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--- a/main.tex
+++ b/main.tex
@@ -88,6 +88,43 @@
\end{frame}
\section{Introduction}
+
+\begin{frame}
+ \frametitle{Introduction}
+ {\bf Aims of computational statistical physics}
+ \begin{itemize}
+ \item {\red numerical microscope}
+ \item computation of {\blue average properties}, static or dynamic
+ \end{itemize}
+ \begin{center}
+ \begin{minipage}[t]{.6\textwidth}
+ \begin{figure}[ht]
+ \centering
+ \resizebox{\textwidth}{!}{%
+ \begin{tikzpicture}
+ \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=\textwidth]{figures/nanotube.png}};
+ \node [draw, color=red!60, fill=red!5, very thick, rectangle, minimum height=1cm] (warm) at (0,1) {$T_+$};
+ \node [draw, color=blue!60, fill=blue!5, very thick, rectangle, minimum height=1cm] (cold) at (7.3,4.1) {$T_-$};
+ % \node [draw=none] (flux) at (5,2) {$J$};
+ \draw[->, line width=1mm] (3.5,0.9) to node[below]{$J$} ++(2.5,2);
+ \node [draw=none, minimum height=1cm] (unknown) at (0,4) {%
+ \begin{minipage}{4cm}
+ \centering
+ \textbf{Fourier's law:}%
+ \[
+ J = - \alert{\kappa} \grad T
+ \]
+ \end{minipage}
+ };
+ \end{tikzpicture}}
+ \end{figure}
+ \end{minipage}
+ \end{center}
+
+``Given the structure and the {\red laws of interaction} of the particles, what are the {\blue macroscopic properties} of the systems composed of these particles?''
+
+\end{frame}
+
\begin{frame}
{Transport coefficients}
At the \alert{macroscopic} level,
@@ -113,9 +150,11 @@
\item Equilibrium vs nonequilibrium dynamics;
\item Definition and computation of the mobility;
\item Computation of other transport coefficients.
+ \item Error analysis
\end{itemize}
\end{frame}
+
\begin{frame}
\begin{center}
\Large
@@ -161,7 +200,7 @@
\end{frame}
\begin{frame}
- {Examples of nonequilibrium dynamics}
+ {Paradigmatic examples of nonequilibrium dynamics}
\begin{block}{Overdamped Langevin dynamics perturbed by a constant force term}
\begin{equation}
\label{eq:overdamped_Langevin_F}
@@ -182,7 +221,7 @@
\end{aligned}
\right.
\end{equation}
- In the rest of the presentation we take {\blue $ M = \I$} for simplicity.
+ In the rest of this section, we take {\blue $ M = \I$} for simplicity.
\end{block}
where
\begin{itemize}
@@ -194,41 +233,14 @@
\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
\[
\mathcal L_{\rm ovd} \Big\vert_{\red \eta = 0} = - \grad V \cdot \grad + \laplacian
= - \grad^* \grad,
+ \qquad
+ \mu(\d q) = \frac{1}{Z} \e^{-V(q)} \, \d q.
\]
where $\grad^* := (\grad V - \grad) \cdot $.
For any $f, g \in C^{\infty}_{\rm c}(\mathcal E)$, we have
@@ -256,6 +268,35 @@
\end{frame}
\begin{frame}
+ {Another example useful for thermal transport}
+ \begin{block}{Langevin dynamics with modified fluctuation}
+ \[
+ \left\{
+ \begin{aligned}
+ \d q_t & = M^{-1} p_t \, \d t, \\*
+ \d p_t & = -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t \, \d t
+ + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t,
+ \end{aligned}
+ \right.
+ \]
+ \end{block}
+ with non-negative temperature
+ \[
+ T_\eta(q) = T_{\rm ref} + \eta \widetilde{T}(q)
+ \]
+ Typically, $\widetilde{T}$ constant and positive on $\mathcal D_+ \subset \mathcal C$,
+ and constant and negative on $\mathcal D_- \subset \mathcal D$.
+ \begin{itemize}
+ \item
+ Non-zero energy flux from $\mathcal D_+$ to $\mathcal D_-$ expected in the steady-state
+
+ \item
+
+ Simplified model of thermal transport (in 3D materials or atom chains)
+ \end{itemize}
+\end{frame}
+
+\begin{frame}
{Worked example in dimension one}
Consider the perturbed overdamped Langevin dynamics with~$q_t \in \torus$
\[
@@ -335,7 +376,7 @@
\emph{Sketch of proof.} Define the Markov transition kernel
\[
- \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder),
+ \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \pi(\placeholder),
\]
Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$.
We have
@@ -373,7 +414,7 @@
\[
L^{\infty}_{*} := \left\{ \phi \in L^{\infty}(\mathcal E) : \int_{\mathcal E} \phi \, \d \mu_{*} = 0 \right\}.
\]
- Thus $\I - \mathcal P$ is invertible and
+ Thus $\I - \mathcal P$ is invertible on~$L^{\infty}_{*}$ and
\[
(\I - \mathcal P)^{-1} = \I + \mathcal P + \mathcal P^2 + \dotsb
\]
@@ -404,7 +445,7 @@
and additionally ${\rm Law} (q_t)$ is smooth by parabolic regularity.
\item
\textbf{Decay of the semigroup}:
- For $t \in [0, \infty)$ and bounded $\varphi$, it holds that
+ For $t \in [0, \infty)$ and $\varphi \in L^{\infty}_*$, it holds that
\begin{align*}
\lVert \e^{t \mathcal L_{\rm ovd}} \varphi \rVert_{L^{\infty}}
&= \left\lVert \e^{(t- \lfloor t \rfloor) \mathcal L_{\rm ovd}} \left( \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi \right) \right\rVert_{L^{\infty}} \\
@@ -413,7 +454,7 @@
\end{align*}
\item
- \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{\eta}$,
+ \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{*}$,
and
\[
\mathcal L_{\rm ovd}^{-1}
@@ -527,7 +568,7 @@
\[
\mathcal L_\eta^* f_\eta = 0
\]
- Observe that
+ Observe that $\mathcal L_{\eta}^* = \mathcal L_0^* + \widetilde {\mathcal L}^*$ with
\[
\mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p - \gamma \grad_p^* \grad_p^*,
\qquad \widetilde {\mathcal L}^* \placeholder = \grad_p^* (F \placeholder)
@@ -549,8 +590,8 @@
\mathcal L_{\eta}^* f_{\eta}
&= \mathcal L_0^* \mathfrak f_0 \\
&\quad + \eta \left(\widetilde {\mathcal L}^* \mathfrak f_0 + \mathcal L_0^* \mathfrak f_1\right) \\
- &\quad + \eta^2 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_2\right) \\
- &\quad + \eta^3 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_2\right) + \dotsb
+ &\quad + \eta^2 \left(\widetilde {\mathcal L}^* \mathfrak f_1 + \mathcal L_0^* \mathfrak f_2\right) \\
+ &\quad + \eta^3 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_3\right) + \dotsb
\end{align*}
This suggests that $\mathfrak f_{i+1} = -(\mathcal L_0^*)^{-1} (\widetilde {\mathcal L}^* \mathfrak f_i)$ and so
\[
@@ -640,11 +681,11 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
\end{center}
\centering
- \begin{minipage}{.8\textwidth}
+ \begin{minipage}{.6\textwidth}
\begin{itemize}
\item Definition through linear response
\item Green--Kubo reformulation
- \item Numerical approximation
+ \item Link with effective diffusion
\end{itemize}
\end{minipage}
\end{frame}
@@ -709,10 +750,26 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
Therefore
\[
\rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0
- = \int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L} \mathbf 1) \, \psi_0
+ = -\int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L}^* \mathbf 1) \, \psi_0
\]
\end{frame}
+\begin{frame}
+ {Numerical results (1)}
+ \begin{figure}
+ \centering
+ \includegraphics[width=.75\textwidth]{figures/LR.eps}
+ \end{figure}
+\end{frame}
+
+\begin{frame}
+ {Numerical results (2)}
+ \begin{figure}
+ \centering
+ \includegraphics[width=.75\textwidth]{figures/mobilityFctGamma.pdf}
+ \caption{Mobility as a function of~$\gamma$~\footnote{See J.~Roussel and G.~Stoltz, \emph{ESAIM: M2AN} (2018)}}
+ \end{figure}
+\end{frame}
\begin{frame}
{Reformulation as integrated correlation functions}
@@ -734,37 +791,120 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
\end{block}
For the mobility,
- it holds $S(q,p) = \beta R(q,p) = F^T p$ and so
+ it holds $S(q,p) = R(q,p) = F^\t p$ and so
\[
- \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p )}{\eta}
- = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0) \Big) d t
+ \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta \bigl(F^\t p \bigr)}{\eta}
+ = \int_0^{+\infty} \expect_0 \Big( \bigl(F^\t p_t\bigr) \bigl(F^\t p_0\bigr) \Big) \, \d t
\]
-
\end{frame}
+\begin{frame}
+ {Elements of proof}
-\begin{frame}\frametitle{Elements of proof}
-
-\bu Proof based on the following equality on $\mathcal{B}(L_0^2(\psi_0))$
+\bu Proof based on the following equality on $\mathcal{B}\bigl(L_0^2(\psi_0)\bigr)$
\[
--\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, d t
+-\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, \d t.
\]
\bu Then,
\begin{align*}
\lim_{\eta \to 0} \frac{\expect_\eta(R)}{\eta} & = -\int_{\mathcal E} R \left[(\wcL \mathcal L_0^{-1})^* \mathbf{1}\right] \psi_0
-= -\int_{\mathcal{E}} [\mathcal L_0^{-1}R ] [\wcL^* \mathbf{1} ] \, \psi_0 \notag \\*
-& = \int_0^{+\infty} \left( \int_{\mathcal{E}} \left(\mathrm{e}^{t \mathcal L_0} R\right) \, S \, \psi_0\right)dt \notag \\
-& = \int_0^{+\infty} \expect \Big( R(q_t,p_t)S(q_0,p_0) \Big) d t
+= -\int_{\mathcal{E}} [\mathcal L_0^{-1}R ] [\wcL^* \mathbf{1} ] \, \psi_0 \\
+& = \int_0^{+\infty} \left( \int_{\mathcal{E}} \left(\mathrm{e}^{t \mathcal L_0} R\right) \, S \, \psi_0\right) \d t \\
+& = \int_0^{+\infty} \expect \Big( R(q_t,p_t)S(q_0,p_0) \Big) \, \d t
\end{align*}
-\bu Note also that $S$ has average 0 w.r.t. invariant measure since
+\bu Note also that $S$ has average 0 w.r.t.\ invariant measure since
\[
-\int_\cX S \, d\pi = \int_\cX \wcL^* \mathbf{1} \, d\pi = \int_\cX \wcL\mathbf{1} \, d\pi = 0
+\int_\cX S \, \d\pi = \int_\cX \wcL^* \mathbf{1} \, \d\pi = \int_\cX \wcL\mathbf{1} \, \d\pi = 0
\]
\end{frame}
+\begin{frame}
+ {Connection with effective diffusion}
+ It is possible to show a {\blue functional central limit theorem} for the Langevin dynamics:
+ \begin{equation*}
+ \varepsilon \widetilde {q}_{s/\varepsilon^2} \xrightarrow[\varepsilon \to 0]{} \sqrt{2 \mat D} \, W_s
+ \qquad \text{weakly on } C([0, \infty)), \qquad \widetilde {q}_t := q_0 + \int_{0}^{t} p_s \, \d s \in {\blue \real^{d}}.
+ \end{equation*}
+ In particular, $\widetilde {q}_t /\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ weakly.
+
+ \vspace{-.25cm}
+ \begin{figure}[ht]
+ \centering
+ \href{run:videos/gle/effective-diffusion.webm?autostart&loop}%
+ {\includegraphics[width=0.75\textwidth]{videos/gle/effective-diffusion.png}}%
+ \caption{Histogram of $q_t/\sqrt{t}$. The potential $V(q) = - \cos(q) / 2$ is illustrated in the background.}
+ \end{figure}
+\end{frame}
+
+\begin{frame}
+ {Mathematical expression for the effective diffusion (dimension 1)}
+ \vspace{.2cm}
+ \begin{block}{Expression of $D$ in terms of the solution to a Poisson equation}
+ Effective diffusion tensor given by $D = \ip{\phi}{p}_{L^2(\mu)}$ and $\phi$ is the solution to
+ \[
+ - \mathcal L \phi = p,
+ \qquad \phi \in L^2_0(\mu).
+ \]
+ \end{block}
+ \textbf{Key idea of the proof:} Apply It\^o's formula to $\phi$
+ \begin{align*}
+ \d \phi(q_s, p_s)
+ % &= \frac{1}{\varepsilon^2} \mathcal L_{L} \phi (q_t, p_t) + \frac{1}{\varepsilon} \, \sqrt{2 \gamma \beta^{-1}} \, \derivative{1}[\phi]{p}(q_t, p_t) \, \d W_t, \\
+ &= - p_s \, \d s + \sqrt{2} \, \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s
+ \end{align*}
+ and then rearrange:
+ \begin{align*}
+ \alert\varepsilon (\widetilde q_{t/\alert\varepsilon^2} - \widetilde q_{0}) &= \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} p_s \, \d s \\
+ &= \underbrace{\alert\varepsilon \bigl(\phi(q_0, p_0) - \phi(q_{t/\alert\varepsilon^2}, p_{t/\alert\varepsilon^2})\bigr)}_{\to 0
+ % ~\text{in $L^p(\Omega, C([0, T], \real))$}
+ }
+ + \underbrace{\sqrt{2 \gamma} \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s}_{\to \sqrt{2 D} W_t~\text{weakly by MCLT}}.
+ \end{align*}
+ % where
+ % \begin{align*}
+ % D &= \gamma \beta^{-1} \, \int \abs{\textstyle \derivative{1}[\phi]{p}(q, p)}^2 \, \mu(\d q \, \d p)
+ % = - \int \phi (\mathcal L \phi) \, \d \mu
+ % = \ip{\phi}{p}.
+ % \end{align*}
+
+ \textbf{In the multidimensional setting}, $D_{F} = \ip{\phi_{F}}{F^\t p}$ with $- \mathcal L \phi_{F} = F^\t p$.
+
+ \textbf{Einstein's relation:} we just showed
+ \(
+ D_F = \beta^{-1} \rho_F.
+ \)
+\end{frame}
+
+
+\begin{frame}
+ {Summary: numerical approaches for calculating the mobility}
+ \begin{itemize}
+ \itemsep.5cm
+ \item {\blue Linear response approach}:
+ \begin{equation*}
+ \rho_F = \lim_{\eta \to 0} \frac{1}{\alert{\eta}} \expect_{\alert{\eta}} \, \bigl[F^\t p\bigr].
+ \end{equation*}
+ where $\mu_{\eta}$ is the invariant distribution of the system with external forcing.
+
+ \item {\blue Einstein's relation}:
+ \[
+ \rho_F = \lim_{t \to \infty} \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl| F^\t (\widetilde {q}_t - q_0)\bigr|^2 \Bigr].
+ \]
+
+ \item Deterministic method, e.g. {\blue Fourier/Hermite Galerkin}, for the Poisson equation
+ \[
+ - \mathcal L_0 \phi_{F} = F^\t p, \qquad \rho_F = \ip{\phi_F}{F^\t p}.
+ \]
+
+ \item {\blue Green--Kubo formula}:
+ \begin{align*}
+ \rho_F &= \int_{0}^{\infty} \expect_{\blue 0}\bigl((F^\t p_0) (F^\t p_t)\bigr) \, \d t.
+ \end{align*}
+ \end{itemize}
+\end{frame}
\begin{frame}
\begin{center}
@@ -820,7 +960,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$.
The Hamiltonian of the system is the sum of the potential and kinetic energies:
\begin{equation*}
- H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2},
+ H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2},
\quad V(r) = \sum_{n=1}^{N-1} v(r_n).
\end{equation*}
\end{frame}
@@ -904,57 +1044,62 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
{Shear viscosity in fluids (2)}
Macroscopic description by Navier--Stokes equation
\[
- \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \eta \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x}
+ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \nu \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x}
\]
Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives
\[
- - \eta U_x''(y) = \overline{\rho} F(y)
+ - \nu U_x''(y) = \overline{\rho} F(y), \qquad \overline \rho := \frac{\rho}{m} = \frac{N}{|\mathcal D|}
+ \]
+ Therefore, for the test function~$g(y) = \e^{2i\pi \frac{y}{L_y}}$
+ \[
+ \nu \int_0^{L_y} U_x(y) g''(y) \, \d y = \overline{\rho} \int_{0}^{L_y} F(y) g(y) \, \d y
+ \]
+
+ $\rightarrow$ Suggests estimating the shear viscosity from molecular dynamics as
+ \[
+ \nu = \frac{\dps \frac{\overline{\rho}}{L_y}\int_{0}^{L_y} F(y) g(y) \, \d y}
+ {\dps \expect_{F} \left[ \frac{1}{N}\sum_{n=1}^{N} \frac{p_{xi}}{m} g''(q_{yi}) \right]}.
\]
\end{frame}
\begin{frame}
{Shear viscosity in fluids (2)}
-
- pairwise interactions
+ Assume pairwise interactions
\[
V(q) = \sum_{1 \leq i < j \leq N} \mathcal V(\abs{q_i - q_j}).
\]
\bu Add a smooth {\blue nongradient force} in the $x$ direction, depending on~$y$
\begin{block}{Langevin dynamics under flow}
\centerequation{\left \{ \begin{aligned}
- d q_{i,t} &= \frac{p_{i,t}}{m} \, dt,\\
- d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, dt + {\red \eta F(q_{yi,t}) \, dt}
- - \gamma \frac{p_{xi,t}}{m} \, dt + \sqrt{\frac{2\gamma}{\beta}} \, dW^{xi}_t, \\
- d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, dt - \gamma \frac{p_{yi,t}}{m} \, dt
- + \sqrt{\frac{2\gamma}{\beta}} \, dW^{yi}_t,
+ \d q_{i,t} &= \frac{p_{i,t}}{m} \, \d t,\\
+ \d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, \d t + {\red \eta F(q_{yi,t}) \, \d t}
+ - \gamma \frac{p_{xi,t}}{m} \, \d t + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{xi}_t, \\
+ \d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, \d t - \gamma \frac{p_{yi,t}}{m} \, \d t
+ + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{yi}_t.
\end{aligned} \right.
}
\end{block}
\smallskip
-\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma,\gamma>0$
+\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$
\smallskip
\bu The perturbation $\dps \wcL = \sum_{i=1}^N \! F(q_{y,i}) \partial_{p_{x,i}}$ is $\mathcal{L}_0$-bounded
-
-\smallskip
-
-\bu {\blue Linear response}:
- $\dps
- \lim_{\eta \rightarrow 0} \frac{\left\langle \mathcal L_0 h \right\rangle_\eta}{\eta}
- = - \frac{\beta}{m} \!
- \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)}
- $
-\medskip
-
\end{frame}
-\begin{frame}\frametitle{Shear viscosity in fluids (3)}
+\begin{frame}
+ {Shear viscosity in fluids (3)}
+\bu {\blue Linear response}:
+\[
+ \lim_{\eta \rightarrow 0} \frac{\expect_{\eta} [\mathcal L_0 h]}{\eta}
+ = - \frac{\beta}{m} \!
+ \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)}.
+\]
\bu Average {\red longitudinal velocity}
@@ -970,11 +1115,12 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
\bu Average {\red off-diagonal stress}
$\dps \sigma_{xy}(Y) = \lim_{\varepsilon \to 0}
- \lim_{\eta \to 0} \frac{\left\langle ... \right\rangle_\eta}{\eta}$,
- where $... =$
+ \lim_{\eta \to 0} \frac{\expect_{\eta} [...]}{\eta}$,
+ where
\vspace{-0.4cm}
\[
\hspace{-0.1cm}
+ ... =
\frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right)
- \! \! \! \! \! \! \! \!
\sum_{1 \leq i < j \leq N} \! \! \! \!
@@ -984,15 +1130,12 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
\bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ (with $\overline{\rho} = N/|\mathcal{D}|$)
\[
-\frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y)
+ \frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y)
\]
-
\end{frame}
-
-
\begin{frame}
-\frametitle{Shear viscosity in fluids (4)}
+ {Shear viscosity in fluids (4)}
\bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$
@@ -1000,62 +1143,61 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
\centerequation{-\eta u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)}
\end{block}
-\bigskip
-
-\hspace{-0.5cm}
-\begin{minipage}{6cm}
-\psfrag{F}{{\scriptsize $F$}}
-\psfrag{U}{{\scriptsize $u$}}
-\psfrag{Y}{{\scriptsize $\ \ Y$}}
-\psfrag{v}{{\scriptsize value}}
-\includegraphics[width=6cm]{figures/ux5.eps}
-\end{minipage}
-\hspace{-0.5cm}
-\begin{minipage}{6cm}
-\psfrag{Y}{}
-\psfrag{v}{{\scriptsize value}}
-\psfrag{S}{{\scriptsize $\sigma_{xy}$}}
-\psfrag{D}{{\scriptsize $-\nu u'$}}
-\includegraphics[width=6cm]{figures/dux5.eps}
-\end{minipage}
-
+\begin{figure}[ht]
+ \centering
+ \includegraphics[width=\linewidth]{figures/shear1.png}
+\end{figure}
\end{frame}
-
+% \begin{frame}
+% \end{frame}
\begin{frame}
\begin{center}
-\Huge{Error estimates on} \\
-\bigskip
-\Huge{the computation of} \\
-\bigskip
-\Huge{transport coefficients}
-\end{center}
+ \Large
+ \color{blue}
+ Part IV: Error estimates on the estimation of transport coefficients
+ \end{center}
+
+ \centering
+ \begin{minipage}{.8\textwidth}
+ \begin{itemize}
+ \item Reminders: strong order, weak order
+ \item Error analysis for the linear response method
+ \item Error analysis for Green--Kubo method
+ \end{itemize}
+ \end{minipage}
\end{frame}
-\begin{frame}\frametitle{Reminder: Error estimates in Monte Carlo simulations}
+\begin{frame}
+ {Reminder: Error estimates in Monte Carlo simulations}
-\bu General SDE $dx_t = b(x_t)\,dt + \sigma(x_t) \, dW_t$, invariant measure $\pi$
+Consider the general SDE
+\[
+ \d x_t = b(x_t)\,\d t + \sigma(x_t) \, \d W_t
+\]
+with invariant measure $\pi$.
\bigskip
\bu {\red Discretization} $x^{n} \simeq x_{n\dt}$, {\blue invariant measure $\pi_\dt$}. For instance,
\[
-x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \sim \mathcal{G}(0,{\rm Id}) \ \mathrm{i.i.d.}
+x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \stackrel{\rm{i.i.d.}}{\sim} \mathcal N(0,{\rm Id})
\]
\medskip
\bu {\blue Ergodicity} of the numerical scheme with invariant measure~$\pi_\dt$
\[
-\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(dx)
+\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(\d x)
\]
\begin{block}{Error estimates for {\red finite} trajectory averages}
\[
-\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) = \expect_\pi(A) + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error}
+\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n)
+= \expect_\pi(A) + \underbrace{\frac{C}{N_{\rm iter} \dt}}_{\rm bias} + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error}
\]
\end{block}
@@ -1076,105 +1218,127 @@ P_\dt \varphi(x) = \expect\Big( \varphi\left(x^{n+1}\right)\Big| x^n = x\Big)
\]
where $(x^n)$ is an approximation of $(x_{n \dt})$
-\bigskip
-
-\bu (Infinitely) Many possibilities! Numerical analysis allows to {\blue discriminate}
-
\medskip
\bu Standard notions of error: {\red fixed integration time $T < +\infty$}
\begin{itemize}
-\item {\blue Strong error} $\dps \sup_{0 \leq n \leq T/\dt} \expect | X^n - X_{n\dt} | \leq C \dt^p$
-\item {\blue Weak error}: $\dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(X^n\right)\right] - \expect\left[\varphi\left(X_{n\dt}\right)\right] \Big| \leq C \dt^p$ (for any $\varphi$)
+\item {\blue Strong error}:
+ \[
+ \dps \sup_{0 \leq n \leq T/\dt} \expect | x^n - x_{n\dt} | \leq C \dt^p
+ \]
+\item {\blue Weak error}: for any $\varphi$,
+ \[
+ \dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(x^n\right)\right] - \expect\left[\varphi\left(x_{n\dt}\right)\right] \Big| \leq C \dt^p
+ \]
%\item ``mean error'' \emph{vs.} ``error of the mean''
\end{itemize}
-%\medskip
-%\bu Example: for Euler-Maruyama, weak order~1, strong order $1/2$ (1 when $\sigma$ constant)
-%\medskip
-
\begin{block}{$\dt$-expansion of the evolution operator}
-\centerequation{P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}}
+ \[
+ P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}
+ \]
\end{block}
\smallskip
-\bu {\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$
+{\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$.
\end{frame}
+\begin{frame}
+ {Elements of proof}
+ \begin{itemize}
+ \item
+ Since $u(t, x) := \e^{t \mathcal L} \varphi(x)$ solves the backward Kolmogorov equation
+ \begin{align*}
+ \partial_t u = \mathcal L u,
+ \qquad u(0, x) = \varphi.
+ \end{align*}
+ we can write formally
+ \[
+ \e^{\dt \mathcal L} \varphi = \I + \dt \mathcal L \varphi + \frac{\dt^2}{2} \mathcal L^2\varphi + \dotsb
+ \]
+ \item
+ Introduce a telescopic sum
+ \begin{align*}
+ \expect \bigl[\varphi(x^N)\bigr] - \expect \bigl[\varphi(x_{N \dt})\bigr]
+ &= P_{\dt}^N \varphi (x_0) - \e^{N \dt \mathcal L} \varphi(x_0) \\
+ &= \sum_{n=0}^{N-1} \left( P_{\dt}^{N-n} \e^{n \dt \mathcal L} \varphi(x_0) - P_{\dt}^{N-(n+1)} \e^{(n+1) \dt \mathcal L} \varphi (x_0) \right) \\
+ &= \sum_{n=0}^{N-1} P_{\dt}^{N-(n+1)} \left( P_{\dt} - \e^{\dt \mathcal L} \right) \e^{n \dt \mathcal L} \varphi (x_0)
+ \end{align*}
+ \end{itemize}
+\end{frame}
-\begin{frame}\frametitle{Example: Euler-Maruyama, weak order~1}
-
-\medskip
-\bu Scheme $x^{n+1} = \Phi_\dt(x^n,G^n) = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n$
-
-\bigskip
+\begin{frame}
+ {Example: Euler-Maruyama, weak order~1}
+ Consider the scheme
+ \[
+ x^{n+1} = \Phi_\dt(x^n,G^n) = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n
+ \]
-\bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$
+ \bigskip
-\bigskip
+ \bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$
-\bu Technical tool: {\blue Taylor expansion}
-\vspace{-0.2cm}
-\[
-\varphi(x + \delta) = \varphi(x) + \delta^T \nabla \varphi(x) + \frac12 \delta^T \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots
-\]
+ \bigskip
-\medskip
+ \bu Technical tool: {\blue Taylor expansion}
+ \vspace{-0.2cm}
+ \[
+ \varphi(x + \delta) = \varphi(x) + \delta^\t \nabla \varphi(x) + \frac12 \delta^\t \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots
+ \]
-\bu Replace $\delta$ with $\sqrt{\dt}\, \sigma(x)\,G + \dt\,b(x)$ and {\blue gather in powers of $\dt$}
-\[
-\begin{aligned}
-\varphi\big(\Phi_\dt(x,G)\big) & = \varphi(x) + \sqrt{\dt}\, \sigma(x)\,G \cdot \nabla \varphi(x) \\
-& \ \ \ + \dt \left(\frac{\sigma(x)^2}{2} G^T \left[\nabla^2\varphi(x)\right]G + b(x)\cdot\nabla \varphi(x) \right) + \dots
-\end{aligned}
-\]
+ \medskip
-\medskip
+ \bu Replace $\delta$ with $\sqrt{\dt}\, \sigma(x)\,G + \dt\,b(x)$ and {\blue gather in powers of $\dt$}
+ \[
+ \begin{aligned}
+ \varphi\big(\Phi_\dt(x,G)\big) & = \varphi(x) + \sqrt{\dt}\, \sigma(x)\,G \cdot \nabla \varphi(x) \\
+ & \ \ \ + \dt \left(\frac{\sigma(x)^2}{2} G^\t \left[\nabla^2\varphi(x)\right]G + b(x)\cdot\nabla \varphi(x) \right) + \dots
+ \end{aligned}
+ \]
-\bu Taking {\blue expectations w.r.t. $G$} leads to
-\[
-P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathrm{O}(\dt^2)
-\]
+ \medskip
+ \bu Taking {\blue expectations w.r.t. $G$} leads to
+ \[
+ P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathcal O(\dt^2)
+ \]
\end{frame}
-
-\begin{frame}\frametitle{Error estimates on the invariant measure (equilibrium)}
+\begin{frame}
+ {Error estimates on the invariant measure (equilibrium)}
\bu {\red Assumptions} on the operators in the weak-type expansion
-\begin{itemize}
-\item invariance of $\pi$ by $\mathcal A_k$ for $1 \leq k \leq p$, namely
-$\dps \int_\cX \mathcal A_k \varphi \, d\pi = 0$
-\item $\dps \int_\cX \mathcal A_{p+1}\varphi \, d\pi = \int_\cX g_{p+1} \varphi \, d\pi$
-(\textit{i.e.} $g_{p+1} = \mathcal A_{p+1}^* \mathbf{1}$)
-\end{itemize}
\begin{block}{Error estimates on $\pi_\dt$}
-\centerequation{
-\int_\cX \varphi \, d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) d\pi + \dt^{p+1} R_{\varphi,\dt}
-}
+ Suppose that
+ \begin{itemize}
+ \item
+ For all smooth $\varphi$, the following expansion holds
+ \[
+ P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}
+ \]
+ \item The probability measure $\pi$ is invariant by $\mathcal A_k$ for $1 \leq k \leq p$, namely
+ \[
+ \int_\cX \mathcal A_k \varphi \, d\pi = 0
+ \]
+ \end{itemize}
+ Then
+ \[
+ \int_\cX \varphi \, d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) d\pi + \dt^{p+1} R_{\varphi,\dt},
+ \]
+ where $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$.
\end{block}
-\medskip
-
-\bu In fact, $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$
-\begin{itemize}
-\item when $\mathcal A_1 = \mathcal L$, the first order correction can be {\red estimated} by some integrated correlation function as $\dps \int_0^{+\infty} \expect\Big(\varphi(x_t)g_{p+1}(x_0)\Big) \, dt$
-\item in general, first order term can be removed by Romberg extrapolation
-\end{itemize}
-
-\medskip
-
-\bu Error on invariant measure can be {\blue (much) smaller} than the weak error
+Error on invariant measure can be {\blue (much) smaller} than the weak error
\end{frame}
-%-----------------------------------------------------------
-\begin{frame}\frametitle{Sketch of proof (1)}
+\begin{frame}
+ {Motivation of the result}
-{\bf Step~1: Establish the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$}
+We verify the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$.
\medskip
@@ -1190,8 +1354,8 @@ $\dps \int_\cX \mathcal A_k \varphi \, d\pi = 0$
\[
\begin{aligned}
& \int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right)\psi\right] (1+\dt^{p}f_{p+1})\, d\pi \\
-& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathrm{O}\left(\dt^{p+1}\right) \\
-& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathrm{O}\left(\dt^{p+1}\right)
+& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathcal O\left(\dt^{p+1}\right) \\
+& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathcal O\left(\dt^{p+1}\right)
\end{aligned}
\]
@@ -1201,40 +1365,50 @@ Suggests $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$
\end{frame}
-%-----------------------------------------------------------
-\begin{frame}\frametitle{Sketch of proof (2)}
+\begin{frame}
+ {Numerical estimators and associated challenges}
+ \begin{itemize}
+ \item
+ Estimator of linear response (observable~$R$ with equilibrium average~0)
+ \[
+ \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, ds \xrightarrow[t\to+\infty]{\mathrm{a.s.}}
+ \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, d\mu = \alpha + \mathcal O(\eta)
+ \]
+ {\bf Issues with linear response methods:}
+ \begin{itemize}
+ \item Statistical error with {\red asymptotic variance $\mathcal O(\eta^{-2})$}
+ \item Bias $\mathcal O(\eta)$ due to $\eta \neq 0$
+ \item Bias from finite integration time
+ \end{itemize}
-{\bf Step~2: Define an approximate inverse}
+ \end{itemize}
+\end{frame}
-\medskip
+\begin{frame}\frametitle{Analysis of variance / finite integration time bias}
-\bu Issue: derivatives of $(\I-P_\dt)^{-1}\varphi$ are not controlled
+ \bu {\bf Statistical error} dictated by {\blue Central Limit Theorem}:
+ \[
+ \sqrt{t} \left(\widehat{A}_{\eta,t} - \alpha_\eta \right) \xrightarrow[t \to +\infty]{\mathrm{law}} \mathcal{N}\left(0,\frac{\sigma_{R,\eta}^2}{\eta^2}\right),
+ \qquad
+ \sigma_{R,\eta}^2 = \sigma_{R,0}^2 + \mathcal O(\eta)
+ \]
+ so $\dps \widehat{A}_{\eta,t} = \alpha_\eta + \mathcal O_{\rm P}\left(\frac{1}{\eta \sqrt{t}}\right)$ $\to$ requires {\red long simulation times} $t \sim \eta^{-2}$
+
+ \bigskip
+
+ \bu {\bf Finite time integration bias}: $\dps \left| \mathbb{E}\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\
+ Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error}
-\bigskip
+%\bigskip
+ %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$
-\bu Consider $\dps \left(\Pi \frac{P_\dt-\I}{\dt} \Pi\right) Q_\dt\psi = \psi + \dt^{p+1} \widetilde{r}_{\psi,\dt}$ where
-\vspace{-0.2cm}
-\[
-\Pi \varphi = \varphi - \int_\cX \varphi \, d\pi
-\]
+\bigskip
-\bu Idea of the construction: truncate the formal series expression
+\bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, d\mu$
\[
-(A + \dt \, B)^{-1} = A^{-1} - \dt \, A^{-1}B A^{-1} + \dt^{2} \, A^{-1}B A^{-1}B A^{-1} + \dots
+\widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T dW_s
\]
-\bigskip
-
-{\bf Step~3: Conclusion}
-
-\medskip
-
-\bu Write the invariances with $\dps \Pi \left(\frac{P_\dt-\I}{\dt}\right) \Pi \psi$ instead of $\dps \left(\frac{P_\dt-\I}{\dt}\right) \psi$
-
-\medskip
-
-\bu Replace $\psi$ by $Q_\dt \varphi$, and gather in~$R_{\varphi,\dt}$ all the higher order terms
-
\end{frame}
@@ -1256,12 +1430,12 @@ C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p
\bu Trotter splitting $\to$ weak order 1
\[
-P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^2)
+P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathcal O(\dt^2)
\]
\bu Strang splitting $\to$ {\blue weak order 2}
\[
-P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^3)
+P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathcal O(\dt^3)
\]
\bu Other category: {\red Geometric Langevin}\footnote{N.~Bou-Rabee and H.~Owhadi, {\em SIAM J. Numer. Anal.} (2010)} algorithms, \textit{e.g.} $P_\dt^{\gamma C,A,B_\eta,A}$ \\
@@ -1323,8 +1497,8 @@ where the remainder is compatible with linear response
\bu Corollary: error estimates on the {\blue numerically computed mobility}
\[
\begin{aligned}
-\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\
-& = \rho_{F} + \dt^\alpha \int_{\mathcal E} F^T M^{-1} p \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt}
+\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\
+& = \rho_{F} + \dt^\alpha \int_{\mathcal E} F^\t M^{-1} p \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt}
\end{aligned}
\]
@@ -1339,17 +1513,11 @@ where the remainder is compatible with linear response
\begin{figure}
\begin{center}
-\includegraphics[width=6.2cm]{figures/LR.eps}
-\includegraphics[width=6.2cm]{figures/mobility_Langevin.eps}
+\includegraphics[width=.8\textwidth]{figures/mobility_Langevin.eps}
\end{center}
\end{figure}
-\small
-{\bf Left:} Linear response of the average velocity as a function of $\eta$ for the scheme associated with $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$ and $\dt = 0.01, \gamma = 1$. \\
-
-\smallskip
-
-{\bf Right:} Scaling of the mobility $\nu_{F,\gamma,\dt}$ for the first order scheme $P_\dt^{A,B_\eta,\gamma C}$ and the second order scheme $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$.
+Scaling of the mobility for the first order scheme $P_\dt^{A,B_\eta,\gamma C}$ and the second order scheme $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$.
\end{frame}
@@ -1490,20 +1658,6 @@ with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \d
\end{frame}
-\begin{frame}\frametitle{Numerical results}
-
-\vspace{-0.5cm}
-\begin{figure}
-\begin{center}
-\includegraphics[width=11.8cm]{figures/error_diffusion.eps}
-%\includegraphics[width=8.2cm]{figures/error_diffusion_zoom.eps}
-\end{center}
-\end{figure}
-
-\end{frame}
-
-
-
\begin{frame}
\begin{center}
\Huge{Conclusion and perspectives}
generated by cgit v1.2.3 (git 2.45.2) at 2024-11-21 13:05:30 +0000