From 7b30fa87b20de3940b4c92f648381c422135fddb Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Sun, 24 Sep 2023 23:16:38 +0200
Subject: Minor changes

---
 main.tex | 616 +++++++++++++++++++++++++++++++++++++++------------------------
 1 file changed, 385 insertions(+), 231 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index b8e83e2..c89b161 100755
--- a/main.tex
+++ b/main.tex
@@ -88,6 +88,43 @@
 \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,
@@ -113,9 +150,11 @@
     \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
@@ -161,7 +200,7 @@
 \end{frame}
 
 \begin{frame}
-    {Examples of nonequilibrium dynamics}
+    {Paradigmatic examples of nonequilibrium dynamics}
     \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term}
         \begin{equation}
             \label{eq:overdamped_Langevin_F}
@@ -182,7 +221,7 @@
                 \end{aligned}
             \right.
         \end{equation}
-        In the rest of the presentation we take {\blue $ M = \I$} for simplicity.
+        In the rest of this section, we take {\blue $ M = \I$} for simplicity.
     \end{block}
     where
     \begin{itemize}
@@ -193,35 +232,6 @@
     Is there an invariant probability measure?
 \end{frame}
 
-\begin{frame}
-    {Another example useful for thermal transport}
-    \begin{block}{Langevin dynamics with modified fluctuation}
-        \[
-            \left\{
-                \begin{aligned}
-                    \d q_t & = M^{-1} p_t  \, \d t, \\*
-                    \d p_t & =  -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t  \, \d t
-                    + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t,
-                \end{aligned}
-            \right.
-        \]
-    \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}
     {When {\yellow $\eta = 0$}, these dynamics are reversible}
     \begin{itemize}
@@ -229,6 +239,8 @@
             \[
                 \mathcal L_{\rm ovd} \Big\vert_{\red \eta = 0} = - \grad V \cdot \grad + \laplacian
                 = - \grad^* \grad,
+                \qquad
+                \mu(\d q) = \frac{1}{Z} \e^{-V(q)} \, \d q.
             \]
             where $\grad^* := (\grad V - \grad) \cdot $.
             For any $f, g \in C^{\infty}_{\rm c}(\mathcal E)$, we have
@@ -255,6 +267,35 @@
     \end{itemize}
 \end{frame}
 
+\begin{frame}
+    {Another example useful for thermal transport}
+    \begin{block}{Langevin dynamics with modified fluctuation}
+        \[
+            \left\{
+                \begin{aligned}
+                    \d q_t & = M^{-1} p_t  \, \d t, \\*
+                    \d p_t & =  -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t  \, \d t
+                    + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t,
+                \end{aligned}
+            \right.
+        \]
+    \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~$q_t \in \torus$
@@ -335,7 +376,7 @@
 
   \emph{Sketch of proof.} Define the Markov transition kernel
   \[
-      \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder),
+      \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \pi(\placeholder),
   \]
   Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$.
   We have
@@ -373,7 +414,7 @@
           \[
               L^{\infty}_{*} := \left\{ \phi \in L^{\infty}(\mathcal E) : \int_{\mathcal E} \phi \, \d \mu_{*} = 0 \right\}.
           \]
-          Thus $\I - \mathcal P$ is invertible and
+          Thus $\I - \mathcal P$ is invertible on~$L^{\infty}_{*}$ and
           \[
               (\I - \mathcal P)^{-1} = \I + \mathcal P + \mathcal P^2 + \dotsb
           \]
@@ -404,7 +445,7 @@
             and additionally ${\rm Law} (q_t)$ is smooth by parabolic regularity.
         \item
             \textbf{Decay of the semigroup}:
-            For $t \in [0, \infty)$ and bounded $\varphi$, it holds that
+            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}} \\
@@ -413,7 +454,7 @@
             \end{align*}
 
         \item
-            \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{\eta}$,
+            \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{*}$,
             and
             \[
                 \mathcal L_{\rm ovd}^{-1}
@@ -527,7 +568,7 @@
         \[
             \mathcal L_\eta^* f_\eta = 0
         \]
-        Observe that
+        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)
@@ -549,8 +590,8 @@
             \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_2 + \mathcal L_0^* \mathfrak f_2\right) \\
-            &\quad + \eta^3 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_2\right) + \dotsb
+            &\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
         \[
@@ -640,11 +681,11 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
     \end{center}
 
     \centering
-    \begin{minipage}{.8\textwidth}
+    \begin{minipage}{.6\textwidth}
     \begin{itemize}
         \item Definition through linear response
         \item Green--Kubo reformulation
-        \item Numerical approximation
+        \item Link with effective diffusion
     \end{itemize}
     \end{minipage}
 \end{frame}
@@ -709,10 +750,26 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
     Therefore
     \[
         \rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0
-        = \int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L} \mathbf 1) \, \psi_0
+        = -\int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L}^* \mathbf 1) \, \psi_0
     \]
 \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 functions}
@@ -734,37 +791,120 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
     \end{block}
 
     For the mobility,
-    it holds $S(q,p) = \beta R(q,p) = F^T p$ and so
+    it holds $S(q,p) = R(q,p) = F^\t p$ and so
     \[
-        \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p  )}{\eta}
-        = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0)  \Big) d t
+        \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta \bigl(F^\t p  \bigr)}{\eta}
+        = \int_0^{+\infty} \expect_0 \Big( \bigl(F^\t p_t\bigr) \bigl(F^\t p_0\bigr)  \Big) \, \d t
     \]
-
 \end{frame}
 
+\begin{frame}
+    {Elements of proof}
 
-\begin{frame}\frametitle{Elements of proof}
-
-\bu Proof based on the following equality on $\mathcal{B}(L_0^2(\psi_0))$
+\bu Proof based on the following equality on $\mathcal{B}\bigl(L_0^2(\psi_0)\bigr)$
 \[
--\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, d t
+-\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, \d t.
 \]
 
 \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
+= -\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
+\bu Note also that $S$ has average 0 w.r.t.\ invariant measure since
 \[
-\int_\cX S \, d\pi = \int_\cX \wcL^* \mathbf{1} \, d\pi = \int_\cX \wcL\mathbf{1} \, d\pi = 0
+\int_\cX S \, \d\pi = \int_\cX \wcL^* \mathbf{1} \, \d\pi = \int_\cX \wcL\mathbf{1} \, \d\pi = 0
 \]
 
 \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} \, \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}
@@ -820,7 +960,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
 
 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},
+    H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2},
     \quad V(r) = \sum_{n=1}^{N-1} v(r_n).
 \end{equation*}
 \end{frame}
@@ -904,57 +1044,62 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
     {Shear viscosity in fluids (2)}
     Macroscopic description by Navier--Stokes equation
     \[
-        \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \eta \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x}
+        \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \nu \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x}
     \]
     Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives
     \[
-        - \eta U_x''(y) = \overline{\rho} F(y)
+        - \nu U_x''(y) = \overline{\rho} F(y), \qquad \overline \rho := \frac{\rho}{m} = \frac{N}{|\mathcal D|}
+    \]
+    Therefore, for the test function~$g(y) = \e^{2i\pi \frac{y}{L_y}}$
+    \[
+        \nu \int_0^{L_y} U_x(y) g''(y) \, \d y = \overline{\rho} \int_{0}^{L_y} F(y) g(y) \, \d y
+    \]
+
+    $\rightarrow$ Suggests estimating the shear viscosity from molecular dynamics as
+    \[
+        \nu = \frac{\dps \frac{\overline{\rho}}{L_y}\int_{0}^{L_y} F(y) g(y) \, \d y}
+        {\dps \expect_{F} \left[ \frac{1}{N}\sum_{n=1}^{N} \frac{p_{xi}}{m} g''(q_{yi}) \right]}.
     \]
 \end{frame}
 
 
 \begin{frame}
     {Shear viscosity in fluids (2)}
-
-    pairwise interactions
+    Assume pairwise interactions
     \[
         V(q) = \sum_{1 \leq i < j \leq N} \mathcal V(\abs{q_i - q_j}).
     \]
 \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 \frac{p_{xi,t}}{m} \, dt + \sqrt{\frac{2\gamma}{\beta}} \, dW^{xi}_t, \\
-  d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, dt - \gamma \frac{p_{yi,t}}{m} \, dt
-  + \sqrt{\frac{2\gamma}{\beta}} \, dW^{yi}_t,
+  \d q_{i,t} &= \frac{p_{i,t}}{m} \, \d t,\\
+  \d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, \d t + {\red \eta F(q_{yi,t}) \, \d t}
+  - \gamma \frac{p_{xi,t}}{m} \, \d t + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{xi}_t, \\
+  \d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, \d t - \gamma \frac{p_{yi,t}}{m} \, \d t
+  + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{yi}_t.
 \end{aligned} \right.
 }
 \end{block}
 
 \smallskip
 
-\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma,\gamma>0$
+\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$
 
 \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)}
+\begin{frame}
+    {Shear viscosity in fluids (3)}
 
+\bu {\blue Linear response}:
+\[
+    \lim_{\eta \rightarrow 0} \frac{\expect_{\eta} [\mathcal L_0 h]}{\eta}
+  = - \frac{\beta}{m} \!
+  \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)}.
+\]
 
 
 \bu Average {\red longitudinal velocity}
@@ -970,11 +1115,12 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
 
 \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 $... =$
+  \lim_{\eta \to 0} \frac{\expect_{\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} \! \! \! \!
@@ -984,15 +1130,12 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
 
 \bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ (with $\overline{\rho} = N/|\mathcal{D}|$)
 \[
-\frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y)
+    \frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y)
 \]
-
 \end{frame}
 
-
-
 \begin{frame}
-\frametitle{Shear viscosity in fluids (4)}
+    {Shear viscosity in fluids (4)}
 
 \bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$
 
@@ -1000,62 +1143,61 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
 \centerequation{-\eta u_x''(Y) + \gamma \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}
-
+\begin{figure}[ht]
+    \centering
+    \includegraphics[width=\linewidth]{figures/shear1.png}
+\end{figure}
 \end{frame}
 
-
+% \begin{frame}
+% \end{frame}
 
 \begin{frame}
     \begin{center}
-\Huge{Error estimates on} \\
-\bigskip
-\Huge{the computation of} \\
-\bigskip
-\Huge{transport coefficients}
-\end{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 Green--Kubo method
+    \end{itemize}
+    \end{minipage}
 \end{frame}
 
 
 
-\begin{frame}\frametitle{Reminder: Error estimates in Monte Carlo simulations}
+\begin{frame}
+    {Reminder: Error estimates in Monte Carlo simulations}
 
-\bu General SDE $dx_t = b(x_t)\,dt + \sigma(x_t) \, dW_t$, invariant measure $\pi$
+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,
 \[
-x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \sim \mathcal{G}(0,{\rm Id}) \ \mathrm{i.i.d.}
+x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \stackrel{\rm{i.i.d.}}{\sim} \mathcal N(0,{\rm Id})
 \]
 
 \medskip
 
 \bu {\blue Ergodicity} of the numerical scheme with invariant measure~$\pi_\dt$
 \[
-\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(dx)
+\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(\d x)
 \]
 
 \begin{block}{Error estimates for {\red finite} trajectory averages}
 \[
-\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) = \expect_\pi(A) + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error}
+\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) 
+= \expect_\pi(A) + \underbrace{\frac{C}{N_{\rm iter} \dt}}_{\rm bias} + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error}
 \]
 \end{block}
 
@@ -1076,105 +1218,127 @@ 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 {\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}
 
-%\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}}
+    \[
+        P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}
+    \]
 \end{block}
 
 \smallskip
 
-\bu {\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$
+{\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
+            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
+            \[
+                \e^{\dt \mathcal L} \varphi = \I + \dt \mathcal L \varphi + \frac{\dt^2}{2} \mathcal L^2\varphi + \dotsb
+            \]
+        \item
+            Introduce 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*}
+    \end{itemize}
+\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
+\begin{frame}
+    {Example: Euler-Maruyama, weak order~1}
+    Consider the scheme
+    \[
+        x^{n+1} = \Phi_\dt(x^n,G^n) = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n
+    \]
 
-\bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$
+    \bigskip
 
-\bigskip
+    \bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$
 
-\bu Technical tool: {\blue Taylor expansion}
-\vspace{-0.2cm}
-\[
-\varphi(x + \delta) = \varphi(x) + \delta^T \nabla \varphi(x) + \frac12 \delta^T \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots
-\]
+    \bigskip
 
-\medskip
+    \bu Technical tool: {\blue Taylor expansion}
+    \vspace{-0.2cm}
+    \[
+        \varphi(x + \delta) = \varphi(x) + \delta^\t \nabla \varphi(x) + \frac12 \delta^\t \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots
+    \]
 
-\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
 
-\medskip
+    \bu Replace $\delta$ with $\sqrt{\dt}\, \sigma(x)\,G + \dt\,b(x)$ and {\blue gather in powers of $\dt$}
+    \[
+        \begin{aligned}
+            \varphi\big(\Phi_\dt(x,G)\big) & = \varphi(x) + \sqrt{\dt}\, \sigma(x)\,G \cdot \nabla \varphi(x) \\
+                                           & \ \ \ + \dt \left(\frac{\sigma(x)^2}{2} G^\t \left[\nabla^2\varphi(x)\right]G + b(x)\cdot\nabla \varphi(x) \right) + \dots
+        \end{aligned}
+    \]
 
-\bu Taking {\blue expectations w.r.t. $G$} leads to
-\[
-P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathrm{O}(\dt^2)
-\]
+    \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}\frametitle{Error estimates on the invariant measure (equilibrium)}
+\begin{frame}
+    {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}
-}
+    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
+            \[
+                \int_\cX \mathcal A_k \varphi \, d\pi = 0
+            \]
+    \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 $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$.
 \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
+Error on invariant measure can be {\blue (much) smaller} than the weak error
 
 \end{frame}
 
-%-----------------------------------------------------------
-\begin{frame}\frametitle{Sketch of proof (1)}
+\begin{frame}
+    {Motivation of the result}
 
-{\bf Step~1: Establish the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$}
+We verify the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$.
 
 \medskip
 
@@ -1190,8 +1354,8 @@ $\dps \int_\cX \mathcal A_k \varphi \, d\pi = 0$
 \[
 \begin{aligned}
 & \int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right)\psi\right] (1+\dt^{p}f_{p+1})\, d\pi \\
-& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathrm{O}\left(\dt^{p+1}\right) \\
-& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathrm{O}\left(\dt^{p+1}\right)
+& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathcal O\left(\dt^{p+1}\right) \\
+& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathcal O\left(\dt^{p+1}\right)
 \end{aligned}
 \]
 
@@ -1201,40 +1365,50 @@ Suggests $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$
 
 \end{frame}
 
-%-----------------------------------------------------------
-\begin{frame}\frametitle{Sketch of proof (2)}
+\begin{frame}
+    {Numerical estimators and associated challenges}
+    \begin{itemize}
+        \item
+            Estimator of linear response (observable~$R$ with equilibrium average~0)
+            \[
+                \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, ds \xrightarrow[t\to+\infty]{\mathrm{a.s.}}
+                \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, d\mu = \alpha + \mathcal O(\eta)
+            \]
+            {\bf Issues with linear response methods:}
+            \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
+            \end{itemize}
 
-{\bf Step~2: Define an approximate inverse}
+    \end{itemize}
+\end{frame}
 
-\medskip
+\begin{frame}\frametitle{Analysis of variance / finite integration time bias}
 
-\bu Issue: derivatives of $(\I-P_\dt)^{-1}\varphi$ are not controlled
+  \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}: $\dps \left| \mathbb{E}\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\
+  Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error}
 
-\bigskip
+%\bigskip
+ %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$
 
-\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}
-\[
-\Pi \varphi = \varphi - \int_\cX \varphi \, d\pi
-\]
+\bigskip
 
-\bu Idea of the construction: truncate the formal series expression
+\bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, d\mu$
 \[
-(A + \dt \, B)^{-1} = A^{-1} - \dt \, A^{-1}B A^{-1} + \dt^{2} \, A^{-1}B A^{-1}B A^{-1} + \dots
+\widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T dW_s
 \]
 
-\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}
 
 
@@ -1256,12 +1430,12 @@ C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p
 
 \bu Trotter splitting $\to$ weak order 1
 \[
-P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^2)
+P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathcal O(\dt^2)
 \]
 
 \bu Strang splitting $\to$ {\blue weak order 2}
 \[
-P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^3)
+P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathcal O(\dt^3)
 \]
 
 \bu Other category: {\red Geometric Langevin}\footnote{N.~Bou-Rabee and H.~Owhadi, {\em SIAM J. Numer. Anal.} (2010)} algorithms, \textit{e.g.} $P_\dt^{\gamma C,A,B_\eta,A}$ \\
@@ -1323,8 +1497,8 @@ where the remainder is compatible with linear response
 \bu Corollary: error estimates on the {\blue numerically computed mobility}
 \[
 \begin{aligned}
-\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E}  F^T M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\
-& = \rho_{F} + \dt^\alpha \int_{\mathcal E}  F^T M^{-1} p  \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt}
+\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E}  F^\t M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\
+& = \rho_{F} + \dt^\alpha \int_{\mathcal E}  F^\t M^{-1} p  \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt}
 \end{aligned}
 \]
 
@@ -1339,17 +1513,11 @@ where the remainder is compatible with linear response
 
 \begin{figure}
 \begin{center}
-\includegraphics[width=6.2cm]{figures/LR.eps}
-\includegraphics[width=6.2cm]{figures/mobility_Langevin.eps}
+\includegraphics[width=.8\textwidth]{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 $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$.
+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}
 
@@ -1490,20 +1658,6 @@ with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \d
 
 \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}
-- 
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