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\documentclass[9pt]{beamer}
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\title{Nonequilibrium systems and computation of transport coefficients\\[.3cm]
\small \textcolor{yellow}{SINEQ Summer school}%
}
\author{%
Urbain Vaes \texorpdfstring{\\\texttt{urbain.vaes@inria.fr}}{}
}
\institute{%
MATHERIALS -- Inria Paris
\textcolor{blue}{\&} CERMICS --
École des Ponts ParisTech
}
\date{\today}
\begin{document}
\begin{frame}[plain]
\begin{figure}[ht]
\centering
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\includegraphics[height=1.2cm]{figures/logo_inria.png}
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\end{figure}
\titlepage
\end{frame}
\begin{frame}
{Some references}
\begin{itemize}
\itemsep.2cm
\item \fullcite{MR3509213}
\item \fullcite{pavliotis2011applied}
\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}
At the \alert{macroscopic} level,
transport coefficients relate an external forcing to an average response expressed through some steady-state flux.
\textbf{Examples:}
\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;
\end{itemize}
\vspace{.3cm}
\textbf{Challenges we do not address:}
\begin{itemize}
\item Choose thermodynamical ensemble;
\item Prescribe microscopic dynamics;
\end{itemize}
\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}
\end{frame}
\section{Equilibrium and nonequilibrium dynamics}
\begin{frame}
{Equilibrium and nonequilibrium dynamics}
Consider a general diffusion process of the form
\[
\d x_t = b(x_t) \, \d t + \sigma(x_t) \, \d W_t,
\]
and assume that it admits an invariant distribution $\mu$.
\vspace{.2cm}
\begin{definition}
[Time-reversibility]
A stationary ($x_0 \sim \mu$) stochastic process $(x_t)$ is time-reversible if its law is invariant under time reversal:
the law of the \emph{forward paths} $(x_s)_{0 \leq s \leq t}$
coincides with the law of the \emph{backward paths} $(x_{t-s})_{0 \leq s \leq t}$.
\end{definition}
\vspace{.2cm}
\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}
\end{frame}
\begin{frame}
{Invariant distribution in dimension 1}
For the equilibrium overdamped Langevin dynamics
\[
\d q_t = - V'(q_t) \, \d t + \sqrt{2} \, \d W_t,
\]
the invariant probability distribution is given by~$Z^{-1} \e^{-V(q)} \, \d q$.
For the perturbed dynamics
\[
\d q_t = - V'(q_t) \, \d t + \blue{\eta} + \sqrt{2} \, \d W_t,
\]
the invariant probability distribution~$\rho_{\eta}$ solves the Fokker--Planck equation
\[
\frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0,
\]
which can be solved as
\[
\rho_{\eta}(q) \propto \int_{\torus} \e^{V(q+y) - V(q) - \eta y} \, \d y.
\]
\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.
\begin{figure}[ht]
\centering
\includegraphics[width=0.39\linewidth]{figures/intro_position.pdf}
\includegraphics[width=0.39\linewidth]{figures/intro_velocity.pdf}
\caption{%
Marginals of the steady state solution of the Langevin dynamics with forcing
}
\end{figure}
\end{frame}
\begin{frame}
{Existence of an invariant distribution}
\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,
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
\[
\forall \phi \in L^{\infty}_{\mathcal K_n}(\mathcal E), \qquad
\left\lVert \e^{t \mathcal L_n} \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}}
\leq C \e^{-\lambda t} \left\lVert \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}}
\]
\end{theorem}
\textbf{Idea of the proof.}
Show that
\begin{align*}
\mathcal L \mathcal K_n &\leq - c_1 \mathcal K_n(q, p) + c_2,
\end{align*}
for $c_1 > 0$ and $c_2 > 0$.
Then apply the main theorem from~\footfullcite{MR2857021}.
\end{frame}
\begin{frame}
{Existence of an invariant measure}
\[
d(P \mu, P \nu)
\leq
\]
\end{frame}
\begin{frame}
\end{frame}
\end{document}
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