summaryrefslogtreecommitdiff
path: root/main.tex
diff options
context:
space:
mode:
Diffstat (limited to 'main.tex')
-rwxr-xr-xmain.tex839
1 files changed, 534 insertions, 305 deletions
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}
% vim: ts=4 sw=4
generated by cgit v1.2.3 (git 2.45.2) at 2024-11-21 13:09:10 +0000