambda_s n\dt}
% }
% where $\mathcal{K}_s$ is a Lyapunov function ($1+|p|^{2s}$ for Langevin) and
% \[
% L^\infty_{\Li_s,\dt} = \left\{ \frac{\varphi}{\mathcal{K}_s} \in L^\infty(\cX), \ \int_\cX \varphi \, d\pi_\dt = 0\right\}
% \]
% \end{block}
% 
% \bu Proof: Lyapunov condition + uniform-in-$\dt$ minorization condition\footnote{M. Hairer and J. Mattingly, \emph{Progr. Probab.} (2011)}
% 
% \end{frame}
% 
% %-----------------------------------------------------------
% \begin{frame}\frametitle{Error estimates on Green-Kubo formulas (3)}
% 
% \begin{block}{Error estimates on integrated correlation functions}
% Observables $\varphi,\psi$ with average~0 w.r.t. invariant measure~$\pi$
% \[
% \int_0^{+\infty} \expect \Big( \psi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(x^{n}\right)\varphi\left(x^0\right)\right) + \dt^\alpha r^{\psi,\varphi}_\dt,
% \]
% where $\expect_\dt$ denotes expectations w.r.t. initial conditions $x_0 \sim \pi_\dt$ and over all realizations of the Markov chain $(x^n)$, and
% \[
% \widetilde{\psi}_{\dt,\alpha} = \psi_{\dt,\alpha} - \int_\cX \psi_{\dt,\alpha} \, d\pi_\dt\]
% with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \dots + \dt^{\alpha-1} \mathcal A_{\alpha}\mathcal L^{-1} \Big)\psi$
% \end{block}
% 
% \bu Useful when $\mathcal A_k \mathcal L^{-1}$ can be computed, \emph{e.g.} $\mathcal A_k = a_k \mathcal L^{k}$
% 
% \medskip
% 
% \bu Reduces to trapezoidal rule for second order schemes
% 
% \end{frame}

%\begin{frame}
%    {Sketch of proof (1)}

%\bu Define $\dps \Pi_\dt \varphi = \varphi - \int_\cX \varphi \, d\pi_\dt$

%\smallskip

%\bu Since $\mathcal L^{-1}\psi$ has average~0 w.r.t.~$\pi$, introduce $\pi_\dt$ as
%\begin{align*}
%\int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} & = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi} \nonumber \\
%%& = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt, \nonumber \\
%& = \int_\cX \Pi_\dt \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt,
%\end{align*}

%\bu Rewrite $-\Pi_\dt \mathcal L^{-1}$ in terms of $P_\dt$ as
%\[
%\begin{aligned}
%& -\Pi_\dt \mathcal L^{-1} \psi = -\Pi_\dt \left(\dt\sum_{n=0}^{+\infty} P_\dt^n \right) \Pi_\dt \left(\frac{\I - P_\dt}{\dt}\right) \mathcal L^{-1} \psi \\
%& \ \ = \dt \left(\sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \right) \left(\mathcal L + \dots + \dt^{\alpha-1} S_{\alpha-1} + \dt^\alpha \widetilde{R}_{\alpha,\dt}\right) \mathcal L^{-1} \psi, \\
%& \ \ = \dt \sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \widetilde{\psi}_{\dt,\alpha} + \dt^\alpha \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \Pi_\dt \widetilde{R}_{\alpha,\dt} \mathcal L^{-1} \psi.
%\end{aligned}
%\]

%\end{frame}

%%-----------------------------------------------------------
%\begin{frame}\frametitle{Sketch of proof (2)}

%\bu Uniform resolvent bounds $\dps \left\| \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq \frac{C_s}{\lambda_s}$

%\medskip

%\bu Coming back to the initial equality,
%\[
%\int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} = \dt \int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \left( \Pi_\dt \varphi \right) d{\pi}_\dt + \mathrm{O}\left(\dt^\alpha\right)
%\]

%\bu Rewrite finally
%\[
%\begin{aligned}
%\int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right)\left( \Pi_\dt \varphi \right) d{\pi}_\dt & = \int_\cX \sum_{n=0}^{+\infty} \left(P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \varphi \, d{\pi}_\dt \\
%& = \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\right)
%\end{aligned}
%\]

%\end{frame}

\begin{frame}
    {Summary}

    \bu {\bf Definition and examples of nonequilibrium systems}
    \begin{itemize}
        \item Convergence to invariant measure
        \item Perturbation expansion of invariant measure
    \end{itemize}

    \bigskip

    \bu {\bf Definition and computation of transport coefficients}
    \begin{itemize}
        \item Mobility, heat conductivity, shear viscosity
        \item Linear response theory
        \item Relationship with Green-Kubo formulas
    \end{itemize}

    \bigskip

    \bu {\bf Elements of numerical analysis}
    \begin{itemize}
        \item estimation of biases due to timestep discretization
        \item {\blue (largely) open issue: variance reduction}
    \end{itemize}

\end{frame}

\end{document}

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