Add worked example

1 files changed, 1096 insertions, 38 deletions

diff --git a/main.tex b/main.tex
index ba7d319..c6117c6 100755
--- a/main.tex
+++ b/main.tex
@@ -1,9 +1,27 @@
 \documentclass[9pt]{beamer}
-\renewcommand{\emph}[1]{\textcolor{blue}{#1}}
 \newcommand{\blue}[1]{\textcolor{blue}{#1}}
+% \newcommand{\red}[1]{\color{red}}
 \newif\iflong
 \longfalse
 
+\newcommand{\placeholder}{\mathord{\color{black!33}\bullet}}%
+\newcommand{\bu}{$\bullet \ $}
+\newcommand{\bi}{\begin{itemize}}
+\newcommand{\ei}{\end{itemize}}
+\renewcommand{\leq}{\leqslant}
+\renewcommand{\le}{\leqslant}
+\renewcommand{\geq}{\geqslant}
+\newcommand{\dt}{{\Delta t}}
+\newcommand\centerequation[1]{\par\smallskip\par \centerline{$\displaystyle #1$}\par \smallskip\par}
+\newcommand{\D}{\,\mathrm{d}}
+\newcommand{\cX}{\mathcal{X}}
+\newcommand{\E}{\expect}
+\newcommand{\wcL}{\widetilde{\mathcal{L}}}
+\newcommand{\Li}{\mathcal{K}}
+\newcommand{\I}{\mathrm{Id}}
+\newcommand{\dps}{\displaystyle}
+\newcommand{\red}{\color{red}}
+
 \input{header}
 \input{macros}
 
@@ -86,6 +104,13 @@
   \end{itemize}
 
   \vspace{.3cm}
+  They can be estimated from molecular simulation at the \blue{microscopic level}.
+  \begin{itemize}
+      \item They are defined from \emph{nonequilibrium} dynamics;
+      \item There are three main classes of methods to calculate them.
+  \end{itemize}
+
+  \vspace{.3cm}
   \textbf{Challenges we do not address:}
   \begin{itemize}
     \item Choose thermodynamical ensemble;
@@ -94,26 +119,20 @@
 \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}
+    \begin{center}
+      \Large
+      \color{blue}
+      Part I: Definition and examples of nonequilibrium systems
+    \end{center}
 
-    \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}
+    \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
+    \end{itemize}
+    \end{minipage}
 \end{frame}
 
 \section{Equilibrium and nonequilibrium dynamics}
@@ -140,24 +159,64 @@
 \end{frame}
 
 \begin{frame}
-  {Invariant distribution in dimension 1}
-  For the equilibrium overdamped Langevin dynamics
-  \[
-    \d q_t = - V'(q_t) \, \d t + \sqrt{2} \, \d W_t,
-  \]
-  the invariant probability distribution is given by~$Z^{-1} \e^{-V(q)} \, \d q$.
-  For the perturbed dynamics
-  \[
-    \d q_t = - V'(q_t) \, \d t + \blue{\eta} + \sqrt{2} \, \d W_t,
-  \]
-  the invariant probability distribution~$\rho_{\eta}$ solves the Fokker--Planck equation
+    {Example of nonequilibrium dynamics}
+    \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term}
+        \begin{equation}
+            \label{eq: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{\frac{2\gamma}{\beta}}  \D W_t,
+                \end{aligned}
+            \right.
+        \end{equation}
+    \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}
+  {Worked example in dimension one}
+  Consider the perturbed overdamped Langevin dynamics with~$q_t \in \torus$
   \[
-    \frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0,
+      \d q_t = - V'(q_t) \, \d t + {\red \eta} \, \d t + \sqrt{2} \, \d W_t,
   \]
-  which can be solved as
+  The associated Fokker--Planck equation reads
   \[
-    \rho_{\eta}(q) \propto \int_{\torus} \e^{V(q+y) - V(q) - \eta y} \, \d y.
+      \frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0.
   \]
+  \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}
@@ -181,7 +240,6 @@
 
 \begin{frame}
   {Existence of an invariant distribution}
-
   \begin{theorem}
       Fix~$\eta_* > 0$ and $n \geq 2$,
       and let $\mathcal K_n(q, p) := 1 + \abs{p}^n$.
@@ -206,16 +264,1016 @@
 \end{frame}
 
 \begin{frame}
-  {Existence of an invariant measure}
+  {Existence of an invariant measure (1/2)}
+  For a Markov transition kernel~$\mathcal P\colon \mathcal E \times \mathcal B(\mathcal E) \to [0, 1]$, let
+  \[
+      (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, \mathcal P(x, \d y),
+      \qquad
+      (\mathcal P^{\dagger} \mu)(A)  := \int_{A} \mathcal P(x, A) \, \mu(\d x).
+  \]
+  \begin{theorem}
+      [Doeblin's theorem]
+      If there exists $\alpha \in (0, 1)$ and a probability measure $\eta$ such that
+      \[
+          \mathcal P^{\dagger} \mu \geq \alpha \eta,
+      \]
+      then there exists $\mu_{\infty}$ such that $\mathcal P^{\dagger} \mu_{\infty} = \mu_{\infty}$.
+      Furthermore $d(\mathcal P^{\dagger^n}  \mu, \mu_{\infty}) \leq \alpha^n d(\mu, \mu_{\infty})$.
+  \end{theorem}
+
+  \emph{Sketch of proof.}
+  Use Banach's fixed point theorem. Define the Markov transition
+  \[
+      \widetilde {\mathcal P}(x, \placeholder) := \frac{1}{1-\alpha} \mathcal P(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder).
+  \]
+  Let $V$ 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 V} \int_{\mathcal E} \phi(q) (\mathcal P^{\dagger} \mu - \mathcal P^{\dagger} \nu) (\d q)
+      = \sup_{\phi \in V} \int_{\mathcal E} \mathcal P \phi(q) \bigl(\mu - \nu\bigr) (\d q) \\
+      &= (1 - \alpha) \sup_{\phi \in V} \int_{\mathcal E} \widetilde {\mathcal P} \phi(q) (\mu - \nu) (\d q)
+      \leq (1 - \alpha) \, d(\mu, \nu).
+  \end{align*}
+\end{frame}
+
+\begin{frame}
+  {Existence of an invariant measure (2/2)}
+  \begin{itemize}
+      \item
+          Suppose that $\phi$ is uniformly bounded and let $\overline \phi = \int_{\mathcal E} \phi \, \d \mu_{\infty}$. Then
+          \[
+              \Bigl\lVert \mathcal P \left(\phi - \overline \phi\right) \Bigr\rVert_{L^\infty}
+              = (1 - \alpha) \Bigl\lVert \widetilde {\mathcal P} (\phi - \overline \phi) \Bigr\rVert_{L^{\infty}}
+              \leq (1 - \alpha) \Bigl\lVert  \phi - \overline \phi \Bigr\rVert_{L^{\infty}},
+          \]
+
+
+      \item
+          In molecular dynamics, this theorem can be employed for showing existence of and convergence to the invariant measure,
+          provided that the \blue{state space is compact}.
+
+      \item
+          For \alert{noncompact state spaces}, an extension called \emph{Harris' theorem}
+  \end{itemize}
+\end{frame}
+
+\begin{frame}
+    {Linear response of nonequilibrium dynamics (1)}
+    \bu The force $\eta F$ induces a non-zero velocity in the direction $F$
+    \medskip
+
+    \bu Encoded by $\dps \expect_\eta(R) = \int_{\mathcal E} R \, \psi_\eta$ with $\dps R(q,p) = F^\t  M^{-1}p$
+
+    \begin{block}
+        {Definition of the mobility}
+        \[
+            \rho_F
+            = \lim_{\eta \to 0} \frac{\expect_\eta (R)-\expect_0 (R)}{\eta}
+            = \lim_{\eta \to 0} \frac{\expect_\eta (R)}{\eta}
+        \]
+    \end{block}
+
+    \medskip
+
+    \bu It is {\red expected} that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and
+    \[
+        f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathrm{O}(\eta^2)
+    \]
+
+    \medskip
+
+    \bu In this case, $\dps \rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0$
+
+    \bigskip
+
+    \bu {\bf Questions:} Can the expansion for $f_\eta$ be made rigorous? What is $\mathfrak{f}_1$?
+
+\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}
+
+\iffalse
+\begin{frame}\frametitle{Linear response of nonequilibrium dynamics (2)}
+
+\bu {\red Perturbative framework} where $\mathcal L_0$ considered on $L^2(\psi_0)$ is the reference
+
+\medskip
+
+\bu 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 $L^2(\psi_0)$}
+\centerequation{\mathcal L_\eta^* f_\eta = 0}
+\end{block}
+
+\bigskip
+
+\bu Formally, $\mathcal L_\eta^* f_\eta = (\mathcal L_0)^* \underbrace{\left(\I + \wcL \mathcal L_0^{-1}\right)^*f_\eta}_{=1 ?} = 0$
+
+\medskip
+
+\bu To make the result precise, introduce $L_0^2(\psi_0) = \Pi_0 L^2(\psi_0)$ with
+\[
+\Pi_0 f = f - \int_{\mathcal E} f \, \psi_0
+\]
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Linear response of nonequilibrium dynamics (2)}
+
+\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}
+
+\medskip
+
+\bu Note that $\dps \int_{\mathcal E} \psi_\eta = 1$
+
+\medskip
+
+\bu Linear response result: $\dps \rho_F = -\int_{\mathcal E} R \left[(\wcL \mathcal L_0^{-1})^* \mathbf{1}\right] \psi_0 $
+
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Elements of proof}
+
+\bu Since $\dps \frac{\gamma}{\beta} \| \nabla_p \varphi \|^2_{L^2(\psi_0)} = -\langle \mathcal L_0 \varphi,\varphi \rangle_{L^2(\psi_0)}$, it follows that
+\vspace{-0.2cm}
+\[
+\| \wcL \varphi \|^2_{L^2(\psi_0)} \leq \| \nabla_p \varphi \|^2_{L^2(\psi_0)} \leq \frac{\beta}{\gamma} \| \mathcal L_0 \varphi \|_{L^2(\psi_0)} \| \varphi \|_{L^2(\psi_0)}
+\]
+
+\bu {\red $\mathcal L_0^{-1}$ is a well defined bounded operator on $L_0^2(\psi_0)$} (hypocoercivity + hypoelliptic regularization)
+\[
+\| \wcL \mathcal L_0^{-1} \varphi \|^2_{L^2(\psi_0)}\leq \frac{\beta}{\gamma} \| \varphi \|_{L^2(\psi_0)} \| \mathcal L_0^{-1} \varphi \|_{L^2(\psi_0)}.
+\]
+
+\bu {\blue $\Pi_0 \wcL \mathcal L_0^{-1}$ is bounded on $L^2_0(\psi_0)$}, so $(\wcL \mathcal L_0^{-1})^* \Pi_0 = (\wcL \mathcal L_0^{-1})^*$ is also bounded on $L^2_0(\psi_0)$
+
+\medskip
+
+\bu Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L^* + \eta \wcL^*$
+\vspace{-0.2cm}
+\[
+\mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\wcL \mathcal L_0^{-1})^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0
+\]
+
+\bu {\red Prove that $f_\eta \geq 0$} (use some ergodicity result to show that $\psi_\eta = f_\eta \psi_0$)
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Reformulation as integrated correlation functions}
+
+\bu Conjugate response $S = \wcL^* \mathbf{1}$, equivalently $\dps \int_{\mathcal E} \left(\wcL \varphi\right) \psi_0 = \int_{\mathcal E} \varphi \, S\, \psi_0$
+
+\medskip
+
+\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 $\psi_\eta(q,p)\,d q\, p$, while $\expect_0$ is taken over initial conditions $(q_0,p_0) \sim \psi_0$ and over all realizations of the equilibrium dynamics.
+\end{block}
+
+\medskip
+
+\bu For the dynamics \eqref{eq:Langevin_F}, it holds $S(q,p) = \beta R(q,p) = \beta F^T M^{-1} p$ so that
+\[
+    \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p  )}{\eta}
+  = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0)  \Big) d t
+\]
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Elements of proof}
+
+\bu Proof based on the following equality on $\mathcal{B}(L_0^2(\psi_0))$
+\[
+-\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, d t
+\]
+
+\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 \notag \\*
+& = \int_0^{+\infty} \left( \int_{\mathcal{E}} \left(\mathrm{e}^{t \mathcal L_0} R\right) \, S \, \psi_0\right)dt \notag \\
+& = \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}\frametitle{Generalization to other dynamics}
+
+\bu {\bf Possible assumptions to justify the linear response}
+\begin{itemize}
+\item existence of invariant measure with smooth density $\psi_\eta$
+\item ergodicity $\dps \frac1t \int_0^t \varphi(x_s) \,d s \xrightarrow[t\to+\infty]{} \int_\cX \varphi \, \psi_\eta$
+\item $\mathrm{Ker}(\mathcal L_0^*) = \mathbf{1}$ and $\mathcal L_0^*$ is invertible on~$L_0^2(\psi_0)$
+\item the perturbation $\wcL$ is $\mathcal L_0$-bounded: there exist $a,b \geq 0$ such that
+\[
+\| \wcL \varphi\|_{L^2(\psi_0)} \leq a \| \mathcal L_0 \varphi\|_{L^2(\psi_0)} + b \| \varphi\|_{L^2(\psi_0)}
+\]
+\end{itemize}
+
+\bigskip
+
+\bu {\bf When the perturbation is not sufficiently weak?} (thermal transport)
+\begin{itemize}
+\item compute $\dps \int_\cX  [(\mathcal L_0+\eta\wcL)\varphi ] (1+\eta\mathfrak{f}_1)\psi_0 = \mathrm{O}(\eta^2)$
+\item use a pseudo-inverse $Q_\eta = \Pi_0\mathcal L_0^{-1}\Pi_0 - \eta \Pi_0\mathcal L_0^{-1}\Pi_0\wcL\Pi_0\mathcal L_0^{-1}\Pi_0$
+\item allows to prove that $\dps \int_\cX \varphi \, \psi_\eta = \int_\cX \varphi \, \psi_0 + \eta \int_\cX \varphi \, \mathfrak{f}_1 \, \psi_0 + \eta^2 r_{\varphi,\eta}$
+\end{itemize}
+
+\end{frame}
+
+
+
+\begin{frame}
+    \begin{center}
+\Huge{Other examples}
+\end{center}
+\end{frame}
+
+
+\begin{frame}\frametitle{Shear viscosity in fluids (1)}
+
+\bigskip
+2D system to simplify notation: $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$
+\begin{figure}
+\psfrag{x}{}
+\psfrag{z}{}
+\psfrag{F}{force}
+\center
+\includegraphics[height=7cm]{figures/osc_shear.eps}
+\end{figure}
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Shear viscosity in fluids (2)}
+
+\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_{i,t} &= \frac{p_{i,t}}{m} \, dt,\\
+  d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, dt + {\red \eta F(q_{yi,t}) \, dt}
+  - \gamma_x \frac{p_{xi,t}}{m} \, dt + \sqrt{\frac{2\gamma_x}{\beta}} \, dW^{xi}_t, \\
+  d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, dt - \gamma_y \frac{p_{yi,t}}{m} \, dt
+  + \sqrt{\frac{2\gamma_y}{\beta}} \, dW^{yi}_t,
+\end{aligned} \right.
+}
+\end{block}
+
+\smallskip
+
+\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma_x,\gamma_y>0$
+
+\smallskip
+
+\bu The perturbation $\dps \wcL = \sum_{i=1}^N \! F(q_{y,i}) \partial_{p_{x,i}}$ is $\mathcal{L}_0$-bounded
+
+\smallskip
+
+\bu {\blue Linear response}:
+  $\dps
+  \lim_{\eta \rightarrow 0} \frac{\left\langle \mathcal L_0 h \right\rangle_\eta}{\eta}
+  = - \frac{\beta}{m} \!
+  \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)}
+  $
+\medskip
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Shear viscosity in fluids (3)}
+
+
+
+\bu Average {\red longitudinal velocity}
+  $u_x(Y) = \dps \lim_{\varepsilon \to 0}
+  \lim_{\eta \to 0} \frac{\left\langle U_x^\varepsilon(Y,\cdot)\right\rangle_\eta}{\eta}$
+  where
+  \vspace{-0.3cm}
   \[
-      d(P \mu, P \nu) 
-      \leq 
+  U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{i=1}^N p_{xi}
+  \chi_{\varepsilon}\left(q_{yi}-Y\right)
   \]
+  \vspace{-0.5cm}
+
+\bu Average {\red off-diagonal stress}
+  $\dps \sigma_{xy}(Y) = \lim_{\varepsilon \to 0}
+  \lim_{\eta \to 0} \frac{\left\langle ... \right\rangle_\eta}{\eta}$,
+  where $... =$
+  \vspace{-0.4cm}
+  \[
+  \hspace{-0.1cm}
+  \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right)
+  - \! \! \! \! \! \! \! \!
+  \sum_{1 \leq i < j \leq N} \! \! \! \!
+  \mathcal{V}'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|}
+  \!\int_{q_{yj}}^{q_{yi}} \!\chi_{\varepsilon}(s-Y) \, ds \right)
+  \]
+
+\bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ (with $\overline{\rho} = N/|\mathcal{D}|$)
+\[
+\frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y)
+\]
+
+\end{frame}
+
+
+
+\begin{frame}
+\frametitle{Shear viscosity in fluids (4)}
+
+\bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$
+
+\begin{block}{Velocity profile in Langevin dynamics under flow}
+\centerequation{-\eta u_x''(Y) + \gamma_x \overline{\rho} u_x(Y) = \overline{\rho} F(Y)}
+\end{block}
+
+\bigskip
+
+\hspace{-0.5cm}
+\begin{minipage}{6cm}
+\psfrag{F}{{\scriptsize $F$}}
+\psfrag{U}{{\scriptsize $u$}}
+\psfrag{Y}{{\scriptsize $\ \ Y$}}
+\psfrag{v}{{\scriptsize value}}
+\includegraphics[width=6cm]{figures/ux5.eps}
+\end{minipage}
+\hspace{-0.5cm}
+\begin{minipage}{6cm}
+\psfrag{Y}{}
+\psfrag{v}{{\scriptsize value}}
+\psfrag{S}{{\scriptsize $\sigma_{xy}$}}
+\psfrag{D}{{\scriptsize $-\nu u'$}}
+\includegraphics[width=6cm]{figures/dux5.eps}
+\end{minipage}
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Thermal transport in one-dimensional chains (1)}
+
+\bu Atoms at positions $q_0,\dots,q_N$ with $q_0 = 0$ fixed
+
+\medskip
+
+\bu Hamiltonian $\dps H(q,p) = \sum_{i=1}^N \frac{p_i^2}{2} + \sum_{i=1}^{N-1} v(q_{i+1} - q_i) + v(q_1)$
+
+\begin{block}{Hamiltonian dynamics with Langevin thermostats at the boundaries}
+\centerequation{ \left\{ \begin{aligned}
+dq_i & = p_i \, dt \\
+dp_i & = \Big( v'(q_{i+1}-q_i) - v'(q_i-q_{i-1}) \Big) dt,\qquad i\neq
+1, N \\[-3pt]
+dp_1 & = \Big( v'(q_2-q_1) - v'(q_1) \Big) dt
+- \gamma p_1 \, dt + \sqrt{2\gamma (T{\red +\Delta T})} \, dW^1_t\\[-3pt]
+dp_N & = - v'(q_N-q_{N-1}) \, dt
+- \gamma p_N \, dt + \sqrt{2\gamma (T{\red -\Delta T})} \, dW^N_t\\[-5pt]
+\end{aligned} \right. }
+\end{block}
+
+\medskip
+
+\bu {\red Perturbation} $\wcL = \gamma( \partial_{p_1}^2 - \partial_{p_N}^2)$ (not $\mathcal L_0$-bounded...)
+
+\medskip
+
+\bu Proving the existence/uniqueness of the invariant measure already requires quite some work\footnote{P. Carmona, {\emph Stoch. Proc. Appl.} (2007)}
+
+\bigskip
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Thermal transport in one-dimensional chains (2)}
+
+\bu Response function: {\blue Total energy current}
+\begin{block}{}
+\centerequation{J = \frac{1}{N-1}\sum_{i=1}^{N-1} j_{i+1,i},
+\qquad
+j_{i+1,i} = -v'(q_{i+1}-q_i)\frac{p_i+p_{i+1}}{2}}
+\end{block}
+\smallskip
+
+\bu Motivation: Local conservation of the energy (in the bulk)
+\[
+\frac{d\varepsilon_i}{dt} = j_{i-1,i} - j_{i,i+1},
+\qquad
+\varepsilon_i = \frac{p_i^2}{2} + \frac12 \Big( v(q_{i+1}-q_{i}) + v(q_i-q_{i-1}) \Big)
+\]
+
+\bu Definition of the {\blue thermal conductivity}: linear response
+\[
+\kappa_N = \lim_{\Delta T \to 0} \frac{\langle J \rangle_{\Delta T}}{\Delta T/N}
+= 2\beta^2 \frac{N}{N-1}\int_0^{+\infty} \sum_{i=1}^{N-1}
+  \expect\Big(j_{2,1}(q_t,p_t)j_{i+1,i}(q_0,p_0)\Big)\, dt
+\]
+
+\medskip
+
+\bu {\blue Synthetic dynamics}: fixed temperatures of the thermostats but external forcings
+$\to$ {\red bulk driven dynamics} with $\wcL^* = -\wcL + c J$
+
+\end{frame}
+
+
+
+\begin{frame}
+    \begin{center}
+\Huge{Error estimates on} \\
+\bigskip
+\Huge{the computation of} \\
+\bigskip
+\Huge{transport coefficients}
+\end{center}
+\end{frame}
+
+
+
+\begin{frame}\frametitle{Reminder: Error estimates in Monte Carlo simulations}
+
+\bu General SDE $dx_t = b(x_t)\,dt + \sigma(x_t) \, dW_t$, invariant measure $\pi$
+
+\bigskip
+
+\bu {\red Discretization} $x^{n} \simeq x_{n\dt}$, {\blue invariant measure $\pi_\dt$}. For instance,
+\[
+x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \sim \mathcal{G}(0,{\rm Id}) \ \mathrm{i.i.d.}
+\]
+
+\medskip
+
+\bu {\blue Ergodicity} of the numerical scheme with invariant measure~$\pi_\dt$
+\[
+\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(dx)
+\]
+
+\begin{block}{Error estimates for {\red finite} trajectory averages}
+\[
+\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) = \expect_\pi(A) + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error}
+\]
+\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 $(x_{n \dt})$
+
+\bigskip
+
+\bu (Infinitely) Many possibilities! Numerical analysis allows to {\blue discriminate}
+
+\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}: $\dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(X^n\right)\right] - \expect\left[\varphi\left(X_{n\dt}\right)\right] \Big| \leq C \dt^p$ (for any $\varphi$)
+%\item ``mean error'' \emph{vs.} ``error of the mean''
+\end{itemize}
+
+%\medskip
+%\bu Example: for Euler-Maruyama, weak order~1, strong order $1/2$ (1 when $\sigma$ constant)
+%\medskip
+
+\begin{block}{$\dt$-expansion of the evolution operator}
+\centerequation{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
+
+\bu {\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Example: Euler-Maruyama, weak order~1}
+
+\medskip
+\bu 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
+\[
+P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathrm{O}(\dt^2)
+\]
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Error estimates on the invariant measure (equilibrium)}
+
+\bu {\red Assumptions} on the operators in the weak-type expansion
+\begin{itemize}
+\item invariance of $\pi$ by $\mathcal A_k$ for $1 \leq k \leq p$, namely
+$\dps \int_\cX \mathcal A_k \varphi \, d\pi = 0$
+\item $\dps \int_\cX \mathcal A_{p+1}\varphi \, d\pi = \int_\cX g_{p+1} \varphi \, d\pi$
+(\textit{i.e.} $g_{p+1} = \mathcal A_{p+1}^* \mathbf{1}$)
+\end{itemize}
+
+\begin{block}{Error estimates on $\pi_\dt$}
+\centerequation{
+\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}
+}
+\end{block}
+
+\medskip
+
+\bu In fact, $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$
+\begin{itemize}
+\item  when $\mathcal A_1 = \mathcal L$, the first order correction can be {\red estimated} by some integrated correlation function as $\dps \int_0^{+\infty} \expect\Big(\varphi(x_t)g_{p+1}(x_0)\Big) \, dt$
+\item in general, first order term can be removed by Romberg extrapolation
+\end{itemize}
+
+\medskip
+
+\bu Error on invariant measure can be {\blue (much) smaller} than the weak error
+
+\end{frame}
+
+%-----------------------------------------------------------
+\begin{frame}\frametitle{Sketch of proof (1)}
+
+{\bf Step~1: Establish 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
+\[
+\begin{aligned}
+& \int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right)\psi\right] (1+\dt^{p}f_{p+1})\, d\pi \\
+& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathrm{O}\left(\dt^{p+1}\right) \\
+& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathrm{O}\left(\dt^{p+1}\right)
+\end{aligned}
+\]
+
+\begin{block}{}
+Suggests $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$
+\end{block}
+
+\end{frame}
+
+%-----------------------------------------------------------
+\begin{frame}\frametitle{Sketch of proof (2)}
+
+{\bf Step~2: Define an approximate inverse}
+
+\medskip
+
+\bu Issue: derivatives of $(\I-P_\dt)^{-1}\varphi$ are not controlled
+
+\bigskip
+
+\bu Consider $\dps \left(\Pi \frac{P_\dt-\I}{\dt} \Pi\right) Q_\dt\psi = \psi + \dt^{p+1} \widetilde{r}_{\psi,\dt}$ where
+\vspace{-0.2cm}
+\[
+\Pi \varphi = \varphi - \int_\cX \varphi \, d\pi
+\]
+
+\bu Idea of the construction: truncate the formal series expression
+\[
+(A + \dt \, B)^{-1} = A^{-1} - \dt \, A^{-1}B A^{-1} + \dt^{2} \, A^{-1}B A^{-1}B A^{-1} + \dots
+\]
+
+\bigskip
+
+{\bf Step~3: Conclusion}
+
+\medskip
+
+\bu Write the invariances with $\dps \Pi \left(\frac{P_\dt-\I}{\dt}\right) \Pi \psi$ instead of $\dps \left(\frac{P_\dt-\I}{\dt}\right) \psi$
+
+\medskip
+
+\bu Replace $\psi$ by $Q_\dt \varphi$, and gather in~$R_{\varphi,\dt}$ all the higher order terms
+
+\end{frame}
+
+
+
+\begin{frame}\frametitle{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
+\[
+P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^2)
+\]
+
+\bu Strang splitting $\to$ {\blue weak order 2}
+\[
+P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^3)
+\]
+
+\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}\frametitle{Error estimates on linear response}
+
+\begin{block}{Error estimates for nonequilibrium dynamics}
+There exists a function $f_{\alpha,1,\gamma} \in H^1(\mu)$ such that
+\vspace{-0.3cm}
+\[
+\int_{\mathcal E} \psi \, d{\mu}_{\gamma,\eta,\dt} = \int_{\mathcal E} \psi \Big(1+ \eta f_{0,1,\gamma} + \dt^\alpha f_{\alpha,0,\gamma} + \eta \dt^\alpha f_{\alpha,1,\gamma} \Big) d{\mu} + r_{\psi,\gamma,\eta,\dt},
+\]
+where the remainder is compatible with linear response
+\vspace{-0.1cm}
+\[
+\left|r_{\psi,\gamma,\eta,\dt}\right| \leq K(\eta^2 + \dt^{\alpha+1}),
+\qquad
+\left|r_{\psi,\gamma,\eta,\dt} - r_{\psi,\gamma,0,\dt}\right| \leq K \eta (\eta + \dt^{\alpha+1})
+\]
+\end{block}
+
+\medskip
+
+\bu Corollary: error estimates on the {\blue numerically computed mobility}
+\[
+\begin{aligned}
+\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E}  F^T M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\
+& = \rho_{F} + \dt^\alpha \int_{\mathcal E}  F^T M^{-1} p  \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt}
+\end{aligned}
+\]
+
+\bu Results in the {\red overdamped} limit\footnote{B.~Leimkuhler, C.~Matthews and G.~Stoltz, {\em IMA J. Numer. Anal.} (2015)}
+
+\bigskip
+
+\end{frame}
+
+
+\begin{frame}\frametitle{Numerical results}
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=6.2cm]{figures/LR.eps}
+\includegraphics[width=6.2cm]{figures/mobility_Langevin.eps}
+\end{center}
+\end{figure}
+
+\small
+{\bf Left:} Linear response of the average velocity as a function of $\eta$ for the scheme associated with $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$ and $\dt = 0.01, \gamma = 1$. \\
+
+\smallskip
+
+{\bf Right:} Scaling of the mobility $\nu_{F,\gamma,\dt}$ 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}\frametitle{Error estimates on Green-Kubo formulas (1)}
+
+\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 Correlation approximated in practice using $K$ independent realizations
+%\bi
+%\item truncating the integration (decay estimates)
+%\item using empirical averages ($K$ independent realizations)
+\vspace{-0.2cm}
+\[
+\begin{aligned}
+& \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right)
+\simeq
+\frac{1}{K}\sum_{m=1}^{K} \left( \phi(x^{n,k}) - \overline{\phi}^{n,K} \right)\left( \varphi(x^{n,k}) - \overline{\varphi}^{n,K} \right) \\[-10pt]
+& \mathrm{where} \ \overline{\phi}^{n,K} = \frac1K \sum_{m=1}^{K} \phi(x^{n,k})
+\end{aligned}
+\]
+
+\bu For methods of {\bf weak order} 2, {\blue trapezoidal rule}
+\vspace{-0.1cm}
+\[
+\begin{aligned}
+\int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} & = \frac{\dt}{2} \expect_\dt \left(\Pi_\dt \phi\left(x^{0}\right)\varphi\left(x^0\right)\right) \\
+& \ \ + \dt \sum_{n=1}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt^2)
+\end{aligned}
+\]
+
+
+%\bu Allows to quantify the variance $\dps \frac{\sigma^2_{A,\dt}}{N_{\rm iter}\dt} \simeq \frac{\dps 2 \int_0^{+\infty} \expect\left[\delta A(x_t)\delta A(x_0)\right] \, dt}{T}$ where $T = N_{\rm iter}\dt$
+
+\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}\frametitle{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}\frametitle{Numerical results}
+
+\vspace{-0.5cm}
+\begin{figure}
+\begin{center}
+\includegraphics[width=11.8cm]{figures/error_diffusion.eps}
+%\includegraphics[width=8.2cm]{figures/error_diffusion_zoom.eps}
+\end{center}
+\end{figure}
+
+\end{frame}
+
+
+
+\begin{frame}
+    \begin{center}
+\Huge{Conclusion and perspectives}
+\end{center}
+\end{frame}
+
+
 \begin{frame}
+    {Main points: recall the outline!}
+
+\bu {\bf Definition and examples of nonequilibrium systems}
+
+\bigskip
+
+\bu {\bf Computation of transport coefficients}
+\begin{itemize}
+\item a survey of computational techniques
+\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}
+\item {\red (not discussed) use of non-reversible dynamics to enhance sampling}
+\end{itemize}
+
+\end{frame}
+
+\begin{frame}
+    {Variance reduction techniques?}
+
+\bu {\blue Importance sampling?} Invariant probability measures $\psi_\infty$, $\psi_\infty^A$ for
+\[
+dq_t = b(q_t) \, dt + \sigma dW_t,
+\qquad
+dq_t = \Big( b(q_t) + \nabla A(q_t) \Big) dt + \sigma dW_t
+\]
+In general {\red $\psi_\infty^A \neq Z^{-1} \psi_\infty \mathrm{e}^{A}$}
+(consider $b(q) = F$ and $A = \widetilde{V}$)
+
+\bigskip
+
+\bu {\blue Stratification?} (as in TI...) Consider $q \in \mathbb{T}^2$, $\psi_\infty = \mathbf{1}_{\mathbb{T}^2}$
+\[
+\left \{ \begin{aligned}
+  dq^1_t & = \partial_{q_2}U(q^1_t,q^2_t) + \sqrt{2} \, dW_t^1 \\
+  dq^2_t & = - \partial_{q_1}U(q^1_t,q^2_t) + \sqrt{2} \, dW_t^2
+\end{aligned} \right.
+\]
+Constraint $\xi(q) = q_2$, {\red constrained dynamics}
+\[
+dq^1_t = f(q^1_t) \, dt + \sqrt{2} \, dW_t^1,
+\qquad
+f(q^1) = \partial_{q_2}U(q^1,0).
+\]
+Then $\dps
+\psi_\infty(q^1) = Z^{-1} \int_0^{1} \e^{V(q^1+y)-V(q^1)-Fy} \, dy \neq \mathbf{1}_{\mathbb{T}}(q^1)$ \\
+where $\dps F = \int_0^1 f$ and $\dps V(q^1) = -\int_0^{q^1} (f(s) - F) \, ds$
 
 \end{frame}
+\fi
 
 \end{document}