From 4d132f781f576645367e7adf3d75efcf84a3301a Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Tue, 26 Sep 2023 01:20:33 +0200
Subject: Shorten presentation

---
 main.tex | 655 ++++++++++++++++++++++++++++++++++-----------------------------
 1 file changed, 354 insertions(+), 301 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index ceb383b..1c84dbc 100755
--- a/main.tex
+++ b/main.tex
@@ -240,7 +240,7 @@
                 \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.
+                \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
@@ -249,14 +249,13 @@
                 = - \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
+        \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^*,
+                = \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 adjoints.
-            We have
+            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  \\
@@ -273,8 +272,8 @@
         \[
             \left\{
                 \begin{aligned}
-                    \d q_t & = M^{-1} p_t  \, \d t, \\*
-                    \d p_t & =  -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t  \, \d t
+                    \d q_t & = p_t  \, \d t, \\*
+                    \d p_t & =  -\nabla V(q_t) \, \d t - \gamma p_t  \, \d t
                     + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t,
                 \end{aligned}
             \right.
@@ -370,8 +369,8 @@
           \mathcal P^{\dagger} \mu \geq \alpha \pi,
           \qquad \text{ (Minorization condition) }
       \]
-      then there exists $\mu_*$ such that $\mathcal P^{\dagger} \mu_* = \mu_*$.
-      Furthermore $d(\mathcal P^{\dagger^n}  \mu, \mu_*) \leq \alpha^n d(\mu, \mu_*)$.
+      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
@@ -437,12 +436,11 @@
             \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) \, 
+                =  \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 achieved by parabolic regularity, 
-            and achieved by {\blue Harnack's inequality}.
+            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
@@ -467,7 +465,7 @@
   {Existence of an invariant measure for perturbed Langevin dynamics}
   Consider the paradigmatic dynamics
   \begin{align*}
-    \d q_t &= M^{-1} p_t \, \d t, \\
+    \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.
@@ -483,13 +481,14 @@
 \end{frame}
 
 \begin{frame}
-    {Harris' theorem}
+    {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:
@@ -516,6 +515,7 @@
         \qquad \overline \phi := \int_{\mathcal E} \phi \, \d \mu_*.
     \]
   \end{theorem}
+  
 \end{frame}
 
 \begin{frame}
@@ -549,7 +549,7 @@
 \end{frame}
 
 \begin{frame}
-{Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/2)}
+{Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/3)}
     Consider the perturbed Langevin dynamics and  write
     \[
         \mathcal L_{\eta} = \mathcal L_0 + {\red \eta \widetilde {\mathcal L}},
@@ -570,7 +570,7 @@
         \]
         Observe that $\mathcal L_{\eta}^* = \mathcal L_0^* + \widetilde {\mathcal L}^*$ with
         \[
-            \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p  - \gamma \grad_p^* \grad_p^*,
+            \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p  - \gamma \grad_p^* \grad_p,
             \qquad \widetilde {\mathcal L}^* \placeholder = \grad_p^* (F  \placeholder)
         \]
     \end{block}
@@ -611,7 +611,7 @@
     \vspace{-.3cm}
     \begin{itemize}
         \itemsep.2cm
-        \item 
+        \item
             The operator $\mathcal L_0^{-1}$ is a well defined bounded operator on $L_0^2(\psi_0)$ \\
 
             ({\red Hypocoercivity} + {\red hypoelliptic regularization})
@@ -677,13 +677,13 @@
 %             %     &\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}
@@ -735,7 +735,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
   \begin{itemize}
     \itemsep.2cm
     \item
-      Non-equilibrium techniques.
+      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.
@@ -761,9 +761,9 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
 
 \begin{frame}
     {Linear response of nonequilibrium dynamics}
-    Consider the nonequilibirium dynamics
+    Consider the nonequilibirium dynamics with $V$ periodic:
     \begin{align*}
-        \d q_t &= M^{-1} p_t \, \d t, \\
+        \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*}
 
@@ -784,7 +784,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
 
     We proved that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and
     \[
-        f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathrm{O}(\eta^2), \qquad \mathfrak f_1 = - (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \mathbf 1.
+        f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathcal O(\eta^2), \qquad \mathfrak f_1 = - (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \mathbf 1.
     \]
     Therefore
     \[
@@ -811,7 +811,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
 \end{frame}
 
 \begin{frame}
-    {Reformulation as integrated correlation functions}
+    {Reformulation as integrated correlation function}
     Define the conjugate response
     \[
         S
@@ -892,7 +892,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
   \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
+    &= - 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*}
@@ -962,7 +962,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
 \end{frame}
 
 \begin{frame}
-    {Thermal transport in one-dimensional chain (1)}
+    {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);
@@ -991,21 +991,21 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
     \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 \dt + \sqrt{2 \gamma {\color{red} (T+\Delta T)}} \, \d W_t^L, \\
+                \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 \dt + \sqrt{2 \gamma {\color{blue} (T-\Delta T)}} \, \d W_t^R,
+                \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},
-    \quad V(r) = \sum_{n=1}^{N-1} v(r_n).
+    \qquad V(r) = \sum_{n=1}^{N-1} v(r_n).
 \end{equation*}
 \end{frame}
 
 \begin{frame}
-    {Thermal transport in one-dimensional chains (2)}
+    {Thermal transport in one-dimensional chains (2/3)}
 
     \begin{itemize}
         \item
@@ -1038,7 +1038,7 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
 
 
 \begin{frame}
-    {Thermal transport in one-dimensional chains (3)}
+    {Thermal transport in one-dimensional chains (3/3)}
 
     \bu Response function: {\blue total energy current}
     \begin{block}
@@ -1067,9 +1067,9 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
 \end{frame}
 
 \begin{frame}
-    {Shear viscosity in fluids (1)}
+    {Shear viscosity in fluids (1/4)}
 
-    Consider a fluid $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$ subjected to a sinusoidal forcing
+    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}
@@ -1079,76 +1079,64 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
     each subjected to a force $F$.
 \end{frame}
 
-\begin{frame}
-    {Shear viscosity in fluids (2)}
-    Macroscopic description by Navier--Stokes equation
-    \[
-        \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \nu \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x}
-    \]
-    Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives
-    \[
-        - \nu U_x''(y) = \overline{\rho} F(y), \qquad \overline \rho := \frac{\rho}{m} = \frac{N}{|\mathcal D|}
-    \]
-    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 (2/4)}
+%     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 (3)}
+    {Shear viscosity in fluids (2/4)}
     Assume pairwise interactions
     \[
-        V(q) = \sum_{1 \leq i < j \leq N} \mathcal V(\abs{q_i - q_j}).
+        V(q) = \sum_{1 \leq \ell < n \leq N} \mathcal V(\abs{q_\ell - q_n}).
     \]
-\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_{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}
+    \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
+    \smallskip
 
-\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$
+    \bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$
 
-\smallskip
+    \smallskip
 
-\bu The perturbation $\dps \wcL = \sum_{i=1}^N \! F(q_{y,i}) \partial_{p_{x,i}}$ is $\mathcal{L}_0$-bounded
+    \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 (4)}
-
-\bu {\blue Linear response}:
-\[
-    \lim_{\eta \rightarrow 0} \frac{\expect_{\eta} [\mathcal L_0 h]}{\eta}
-  = - \frac{\beta}{m} \!
-  \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)}.
-\]
+    {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}
   $u_x(Y) = \dps \lim_{\varepsilon \to 0}
-  \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\cdot) \right]}{\eta}$
+  \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\placeholder) \right]}{\eta}$
   where
   \vspace{-0.3cm}
   \[
-  U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{i=1}^N p_{xi}
-  \chi_{\varepsilon}\left(q_{yi}-Y\right)
+  U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{n=1}^N p_{n,x}
+  \, \chi_{\varepsilon}(q_{n,y}-Y)
   \]
   \vspace{-0.5cm}
 
@@ -1160,36 +1148,65 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
   \[
   \hspace{-0.1cm}
   ... =
-  \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right)
+  \frac{1}{L_x} \left( \sum_{n=1}^N \frac{p_{n,x} p_{n,y}}{m}\chi_{\varepsilon}(q_{n,y}-Y)
   - \! \! \! \! \! \! \! \!
-  \sum_{1 \leq i < j \leq N} \! \! \! \!
-  v'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|}
-  \!\int_{q_{yj}}^{q_{yi}} \!\chi_{\varepsilon}(s-Y) \, ds \right)
+  \sum_{1 \leq n < \ell \leq N} \! \! \! \!
+  \mathcal V'(|q_n-q_\ell|)\frac{ q_{n,x}-q_{\ell,x}}{|q_n-q_\ell|}
+  \!\int_{q_{\ell,y}}^{q_{n,y}} \!\chi_{\varepsilon}(s-Y) \, ds \right)
   \]
 
-\bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ (with $\overline{\rho} = N/|\mathcal{D}|$)
+\bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$
 \[
-    \frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y)
+    \frac{\d\sigma_{xy}(Y)}{\d Y} + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y),
+    \qquad \overline{\rho} = \frac{N}{|\mathcal{D}|}.
 \]
 \end{frame}
 
 \begin{frame}
-    {Shear viscosity in fluids (4)}
+    {Shear viscosity in fluids (4/4)}
 
-\bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$
+\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{-\eta u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)}
+\centerequation{-\nu u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)}
 \end{block}
 
-\begin{figure}[ht]
-    \centering
-    \includegraphics[width=\linewidth]{figures/shear1.png}
-\end{figure}
+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}
-% \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}
@@ -1203,7 +1220,7 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
     \begin{itemize}
         \item Reminders: strong order, weak order
         \item Error analysis for the linear response method
-        \item Error analysis for Green--Kubo method
+        \item Error analysis for the Green--Kubo method
     \end{itemize}
     \end{minipage}
 \end{frame}
@@ -1235,7 +1252,7 @@ x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \stackrel{\
 
 \begin{block}{Error estimates for {\red finite} trajectory averages}
 \[
-\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) 
+\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n)
 = \expect_\pi(A) + \underbrace{\frac{C}{N_{\rm iter} \dt}}_{\rm bias} + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error}
 \]
 \end{block}
@@ -1265,7 +1282,7 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$
     \[
         \dps \sup_{0 \leq n \leq T/\dt} \expect | x^n - x_{n\dt} | \leq C \dt^p
     \]
-\item {\blue Weak error}: for any $\varphi$, 
+\item {\blue Weak error}: for any $\varphi$,
     \[
         \dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(x^n\right)\right] - \expect\left[\varphi\left(x_{n\dt}\right)\right] \Big| \leq C \dt^p
     \]
@@ -1287,6 +1304,15 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$
 \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*}
@@ -1297,14 +1323,6 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$
             \[
                 \e^{\dt \mathcal L} \varphi = \I + \dt \mathcal L \varphi + \frac{\dt^2}{2} \mathcal L^2\varphi + \dotsb
             \]
-        \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}
 
@@ -1348,12 +1366,10 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$
 \begin{frame}
     {Error estimates on the invariant measure (equilibrium)}
 
-\bu {\red Assumptions} on the operators in the weak-type expansion
-
 \begin{block}{Error estimates on $\pi_\dt$}
     Suppose that
     \begin{itemize}
-        \item 
+        \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}
@@ -1362,12 +1378,14 @@ where $(x^n)$ is an approximation of $(x_{n \dt})$
             \[
                 \int_\cX \mathcal A_k \varphi \, d\pi = 0
             \]
+        \item
+            + {\red Technical assumptions} usually satisfied
     \end{itemize}
     Then
     \[
-        \int_\cX \varphi \, d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) d\pi + \dt^{p+1} R_{\varphi,\dt},
+        \int_\cX \varphi \, \d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) \d\pi + \dt^{p+1} R_{\varphi,\dt},
     \]
-    where $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$.
+    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
@@ -1421,12 +1439,12 @@ C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p
 
 \bu Trotter splitting $\to$ weak order 1
 \[
-P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathcal O(\dt^2)
+P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \, \mathcal O(\dt^2)
 \]
 
 \bu Strang splitting $\to$ {\blue weak order 2}
 \[
-P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathcal O(\dt^3)
+P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \, \mathcal O(\dt^3)
 \]
 
 \bu Other category: {\red Geometric Langevin}\footnote{N.~Bou-Rabee and H.~Owhadi, {\em SIAM J. Numer. Anal.} (2010)} algorithms, \textit{e.g.} $P_\dt^{\gamma C,A,B_\eta,A}$ \\
@@ -1467,85 +1485,89 @@ p^{n+1} & = \alpha_{\dt/2} \widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha_{\dt}}{\be
 
 
 \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}
+    {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}\frametitle{Analysis of variance / finite integration time bias}
+\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}: $\dps \left| \mathbb{E}\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\
-  Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error}
+    \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 Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$
+    \bigskip
 
-\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}
 
-\bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, d\mu$
-\[
-\widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T dW_s
-\]
+    %\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}\frametitle{Error estimates on linear response}
+\begin{frame}
+    {Error estimates on linear response (3/3)}
 
-\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}
+    \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}
 \[
 \begin{aligned}
-\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E}  F^\t M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\
-& = \rho_{F} + \dt^\alpha \int_{\mathcal E}  F^\t M^{-1} p  \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt}
+\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t p \, \mu_{\eta,\dt}(\d{q}\,\d{p}) - \int_{\mathcal E}  F^\t p \, \mu_{0,\dt}(\d{q}\,\d{p}) \right) \\
+& = \rho_{F} + \dt^\alpha \int_{\mathcal E}  F^\t p  \, f_{\alpha,1} \, \d{\mu} + \dt^{\alpha+1} r_{\dt}
 \end{aligned}
 \]
 
-\bu Results in the {\red overdamped} limit\footnote{B.~Leimkuhler, C.~Matthews and G.~Stoltz, {\em IMA J. Numer. Anal.} (2015)}
-
-\bigskip
-
 \end{frame}
 
 
@@ -1561,172 +1583,203 @@ Scaling of the mobility for the first order scheme $P_\dt^{A,B_\eta,\gamma C}$ a
 
 \end{frame}
 
+\begin{frame}
+    {Error estimates on the Green--Kubo formula (1/3)}
+    \textbf{Aim:} For observable~$R$, approximate
+    \[
+        \alpha = \int_0^{+\infty} \!\! \expect_0\Big(R(q_t,p_t)S(q_0,p_0) \Big) \, \d t
+    \]
+  
 
-%-----------------------------------------------------------
-\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 $T = N_{\rm iter}\dt$
+    \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}\frametitle{Error estimates on Green-Kubo formulas (2)}
+\begin{frame}
+    {Error estimates on the Green--Kubo formula (2/3)}
 
-\bu Error of {\red order~$\alpha$ on invariant measure}: $\dps \int_\cX \psi \, d{\pi}_\dt = \int_\cX \psi \, d{\pi} + \mathrm{O}(\dt^\alpha)$
+    \bu {\bf Truncation bias}: {\blue small} due to generic exponential decay of correlations
+    \[
+        \left|\expect\left(\widehat{A}_{K,T}\right)-\alpha\right| \leq C \e^{-\kappa T}
+    \]
 
-\medskip
+    \bigskip
 
-\bu Expansion of the evolution operator ($p+1 \geq \alpha$ and $\mathcal A_1 = \mathcal L$)
-\[
-P_\dt \varphi = \varphi + \dt \, \mathcal L \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}
-\]
+    \bu {\bf Statistical error}: {\red large}, increases with the integration time
+    \[
+        \forall T \geq 1, \qquad \mathrm{Var}\left(\widehat{A}_{K,T}\right) \leq C \frac{T}{K}
+    \]
 
-\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)}
+    \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}\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.} $\mathcal A_k = a_k \mathcal L^{k}$
+\begin{frame}
+    {Error estimates on Green-Kubo formulas (1/3)}
 
-\medskip
+    \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 Reduces to trapezoidal rule for second order schemes
+    \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{Sketch of proof (1)}
+% \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}
 
-\bu Define $\dps \Pi_\dt \varphi = \varphi - \int_\cX \varphi \, d\pi_\dt$
+%\begin{frame}
+%    {Sketch of proof (1)}
 
-\smallskip
+%\bu Define $\dps \Pi_\dt \varphi = \varphi - \int_\cX \varphi \, d\pi_\dt$
 
-\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*}
+%\smallskip
 
-\bu Rewrite $-\Pi_\dt \mathcal L^{-1}$ in terms of $P_\dt$ as
-\[
-\begin{aligned}
-& -\Pi_\dt \mathcal L^{-1} \psi = -\Pi_\dt \left(\dt\sum_{n=0}^{+\infty} P_\dt^n \right) \Pi_\dt \left(\frac{\I - P_\dt}{\dt}\right) \mathcal L^{-1} \psi \\
-& \ \ = \dt \left(\sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \right) \left(\mathcal L + \dots + \dt^{\alpha-1} S_{\alpha-1} + \dt^\alpha \widetilde{R}_{\alpha,\dt}\right) \mathcal L^{-1} \psi, \\
-& \ \ = \dt \sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \widetilde{\psi}_{\dt,\alpha} + \dt^\alpha \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \Pi_\dt \widetilde{R}_{\alpha,\dt} \mathcal L^{-1} \psi.
-\end{aligned}
-\]
+%\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*}
 
-\end{frame}
+%\bu Rewrite $-\Pi_\dt \mathcal L^{-1}$ in terms of $P_\dt$ as
+%\[
+%\begin{aligned}
+%& -\Pi_\dt \mathcal L^{-1} \psi = -\Pi_\dt \left(\dt\sum_{n=0}^{+\infty} P_\dt^n \right) \Pi_\dt \left(\frac{\I - P_\dt}{\dt}\right) \mathcal L^{-1} \psi \\
+%& \ \ = \dt \left(\sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \right) \left(\mathcal L + \dots + \dt^{\alpha-1} S_{\alpha-1} + \dt^\alpha \widetilde{R}_{\alpha,\dt}\right) \mathcal L^{-1} \psi, \\
+%& \ \ = \dt \sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \widetilde{\psi}_{\dt,\alpha} + \dt^\alpha \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \Pi_\dt \widetilde{R}_{\alpha,\dt} \mathcal L^{-1} \psi.
+%\end{aligned}
+%\]
 
-%-----------------------------------------------------------
-\begin{frame}\frametitle{Sketch of proof (2)}
+%\end{frame}
 
-\bu Uniform resolvent bounds $\dps \left\| \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq \frac{C_s}{\lambda_s}$
+%%-----------------------------------------------------------
+%\begin{frame}\frametitle{Sketch of proof (2)}
 
-\medskip
+%\bu Uniform resolvent bounds $\dps \left\| \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq \frac{C_s}{\lambda_s}$
 
-\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
-\[
-\begin{aligned}
-\int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right)\left( \Pi_\dt \varphi \right) d{\pi}_\dt & = \int_\cX \sum_{n=0}^{+\infty} \left(P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \varphi \, d{\pi}_\dt \\
-& = \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\right)
-\end{aligned}
-\]
+%\medskip
 
-\end{frame}
+%\bu Coming back to the initial equality,
+%\[
+%\int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} = \dt \int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \left( \Pi_\dt \varphi \right) d{\pi}_\dt + \mathrm{O}\left(\dt^\alpha\right)
+%\]
 
-\begin{frame}
-    \begin{center}
-\Huge{Conclusion and perspectives}
-\end{center}
-\end{frame}
+%\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}
-    {Main points: recall the outline!}
+    {Summary}
 
-\bu {\bf Definition and examples of nonequilibrium systems}
+    \bu {\bf Definition and examples of nonequilibrium systems}
+    \begin{itemize}
+        \item Convergence to invariant measure
+        \item Perturbation expansion of invariant measure
+    \end{itemize}
 
-\bigskip
+    \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}
+    \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
+    \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}
+    \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}
 
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