From e64a03583d84c663f302e53886783179496ae517 Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Sat, 16 Sep 2023 19:00:39 +0200
Subject: Add slide

---
 header.tex |  2 +-
 main.tex   | 13 +++++++++++++
 2 files changed, 14 insertions(+), 1 deletion(-)

diff --git a/header.tex b/header.tex
index 2fa0856..98eef2d 100755
--- a/header.tex
+++ b/header.tex
@@ -65,7 +65,7 @@
 
 % Layout
 % \usefonttheme[onlymath]{serif}
-\usefonttheme[stillsansseriflarge]{serif}
+\usefonttheme[stillsansseriflarge,stillsansserifsmall]{serif}
 % \useinnertheme{rounded}
 \usecolortheme{seahorse}
 \usecolortheme{orchid}
diff --git a/main.tex b/main.tex
index 4ad8575..04a91b6 100755
--- a/main.tex
+++ b/main.tex
@@ -58,6 +58,7 @@
 \begin{frame}
   {Some references}
   \begin{itemize}
+    \itemsep.2cm
     \item \fullcite{MR3509213}
     \item \fullcite{pavliotis2011applied}
     \item Lecture notes by Gabriel Stoltz  on computational statistical physics:
@@ -108,6 +109,7 @@
       \[
         \rho = \int_{0}^{\infty} \expect_{\mu} \bigl[\varphi(x_t) \phi(x_0)\bigr] \, \d t.
       \]
+      We will derive this formula from linear response.
     \item
       Transient techniques:
   \end{itemize}
@@ -136,6 +138,17 @@
   \end{theorem}
 \end{frame}
 
+\begin{frame}
+  {Existence of an invariant measure for noneq.\ 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,
+  \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.
+
+\end{frame}
+
 \end{document}
 
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