Minor changes

4 files changed, 561 insertions, 311 deletions

diff --git a/figures.jl b/figures.jl
index ed9b70f..f576d44 100644
--- a/figures.jl
+++ b/figures.jl
@@ -2,16 +2,22 @@ using Plots
 using QuadGK
 using LaTeXStrings
 
-V(x) = 1 - cos(x)
+Plots.default(fontfamily="Computer Modern",
+              titlefontsize=20,
+              xtickfontsize=12,
+              ytickfontsize=12,
+              legendfontsize=12)
+
+V(x) = 1/2 * (1 - cos(x))
 η = .1
 
-# plot(title="Invariant distribution of perturbed OL") 
+plot(title=L"Steady state $\rho_{\eta}$ with forcing") 
 plot(title="") 
 for η ∈ [0, .5, 1, 2]
     ρ(q) = exp(-V(q)) * quadgk(y -> exp(V(q + y) - η*y), 0, 2π)[1]
     Z = quadgk(ρ, - π, π)[1]
     plot!(q -> ρ(q) / Z, -π, π, label=L"\eta = " * string(η), lw=2)
 end
-plot!(xlims = (-π, π), grid=false, ylims=(0, .4), framestyle=:box)
+plot!(xlims = (-π, π), grid=false, framestyle=:box)
 savefig("figures/invariant_perturbed_ol.pdf")
 
diff --git a/header.tex b/header.tex
index 83d0e93..8433786 100755
--- a/header.tex
+++ b/header.tex
@@ -23,6 +23,7 @@
 \usepackage{subcaption}
 \usepackage{appendixnumberbeamer}
 \usepackage{xparse}
+\usepackage{dsfont}
 \usepackage[makeroom]{cancel}
 \renewcommand{\CancelColor}{\color{red}}
 \usepackage[%
@@ -39,9 +40,11 @@
 \usetikzlibrary{quotes}
 \pgfplotsset{compat=1.17}
 
-\definecolor{darkred}{rgb}{.5,0,0}
-\definecolor{darkgreen}{rgb}{0,.5,0}
-\definecolor{darkblue}{rgb}{0,0,.5}
+\definecolor{linea}{RGB}{211,  94,  60}
+\definecolor{lineb}{RGB}{ 78, 144, 204}
+\colorlet{dark_color}{black!67}
+\tikzstyle{dark} =[very thick, color=dark_color]
+\tikzstyle{green} =[very thick, color=darkgreen]
 
 \newtheorem{proposition}{Proposition}
 
diff --git a/main.bib b/main.bib
index 0532ba7..2e637db 100755
--- a/main.bib
+++ b/main.bib
@@ -451,3 +451,15 @@ MRREVIEWER = {Wojciech\ Bartoszek},
        DOI = {10.1007/978-3-0348-0021-1\_7},
        URL = {https://doi.org/10.1007/978-3-0348-0021-1_7},
 }
+
+@book {MR2723222,
+    AUTHOR = {Tuckerman, Mark E.},
+    TITLE = {Statistical Mechanics: Theory and Molecular Simulation},
+    SERIES = {Oxford Graduate Texts},
+    PUBLISHER = {Oxford University Press},
+    YEAR = {2010},
+    ISBN = {978-0-19-852526-4},
+    MRCLASS = {82-01 (81V55 82-08 82C80)},
+    MRNUMBER = {2723222},
+    MRREVIEWER = {Gabriel Stoltz},
+}
diff --git a/main.tex b/main.tex
index 7596b40..b8e83e2 100755
--- a/main.tex
+++ b/main.tex
@@ -1,9 +1,8 @@
 \documentclass[9pt]{beamer}
-\newcommand{\blue}[1]{\textcolor{blue}{#1}}
-% \newcommand{\red}[1]{\color{red}}
 \newif\iflong
 \longfalse
 
+\usepackage{psfrag}
 \newcommand{\placeholder}{\mathord{\color{black!33}\bullet}}%
 \newcommand{\bu}{$\bullet \ $}
 \newcommand{\bi}{\begin{itemize}}
@@ -21,6 +20,8 @@
 \newcommand{\I}{\mathrm{Id}}
 \newcommand{\dps}{\displaystyle}
 \newcommand{\red}{\color{red}}
+\newcommand{\blue}{\color{blue}}
+\newcommand{\yellow}{\color{yellow}}
 
 \input{header}
 \input{macros}
@@ -80,16 +81,12 @@
     \itemsep.2cm
     \item \fullcite{MR3509213}
     \item \fullcite{pavliotis2011applied}
+    \item \fullcite{MR2723222}
     \item Lecture notes by Gabriel Stoltz  on computational statistical physics:
       \url{http://cermics.enpc.fr/~stoltz/Cours/intro_phys_stat.pdf}
   \end{itemize}
 \end{frame}
 
-\begin{frame}
-  {Outline}
-  \tableofcontents
-\end{frame}
-
 \section{Introduction}
 \begin{frame}
   {Transport coefficients}
@@ -100,21 +97,22 @@
   \begin{itemize}
     \item The \emph{mobility} relates an external force to a velocity;
     \item The \emph{heat conductivity} relates a temperature difference to a heat flux;
-    \item The \emph{shear viscosity} relates a shear velocity to a shear stress;
+    \item The \emph{shear viscosity} relates a shear velocity to a shear stress.
   \end{itemize}
 
   \vspace{.3cm}
   They can be estimated from molecular simulation at the \blue{microscopic level}.
   \begin{itemize}
-      \item They are defined from \emph{nonequilibrium} dynamics;
-      \item There are three main classes of methods to calculate them.
+      \item Defined from \emph{nonequilibrium} dynamics;
+      \item Three main classes of methods to calculate them.
   \end{itemize}
 
   \vspace{.3cm}
-  \textbf{Challenges we do not address:}
+  \textbf{\blue Outline of this talk}
   \begin{itemize}
-    \item Choose thermodynamical ensemble;
-    \item Prescribe microscopic dynamics;
+    \item Equilibrium vs nonequilibrium dynamics;
+    \item Definition and computation of the mobility;
+    \item Computation of other transport coefficients.
   \end{itemize}
 \end{frame}
 
@@ -131,6 +129,7 @@
         \item Equilibrium vs nonequilibrium dynamics
         \item Existence of an invariant measure for nonequilibrium dynamics
         \item Convergence to the invariant measure
+        \item Perturbation expansion of the invariant measure
     \end{itemize}
     \end{minipage}
 \end{frame}
@@ -156,13 +155,16 @@
   \begin{theorem}
     A stationary diffusion processes $x_t$ in $\real^d$ with generator $\mathcal L$ and invariant measure~$\mu$ is reversible if and only if $\mathcal L$ is self-adjoint in~$L^2(\mu)$.
   \end{theorem}
+
+  In this course, equilibrium = reversible,
+  possibly up to a one-to-one transformation preserving the invariant measure.
 \end{frame}
 
 \begin{frame}
-    {Example of nonequilibrium dynamics}
+    {Examples of nonequilibrium dynamics}
     \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term}
         \begin{equation}
-            \label{eq:Langevin_F}
+            \label{eq:overdamped_Langevin_F}
             \tag{NO}
             \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F} +  \sqrt{2} \, \d W_t
         \end{equation}
@@ -176,10 +178,11 @@
                 \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,
+                    + \sqrt{2\gamma}  \D W_t,
                 \end{aligned}
             \right.
         \end{equation}
+        In the rest of the presentation we take {\blue $ M = \I$} for simplicity.
     \end{block}
     where
     \begin{itemize}
@@ -191,6 +194,68 @@
 \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}
+        \item For overdamped Langevin dynamics
+            \[
+                \mathcal L_{\rm ovd} \Big\vert_{\red \eta = 0} = - \grad V \cdot \grad + \laplacian
+                = - \grad^* \grad,
+            \]
+            where $\grad^* := (\grad V - \grad) \cdot $.
+            For any $f, g \in C^{\infty}_{\rm c}(\mathcal E)$, we have
+            \[
+                \int_{\mathcal E} (\mathcal L_{\rm ovd} f ) g \, \d \mu
+                = - \int_{\mathcal E} \nabla f \cdot \nabla g \, \d \mu
+                = \int_{\mathcal E} (\mathcal L_{\rm ovd} g ) f \, \d \mu.
+            \]
+        \item For Langevin dynamics
+            \begin{align*}
+                \mathcal L\Big\vert_{\red \eta = 0}
+                = p \cdot \grad_q - \grad V \cdot \grad_p + \gamma \left( - p \cdot \grad_p + \laplacian_p \right)
+                = \grad_p^* \grad_q - \grad_q^* \grad_p  - \gamma \grad_p^* \grad_p^*,
+            \end{align*}
+            where  $\grad_q^* := (\grad V - \grad_q) \cdot $ and $\grad_p^* = (p -\grad_p) \cdot$ are the formal adjoints.
+            We have
+            \begin{align*}
+                \int_{\mathcal E} (\mathcal Lf ) g \, \d \mu
+                &= \int_{\mathcal E} g \left(\grad_p^* \grad_q - \grad_q^* \grad_p\right) f -  \gamma \grad_p f \cdot \grad_p g \, \d \mu  \\
+                &= \int_{\mathcal E} {\red -} f \left(\grad_p^* \grad_q - \grad_q^* \grad_p\right) g -  \gamma \grad_p f \cdot \grad_p g \, \d \mu \\
+                &= \int_{\mathcal E} (f \circ S) \bigl(\mathcal L (g \circ S)\bigr) \, \d \mu
+                \qquad S f(q, p) := f(q, -p).
+            \end{align*}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}
   {Worked example in dimension one}
   Consider the perturbed overdamped Langevin dynamics with~$q_t \in \torus$
   \[
@@ -221,22 +286,22 @@
 
 \begin{frame}
   {Nonequilibrium overdamped Langevin dynamics}
-  In general, can we prove existence of and convergence to an invariant measure for
+  In general, how can we prove existence of an invariant measure for
   \[
       \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F} +  \sqrt{2} \, \d W_t \, ?
   \]
 
   \begin{itemize}
-      \item 
+      \item
           If the state space is compact (e.g. $\torus^d$),
-          then we can apply Doeblin's theorem.
+          apply Doeblin's theorem.
 
       \item
-          If not, we can apply its generization, Harris' theorem.
+          If not, use its generization, Harris' theorem.
   \end{itemize}
 
   \medskip
-  Fix $t > 0$ and denote by $p\colon \mathcal E \times \mathcal B(\mathcal E)$ the Markov transition kernel
+  Fix ${\blue t = 1}$ and denote by $p\colon \mathcal E \times \mathcal B(\mathcal E)$ the Markov transition kernel
   \[
       p(x, A) := \proba \left[ q_t \in A \, \middle| \, q_0 = x \right].
   \]
@@ -254,21 +319,23 @@
 \end{frame}
 
 \begin{frame}
-  {Existence of an invariant measure (1/2)}
-  Remember that the set of probability measures with TV distance $d(\placeholder, \placeholder)$ is complete.
+  {Existence of an invariant measure for compact state space (1/2)}
+  Let $d(\placeholder, \placeholder)$ denote the total variation metric.
   \begin{theorem}
       [Doeblin's theorem]
-      If there exists $\alpha \in (0, 1)$ and a probability measure $\eta$ such that
+      If there exists $\alpha \in (0, 1)$ and a probability measure $\pi$ such that
       \[
-          \mathcal P^{\dagger} \mu \geq \alpha \eta,
+          \forall \mu, \qquad
+          \mathcal P^{\dagger} \mu \geq \alpha \pi,
+          \qquad \text{ (Minorization condition) }
       \]
-      then there exists $\mu_{\infty}$ such that $\mathcal P^{\dagger} \mu_{\infty} = \mu_{\infty}$.
-      Furthermore $d(\mathcal P^{\dagger^n}  \mu, \mu_{\infty}) \leq \alpha^n d(\mu, \mu_{\infty})$.
+      then there exists $\mu_*$ such that $\mathcal P^{\dagger} \mu_* = \mu_*$.
+      Furthermore $d(\mathcal P^{\dagger^n}  \mu, \mu_*) \leq \alpha^n d(\mu, \mu_*)$.
   \end{theorem}
 
   \emph{Sketch of proof.} Define the Markov transition kernel
   \[
-      \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder), 
+      \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder),
   \]
   Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$.
   We have
@@ -283,34 +350,84 @@
 \end{frame}
 
 \begin{frame}
-  {Existence of an invariant measure (2/2)}
+  {Existence of an invariant measure for compact state space (2/2)}
+  Two simple corollaries:
   \begin{itemize}
       \item
           Suppose that $\phi$ is uniformly bounded. Then
           \begin{align*}
               \left\lvert \mathcal P^n \phi(x) - \overline \phi \right\rvert
-              &= \int_{\mathcal E} \mathcal P^n (\phi - \overline \phi) \, \d(\delta_x - \mu_{\infty})
-              = \int_{\mathcal E} (\phi - \overline \phi) \, (\mathcal P^{\dagger n} \delta_x -  \mathcal P^{\dagger n} \mu_{\infty}) (\d q) \\
-              &\leq \norm{\phi - \overline \phi}_{L^{\infty}} (1-\alpha)^n d(\delta_x, \mu_{\infty}) 
+              &= \int_{\mathcal E} \mathcal P^n (\phi - \overline \phi) \, \d(\delta_x - \mu_{*})
+              = \int_{\mathcal E} (\phi - \overline \phi) \, (\mathcal P^{\dagger n} \delta_x -  \mathcal P^{\dagger n} \mu_{*}) (\d q) \\
+              &\leq \norm{\phi - \overline \phi}_{L^{\infty}} (1-\alpha)^n d(\delta_x, \mu_{*})
               \leq 2 \norm{\phi - \overline \phi}_{L^{\infty}} (1 - \alpha)^n.
           \end{align*}
-
+          This shows that
+          \[
+              \left\lVert \mathcal P^n \phi(x) - \overline \phi \right\rVert_{L^{\infty}}
+              \leq 2 (1 - \alpha)^n \norm{\phi - \overline \phi}_{L^{\infty}}.
+          \]
 
       \item
-          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}
+          The Neumann series $\I + \mathcal P + \mathcal P^2 + \dotsb$ is convergent as a bounded operator on
+          \[
+              L^{\infty}_{*} := \left\{ \phi \in L^{\infty}(\mathcal E) : \int_{\mathcal E} \phi \, \d \mu_{*} = 0 \right\}.
+          \]
+          Thus $\I - \mathcal P$ is invertible and
+          \[
+              (\I - \mathcal P)^{-1} = \I + \mathcal P + \mathcal P^2 + \dotsb
+          \]
   \end{itemize}
 \end{frame}
 
 \begin{frame}
-  {Existence of an invariant measure for noneq.\ dynamics}
+    {Connection with the time-continuous setting}
+    Consider the overdamped Langevin dynamics on~$\torus^d$:
+    \[
+        \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F \, \d t} +  \sqrt{2} \, \d W_t,
+        \qquad q_t \in \torus^d.
+    \]
+
+    \begin{itemize}
+        \itemsep.5cm
+        \item
+            The \textbf{minorization condition} is satisfied.
+            Indeed for $t > 0$
+            \begin{align*}
+                p(x, A)
+                &= \expect \left[ q_t \in A \, \middle| \, q_0 = x \right]
+                = \expect \left[ \mathds 1_{A} \left(x +  W_t \right) M_t \right]
+                && M_t = \text{Girsanov weight} \\
+                &= \proba \left[ x + W_t \in A \right] \expect \left[ M_t \, | \, \{x + W_t \in A\} \right] \\
+                &\geq C \proba \left[ x + W_t \in A \right] \geq C \lambda(A) && \lambda := \text{Lebesgue measure}.
+            \end{align*}
+            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
+            \begin{align*}
+                \lVert \e^{t \mathcal L_{\rm ovd}} \varphi \rVert_{L^{\infty}}
+        &= \left\lVert \e^{(t-  \lfloor t \rfloor) \mathcal L_{\rm ovd}} \left( \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi \right) \right\rVert_{L^{\infty}} \\
+        &\leq  \left\lVert   \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi  \right\rVert_{L^{\infty}}
+        \leq 2 \e^{\alpha} \e^{- \alpha t}  \lVert \varphi \rVert_{L^{\infty}}.
+            \end{align*}
+
+        \item
+            \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{\eta}$,
+            and
+            \[
+                \mathcal L_{\rm ovd}^{-1}
+                = - \int_{0}^{\infty} \e^{t \mathcal L_{\rm ovd}} \, \d t.
+            \]
+    \end{itemize}
+\end{frame}
+
+\begin{frame}
+  {Existence of an invariant measure for perturbed Langevin dynamics}
   Consider the paradigmatic dynamics
   \begin{align*}
     \d q_t &= M^{-1} p_t \, \d t, \\
-    \d p_t &= - \bigl(\grad V(q_t) + \eta F\bigr) \, \d t - \gamma M^{-1} p_t \, \d t + \sqrt{\frac{2 \gamma}{\beta}} \, \d W_t,
+    \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.
 
@@ -325,14 +442,55 @@
 \end{frame}
 
 \begin{frame}
-  {Existence of an invariant distribution}
+    {Harris' theorem}
+      Let $p(x, A)$ denote a Markov transition kernel and let
+      \[
+          (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, p(x, \d y),
+          \qquad
+          (\mathcal P^{\dagger} \mu)(A)  := \int_{A} p(x, A) \, \mu(\d x).
+      \]
   \begin{theorem}
-      Fix~$\eta_* > 0$ and $n \geq 2$,
+      [Harris's theorem]
+      Suppose that the following conditions are satisfied:
+      \begin{itemize}
+          \item
+            There exists $\mathcal K\colon \mathcal E \to [1, \infty)$
+            and constants~$a > 0$ and $b \geq 0$ such that
+            \[
+                \forall x \in \mathcal E, \qquad
+                \mathcal L \mathcal K(x) \leq - a \mathcal K(x) + b,
+            \]
+          \item
+            There exists a constant $\alpha \in (0, 1)$ and a probability measure~$\pi$ such that
+            \[
+                \inf_{x \in \mathcal C} p(x, \d y) \geq \, \alpha \, \pi(\d y),
+            \]
+            where $\mathcal C = \{x \in \real \, | \, \mathcal K(x) \leq K_{\max} \}$ for some $K_{\max} \geq 1 + 2 \, \frac{b}{a}$.
+    \end{itemize}
+    Then there $\exists! \, \, \mu_{*}$ such that $\mathcal P^{\dagger} \mu_{*} = \mu_{*}$.
+    Furthermore  there is $\gamma \in (0, 1)$ such that
+    \[
+        \left\lVert \frac{\mathcal P^n \phi - \overline \phi}{\mathcal K} \right\rVert_{L^{\infty}}
+        \leq C \gamma^n \norm{ \frac{\mathcal P^n \phi - \overline \phi}{\mathcal K} }_{L^{\infty}},
+        \qquad \overline \phi := \int_{\mathcal E} \phi \, \d \mu_*.
+    \]
+  \end{theorem}
+\end{frame}
+
+\begin{frame}
+  {Application to perturbed Langevin dynamics}
+  For $\mathcal K \colon \mathcal E \to [1, \infty)$, let
+  \[
+      L^{\infty}_{\mathcal K}
+      := \left\{ \varphi \text{~measureable } : \norm{\frac{\varphi}{\mathcal K}}_{L^{\infty}} < \infty \right\}
+  \]
+
+  \begin{theorem}
+      Fix~$\eta > 0$ and $n \geq 2$,
       and let $\mathcal K_n(q, p) := 1 + \abs{p}^n$.
-      For any $\eta \in [- \eta_*,  \eta_*]$,
-      there exists a unique invariant probability measure,
+      There exists a unique invariant probability measure,
       with a smooth density~$\psi_{\eta}(q, p)$ with respect to the Lebesgue measure.
-      Furthermore there exists $C = C(n, \eta_*) > 0$ and $\lambda = \lambda(n, \eta_*) > 0$ such that
+      Furthermore there exists $C = C(n, \eta) > 0$ and $\lambda = \lambda(n, \eta) > 0$ such that
       \[
           \forall \phi \in L^{\infty}_{\mathcal K_n}(\mathcal E), \qquad
           \left\lVert \e^{t \mathcal L_n} \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}}
@@ -341,102 +499,117 @@
   \end{theorem}
 
   \textbf{Idea of the proof.}
-  Show that
+  Show that the assumptions of Harris' theorem are satisfied,
+  in particular that
   \begin{align*}
-    \mathcal L \mathcal K_n &\leq - c_1 \mathcal K_n(q, p) + c_2,
+    \mathcal L \mathcal K_n &\leq - a \mathcal K_n(q, p) + b,
   \end{align*}
-  for $c_1 > 0$ and $c_2 > 0$.
-  Then apply the main theorem from~\footfullcite{MR2857021}.
+  for $a > 0$ and $b \geq 0$.
 \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 $\dps R(q,p) = F^\t  M^{-1}p$
+{Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/2)}
+    Consider the perturbed Langevin dynamics and  write
+    \[
+        \mathcal L_{\eta} = \mathcal L_0 + {\red \eta \widetilde {\mathcal L}},
+        \qquad \widetilde {\mathcal L} = F \cdot \grad_p
+    \]
 
-    \begin{block}
-        {Definition of the mobility}
+    It is {\red expected} that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and
+    \[
+        f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathcal O(\eta^2)
+    \]
+    The invariance of $\psi_\eta$ can be written as
+    \[
+        \int_{\mathcal E} (\mathcal L_\eta \varphi) \psi_\eta = 0 = \int_{\mathcal E} (\mathcal L_\eta \varphi) f_\eta \psi_0
+    \]
+    \begin{block}{Fokker-Planck equation on $L^2(\psi_0)$}
         \[
-            \rho_F
-            = \lim_{\eta \to 0} \frac{\expect_\eta (R)-\expect_0 (R)}{\eta}
-            = \lim_{\eta \to 0} \frac{\expect_\eta (R)}{\eta}
+            \mathcal L_\eta^* f_\eta = 0
+        \]
+        Observe that
+        \[
+            \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p  - \gamma \grad_p^* \grad_p^*,
+            \qquad \widetilde {\mathcal L}^* \placeholder = \grad_p^* (F  \placeholder)
         \]
     \end{block}
 
-    \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
+    {\bf Questions:} Can the expansion for $f_\eta$ be made rigorous? What is $\mathfrak{f}_1$?
 
-    \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}
+    {Perturbation expansion for {\yellow $\eta$ sufficiently small} (2/3)}
+    \begin{block}
+        {Formal asymptotics}
+        Write $f_\eta = \mathfrak f_0 + \eta \mathfrak{f}_1 + \eta^2 \mathfrak{f}_2 + \dotsb$ and expand
+        \begin{align*}
+            \mathcal L_{\eta}^* f_{\eta}
+            &= \mathcal L_0^* \mathfrak f_0 \\
+            &\quad + \eta \left(\widetilde {\mathcal L}^* \mathfrak f_0 + \mathcal L_0^* \mathfrak f_1\right) \\
+            &\quad + \eta^2 \left(\widetilde {\mathcal L}^* \mathfrak f_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
+        \end{align*}
+        This suggests that $\mathfrak f_{i+1} = -(\mathcal L_0^*)^{-1} (\widetilde {\mathcal L}^* \mathfrak f_i)$ and so
+        \[
+            f_\eta = \sum_{i=0}^{\infty} (-\eta)^i \Bigl((\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^*\Bigr)^i \mathbf 1
+            = \left(\I + \eta(\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right)^{-1} \mathbf 1.
+        \]
+    \end{block}
 \end{frame}
 
-\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
-\[
-\Pi_0 f = f - \int_{\mathcal E} f \, \psi_0
-\]
 
+\begin{frame}
+    {Elements of proof}
+    Let us introduce
+    \[
+        H^1_{p}(\psi_0) =
+        \Bigl\{ \varphi \in L^2(\psi_0) : \grad_p \varphi \in L^2(\psi) \Bigr\},
+        \qquad  \| \varphi \|_{H^1_{p}(\psi_0)}^2 = \| \varphi \|_{L^2(\psi_0)}^2 + \| \nabla_p \varphi \|_{L^2(\psi_0)}^2.
+    \]
+    \vspace{-.3cm}
+    \begin{itemize}
+        \itemsep.2cm
+        \item
+            The operator {\blue $\widetilde {\mathcal L}^*\colon H^1_p(\psi_0) \to L^2_0(\psi)$} is well-defined and bounded.
+            Indeed
+            \[
+                \lVert \widetilde {\mathcal L}^* \varphi \rVert_{L^2_0(\psi_0)}^2
+                = \ip{\nabla_p^* F \varphi}{\nabla_p^* F \varphi}_{L^2_0(\psi_0)}
+                \leq \lVert \varphi \rVert_{H^1_p(\psi_0)}^2
+            \]
+            and
+            \[
+                \int_{\mathcal E} \widetilde {\mathcal L}^* \phi \, \psi_0
+                = \int_{\mathcal E} \nabla_p^* (F \phi) \, \psi_0 = 0.
+            \]
+        \item
+            The operator {\blue $(\mathcal L_0^*)^{-1} \colon L^2_0(\psi_0) \to H^1_p(\psi_0)$} is well-defined and bounded,
+            by {\red hypocoercivity} and {\red hypoelliptic regularization}.
+            % In particular, for $\phi = (\mathcal L_0^*)^{-1} \varphi$
+            % \begin{align*}
+            %     \| \phi \|_{L^2(\psi_0)}^2
+            %     +  \| \nabla_p \phi \|_{L^2(\psi_0)}^2
+            %     &= \|(\mathcal L_0^*)^{-1} \varphi \|_{L^2(\psi_0)}^2
+            %     + \frac{1}{\gamma} \ip{-\mathcal L_0^* \phi}{\phi}_{L^2(\psi_0)} \\
+            %     &\leq \frac{1}{\gamma} \norm{(\mathcal L_0^*)^{-1}}_{\mathcal B\bigl(L^2(\psi_0)\bigr)}^2
+            %     \norm{\varphi}_{L^2(\psi_0)}
+            % \end{align*}
+
+        \item Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L_0^* + \eta \wcL^*$
+            \vspace{-0.2cm}
+            \[
+                \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0.
+            \]
+
+        \item {\red Prove that $f_\eta \geq 0$}.
+    \end{itemize}
 \end{frame}
 
-
-\begin{frame}\frametitle{Linear response of nonequilibrium dynamics (2)}
+\begin{frame}
+    {Perturbation expansion for {\yellow $\eta$ sufficiently small} (3/3)}
 
 \begin{block}{Power expansion of the invariant measure}
 Spectral radius $r$ of the bounded operator
@@ -445,7 +618,7 @@ Spectral radius $r$ of the bounded operator
   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
+  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}
@@ -454,67 +627,118 @@ Spectral radius $r$ of the bounded operator
   \end{equation}
 \end{block}
 
-\medskip
-
-\bu Note that $\dps \int_{\mathcal E} \psi_\eta = 1$
-
-\medskip
+Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
+\end{frame}
 
-\bu Linear response result: $\dps \rho_F = -\int_{\mathcal E} R \left[(\wcL \mathcal L_0^{-1})^* \mathbf{1}\right] \psi_0 $
+\section{Computation of transport coefficients}
 
+\begin{frame}
+    \begin{center}
+      \Large
+      \color{blue}
+      Part II: Definition and calculation of the mobility
+    \end{center}
 
+    \centering
+    \begin{minipage}{.8\textwidth}
+    \begin{itemize}
+        \item Definition through linear response
+        \item Green--Kubo reformulation
+        \item Numerical approximation
+    \end{itemize}
+    \end{minipage}
 \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{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)
-\[
-\| \wcL \mathcal L_0^{-1} \varphi \|^2_{L^2(\psi_0)}\leq \frac{\beta}{\gamma} \| \varphi \|_{L^2(\psi_0)} \| \mathcal L_0^{-1} \varphi \|_{L^2(\psi_0)}.
-\]
+    \item
+      Equilibrium techniques based on the Green--Kubo formula
+      \[
+        \rho = \int_{0}^{\infty} \expect_{\mu} \bigl[\varphi(x_t) \phi(x_0)\bigr] \, \d t.
+      \]
+      We will derive this formula from linear response.
+    \item
+      Transient methods.
+      \begin{itemize}
+          \item System locally perturbed
+          \item Relaxation of this perturbation enables to calibrate macroscopic model.
+      \end{itemize}
+  \end{itemize}
 
-\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 $L^2_0(\psi_0)$
+  We illustrate the first two for the simplest transport coefficient:
+  the {\blue mobility}.
+\end{frame}
 
-\medskip
+\begin{frame}
+    {Linear response of nonequilibrium dynamics}
+    Consider the nonequilibirium dynamics
+    \begin{align*}
+        \d q_t &= M^{-1} 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*}
 
-\bu Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L^* + \eta \wcL^*$
-\vspace{-0.2cm}
-\[
-\mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\wcL \mathcal L_0^{-1})^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0
-\]
+    \begin{itemize}
+        \item The force {\red $\eta F$} induces a non-zero velocity in the direction $F$
+        \item Encoded by $\dps \expect_\eta(R) = \int_{\mathcal E} R \, \psi_\eta$ with $\dps R(q,p) = F^\t  p$
+    \end{itemize}
 
-\bu {\red Prove that $f_\eta \geq 0$} (use some ergodicity result to show that $\psi_\eta = f_\eta \psi_0$)
+    \begin{definition}
+        [Mobility]
+        The mobility in direction $F$ is defined mathematically as
+        \[
+            \rho_{F} =
+            \lim_{\alert{\eta} \to 0} \frac{\expect_{\red \eta} [R] - \expect_{0} [R]}{\red \eta}
+            = \lim_{\eta \to 0} \frac{1}{\alert{\eta}}\expect_{\red \eta} [R]
+        \]
+    \end{definition}
 
+    We proved that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and
+    \[
+        f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathrm{O}(\eta^2), \qquad \mathfrak f_1 = - (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \mathbf 1.
+    \]
+    Therefore
+    \[
+        \rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0
+        = \int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L} \mathbf 1) \, \psi_0
+    \]
 \end{frame}
 
 
-\begin{frame}\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}
+\begin{frame}
+    {Reformulation as integrated correlation functions}
+    Define the conjugate response
+    \[
+        S
+        = \wcL^* \mathbf{1}
+        = \nabla_p^* (F \mathbf 1)
+        = F^\t p.
+    \]
 
-\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 $\psi_\eta(q,p)\, \d q\, \d p$,
+        while $\expect_0$ is w.r.t.\ initial conditions~$(q_0,p_0) \sim \psi_0$ and over all realizations of the equilibrium dynamics.
+    \end{block}
 
-\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
-\[
-    \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p  )}{\eta}
-  = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0)  \Big) d t
-\]
+    For the mobility,
+    it holds $S(q,p) = \beta R(q,p) = F^T p$ and so
+    \[
+        \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p  )}{\eta}
+        = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0)  \Big) d t
+    \]
 
 \end{frame}
 
@@ -542,71 +766,175 @@ Spectral radius $r$ of the bounded operator
 \end{frame}
 
 
-\begin{frame}\frametitle{Generalization to other dynamics}
+\begin{frame}
+    \begin{center}
+      \Large
+      \color{blue}
+      Part III: Computation of other transport coefficients
+    \end{center}
 
-\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}
+    \centering
+    \begin{minipage}{.6\textwidth}
+    \begin{itemize}
+        \item Thermal conductivity
+        \item Shear viscosity
+    \end{itemize}
+    \end{minipage}
+\end{frame}
 
-\bigskip
+\begin{frame}
+    {Thermal transport in one-dimensional chain (1)}
+    Consider a chain of $N$ atoms with nearest-neighbor interactions
+    \begin{tikzpicture}
+        \coordinate (origin) at (0,0);
+        \coordinate (shift) at (1.8,0);
+        \node [draw, color=red!60, fill=red!5, very thick, rectangle, minimum height=1cm] (nc) at (0,0) {$T_L$};
+        \node [draw, color=blue!60,  fill=blue!5,  very thick, rectangle, minimum height=1cm] (nh) at ($ (origin) + 6*(shift) $) {$T_R$};
+        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n1)  at ($ (origin) + 1*(shift) $)  {$p_1$};
+        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n2)  at ($ (origin) + 2*(shift) $)  {$p_2$};
+        \node [draw=none, circle, minimum size=1cm]                                    (n3)  at ($ (origin) + 3*(shift) $)  {$\dotsb$};
+        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n-2) at ($ (origin) + 4*(shift) $) {$p_{N-1}$};
+        \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n-1) at ($ (origin) + 5*(shift) $) {$p_{N}$};
+        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n1) -- node[below=.25cm]{$r_1$} (n2);
+        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n2) -- node[below=.25cm]{$r_2$} (n3);
+        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n3) -- node[below=.25cm]{$r_{N-2}$} (n-2);
+        \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n-2) -- node[below=.25cm]{$r_{N-1}$} (n-1);
+        \draw[red, ->] (nc) to [out=45,in=135] node[above]{$j_0$} (n1);
+        \draw[red, ->] (n1) to [out=45,in=135] node[above]{$j_1$} (n2);
+        \draw[red, ->] (n2) to [out=45,in=135] node[above]{$j_2$} (n3);
+        \draw[red, ->] (n3) to [out=45,in=135] node[above]{$j_{N-2}$} (n-2);
+        \draw[red, ->] (n-2) to [out=45,in=135] node[above]{$j_{N-1}$} (n-1);
+        \draw[red, ->] (n-2) to [out=45,in=135] node[above]{$j_{N-1}$} (n-1);
+        \draw[red, ->] (n-1) to [out=45,in=135] node[above]{$j_{N}$} (nh);
+    \end{tikzpicture}
+
+    Mathematical model:
+    \begin{equation*}
+        \left\{ \begin{aligned}
+                \d r_n &= (p_{n+1} - p_n) \, \d t, \\
+                \d p_1 &= v'(r_1) \, \d t - \gamma p_1 \dt + \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,
+        \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).
+\end{equation*}
+\end{frame}
 
-\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}
+\begin{frame}
+    {Thermal transport in one-dimensional chains (2)}
 
-\end{frame}
+    \begin{itemize}
+        \item
+            When ${\red \Delta T} = 0$,
+            invariant distribution given by
+            \[
+                \pi(\d r \, \d p) = Z_\beta^{-1} \exp\left(- \beta \left( \frac {|p|^2} {2} + V(r) \right)\right) \, \d r \, \d p,
+                \qquad \beta = T^{-1}.
+            \]
+
+        \item
+            Generator of the dynamics:
+            \begin{equation*}
+                \begin{aligned}
+                    \mathcal L
+            &= \sum_{n=1}^{N-1} (p_{n+1} - p_n) \partial_{r_n}
+            + \sum_{n=1}^N \Bigl(v'(r_n) - v'(r_{n-1})\Bigr) \partial_{p_n}  \\
+            &\qquad - \gamma p_1 \partial_{p_1} + \gamma T \partial_{p_1}^2 - \gamma p_N \partial_{p_N} + \gamma T \partial_{p_N}^2
+            + {\red \gamma \Delta T (\partial_{p_1}^2 - \partial_{p_N}^2)}.
+                \end{aligned}
+            \end{equation*}
+
+            The {\red perturbation} $\widetilde {\mathcal L} = \gamma( \partial_{p_1}^2 - \partial_{p_N}^2)$
+            is not bounded relatively to $\mathcal L_0$...
 
+            \vspace{.5cm}
+            $\rightarrow$ Existence/uniqueness of the invariant measure more difficult to prove\footnote{P. Carmona, \emph {Stoch. Proc. Appl.} (2007)}
+    \end{itemize}
+\end{frame}
 
 
 \begin{frame}
-    \begin{center}
-\Huge{Other examples}
-\end{center}
+    {Thermal transport in one-dimensional chains (3)}
+
+    \bu Response function: {\blue total energy current}
+    \begin{block}
+        {Definition of the heat flux}
+        \[
+            J = \frac{1}{N-1}\sum_{n=1}^{N-1} j_{n},
+            \qquad
+            j_{n} = -v'(r_n)\frac{p_n+p_{n+1}}{2}
+        \]
+    \end{block}
+    \smallskip
+
+    \bu Motivation: Local conservation of the energy (in the bulk $2 \leq n \leq N-1$)
+    \[
+        \frac{\d\varepsilon_n}{\d t} =
+        \mathcal L \varepsilon_n = j_{i-1} - j_{i},
+        \qquad
+        \varepsilon_n = \frac{p_n^2}{2} + \frac12 \Big( v(r_{i-1}) + v(r_n) \Big)
+    \]
+
+    \bu Definition of the {\blue thermal conductivity}: linear response
+    \[
+        \kappa_N = \lim_{\Delta T \to 0} \frac{(N-1)}{2\Delta T} \expect_{\Delta  t} [J].
+    \]
+
 \end{frame}
 
+\begin{frame}
+    {Shear viscosity in fluids (1)}
 
-\begin{frame}\frametitle{Shear viscosity in fluids (1)}
+    Consider a fluid $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$ subjected to a sinusoidal forcing
+    \begin{figure}
+        \centering
+        \includegraphics[height=.5\textwidth]{figures/osc_shear.eps}
+    \end{figure}
 
-\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}
+    Suppose that the box contains $N$ particles of mass $m$,
+    each subjected to a force $F$.
+\end{frame}
 
+\begin{frame}
+    {Shear viscosity in fluids (2)}
+    Macroscopic description by Navier--Stokes equation
+    \[
+        \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \eta \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x}
+    \]
+    Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives
+    \[
+        - \eta U_x''(y) = \overline{\rho} F(y)
+    \]
 \end{frame}
 
 
-\begin{frame}\frametitle{Shear viscosity in fluids (2)}
+\begin{frame}
+    {Shear viscosity in fluids (2)}
 
+    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_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,
+  - \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,
 \end{aligned} \right.
 }
 \end{block}
 
 \smallskip
 
-\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma_x,\gamma_y>0$
+\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma,\gamma>0$
 
 \smallskip
 
@@ -631,7 +959,7 @@ Spectral radius $r$ of the bounded operator
 
 \bu Average {\red longitudinal velocity}
   $u_x(Y) = \dps \lim_{\varepsilon \to 0}
-  \lim_{\eta \to 0} \frac{\left\langle U_x^\varepsilon(Y,\cdot)\right\rangle_\eta}{\eta}$
+  \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\cdot) \right]}{\eta}$
   where
   \vspace{-0.3cm}
   \[
@@ -650,7 +978,7 @@ Spectral radius $r$ of the bounded operator
   \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right)
   - \! \! \! \! \! \! \! \!
   \sum_{1 \leq i < j \leq N} \! \! \! \!
-  \mathcal{V}'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|}
+  v'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|}
   \!\int_{q_{yj}}^{q_{yi}} \!\chi_{\varepsilon}(s-Y) \, ds \right)
   \]
 
@@ -669,7 +997,7 @@ Spectral radius $r$ of the bounded operator
 \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)}
+\centerequation{-\eta u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)}
 \end{block}
 
 \bigskip
@@ -694,71 +1022,6 @@ Spectral radius $r$ of the bounded operator
 \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}
@@ -1273,40 +1536,6 @@ with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \d
 
 \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}
 
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