\documentclass[9pt]{beamer} \newif\iflong \longfalse \usepackage{psfrag} \newcommand{\placeholder}{\mathord{\color{black!33}\bullet}}% \newcommand{\bu}{$\bullet \ $} \newcommand{\bi}{\begin{itemize}} \newcommand{\ei}{\end{itemize}} \renewcommand{\leq}{\leqslant} \renewcommand{\le}{\leqslant} \renewcommand{\geq}{\geqslant} \newcommand{\dt}{{\Delta t}} \newcommand\centerequation[1]{\par\smallskip\par \centerline{$\displaystyle #1$}\par \smallskip\par} \newcommand{\D}{\,\mathrm{d}} \newcommand{\cX}{\mathcal{X}} \newcommand{\E}{\expect} \newcommand{\wcL}{\widetilde{\mathcal{L}}} \newcommand{\Li}{\mathcal{K}} \newcommand{\I}{\mathrm{Id}} \newcommand{\dps}{\displaystyle} \newcommand{\red}{\color{red}} \newcommand{\blue}{\color{blue}} \newcommand{\yellow}{\color{yellow}} \input{header} \input{macros} \newcommand{\highlight}[2]{% \colorbox{#1!20}{$\displaystyle#2$}} \newcommand{\hiat}[4]{% \only<#1>{\highlight{#3}{#4}}% \only<#2>{\highlight{white}{#4}}% } \graphicspath{{figures/}} \AtEveryCitekey{\clearfield{pages}} \AtEveryCitekey{\clearfield{eprint}} \AtEveryCitekey{\clearfield{volume}} \AtEveryCitekey{\clearfield{number}} \AtEveryCitekey{\clearfield{month}} \addbibresource{main.bib} \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 % \includegraphics[height=1.5cm]{figures/logo_matherials.png} % \hspace{.5cm} \includegraphics[height=1.2cm]{figures/logo_inria.png} \hspace{.5cm} \includegraphics[height=1.5cm]{figures/logo_ponts.png} \hspace{.5cm} \includegraphics[height=1.5cm]{figures/logo_ERC.jpg} \hspace{.5cm} \includegraphics[height=1.5cm]{figures/logo_EMC2.png} \end{figure} \titlepage \end{frame} \begin{frame} {Some references} \begin{itemize} \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} \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} They can be estimated from molecular simulation at the \blue{microscopic level}. \begin{itemize} \item Defined from \emph{nonequilibrium} dynamics; \item Three main classes of methods to calculate them. \end{itemize} \vspace{.3cm} \textbf{\blue Outline of this talk} \begin{itemize} \item Equilibrium vs nonequilibrium dynamics; \item Definition and computation of the mobility; \item Computation of other transport coefficients. \end{itemize} \end{frame} \begin{frame} \begin{center} \Large \color{blue} Part I: Definition and examples of nonequilibrium systems \end{center} \centering \begin{minipage}{.8\textwidth} \begin{itemize} \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} \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} In this course, equilibrium = reversible, possibly up to a one-to-one transformation preserving the invariant measure. \end{frame} \begin{frame} {Examples of nonequilibrium dynamics} \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term} \begin{equation} \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} \end{block} \begin{block}{Langevin dynamics perturbed by a constant force term} \begin{equation} \label{eq:Langevin_F} \tag{NL} \left\{ \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{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} \item $F \in \real^d$ with $\abs{F} = 1$ is a given direction \item $\eta \in \real$ is the strength of the external forcing. \end{itemize} Is there an invariant probability measure? \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~$q_t \in \torus$ \[ \d q_t = - V'(q_t) \, \d t + {\red \eta} \, \d t + \sqrt{2} \, \d W_t, \] The associated Fokker--Planck equation reads \[ \frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0. \] \begin{minipage}[t]{.45\textwidth} \vspace{.5cm} The solution is unique and given by \[ \rho_{\eta}(q) \propto \e^{-V(q)} \int_{\torus} \e^{V(q+y) - \eta y} \, \d y. \] \textbf{Example:} $\rho_{\eta}$ with $V(q) = \frac{1}{2} (1 - \cos q)$. \end{minipage} \begin{minipage}[t]{.5\textwidth} \end{minipage} \begin{minipage}[t]{.45\textwidth} \begin{figure}[ht] \centering \includegraphics[width=\linewidth]{figures/invariant_perturbed_ol.pdf} \end{figure} \end{minipage} \end{frame} \begin{frame} {Nonequilibrium overdamped Langevin dynamics} 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 If the state space is compact (e.g. $\torus^d$), apply Doeblin's theorem. \item If not, use its generization, Harris' theorem. \end{itemize} \medskip Fix ${\blue t = 1}$ and denote by $p\colon \mathcal E \times \mathcal B(\mathcal E)$ the Markov transition kernel \[ p(x, A) := \proba \left[ q_t \in A \, \middle| \, q_0 = x \right]. \] For an observable $\phi \colon \mathcal E \to \real$ and a probability measure $\mu$, we let \[ (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, p(x, \d y), \qquad (\mathcal P^{\dagger} \mu)(A) := \int_{A} p(x, A) \, \mu(\d x). \] Note that $\mathcal P$ and $\mathcal P^{\dagger}$ are formally $L^2$ adjoints: \[ \int_{\mathcal E} (\mathcal P \phi) \, \d \mu = \int_{\mathcal E} \phi \, \d (\mathcal P^\dagger \mu). \] \end{frame} \begin{frame} {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 $\pi$ such that \[ \forall \mu, \qquad \mathcal P^{\dagger} \mu \geq \alpha \pi, \qquad \text{ (Minorization condition) } \] 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 \[ \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder), \] Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$. We have \begin{align*} d(\mathcal P^\dagger \mu, \mathcal P^\dagger \nu) &= \sup_{\phi \in \mathcal F} \int_{\mathcal E} \phi(q) (\mathcal P^{\dagger} \mu - \mathcal P^{\dagger} \nu) (\d q) = \sup_{\phi \in \mathcal F} \int_{\mathcal E} \mathcal P \phi(q) \bigl(\mu - \nu\bigr) (\d q) \\ &= (1 - \alpha) \sup_{\phi \in \mathcal F} \int_{\mathcal E} \widetilde {\mathcal P} \phi(q) (\mu - \nu) (\d q) \leq (1 - \alpha) \, d(\mu, \nu). \end{align*} Conclude using Banach's fixed point theorem. \end{frame} \begin{frame} {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_{*}) = \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 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} {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 &= - \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. \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} {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} [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 $\mathcal K_n(q, p) := 1 + \abs{p}^n$. 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 the assumptions of Harris' theorem are satisfied, in particular that \begin{align*} \mathcal L \mathcal K_n &\leq - a \mathcal K_n(q, p) + b, \end{align*} for $a > 0$ and $b \geq 0$. \end{frame} \begin{frame} {Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/2)} Consider the perturbed Langevin dynamics and write \[ \mathcal L_{\eta} = \mathcal L_0 + {\red \eta \widetilde {\mathcal L}}, \qquad \widetilde {\mathcal L} = F \cdot \grad_p \] 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 $L^2(\psi_0)$} \[ \mathcal L_\eta^* f_\eta = 0 \] 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} {\bf Questions:} Can the expansion for $f_\eta$ be made rigorous? What is $\mathfrak{f}_1$? \end{frame} \begin{frame} {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} \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} {Perturbation expansion for {\yellow $\eta$ sufficiently small} (3/3)} \begin{block}{Power expansion of the invariant measure} Spectral radius $r$ of the bounded operator $(\wcL \mathcal L_0^{-1})^* \in \mathcal{B}(L_0^2(\psi_0))$: \[ r = \lim_{n \to +\infty} \left\| \left[ \left(\wcL \mathcal L_0^{-1}\right)^* \right]^n \right\|^{1/n}. \] 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 \begin{equation} \label{eq:expansion_psi_xi_general} f_\eta = \left( 1+\eta (\wcL \mathcal L_0^{-1})^* \right)^{-1} \mathbf{1} = \biggl( 1 + \sum_{n=1}^{+\infty} (-\eta)^n [ (\wcL \mathcal L_0^{-1})^* ]^n \biggr) \mathbf{1}. \end{equation} \end{block} Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \end{frame} \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} \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} We illustrate the first two for the simplest transport coefficient: the {\blue mobility}. \end{frame} \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*} \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} \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} {Reformulation as integrated correlation functions} Define the conjugate response \[ S = \wcL^* \mathbf{1} = \nabla_p^* (F \mathbf 1) = F^\t p. \] \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} 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} \begin{frame}\frametitle{Elements of proof} \bu Proof based on the following equality on $\mathcal{B}(L_0^2(\psi_0))$ \[ -\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, d t \] \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 \end{align*} \bu Note also that $S$ has average 0 w.r.t. invariant measure since \[ \int_\cX S \, d\pi = \int_\cX \wcL^* \mathbf{1} \, d\pi = \int_\cX \wcL\mathbf{1} \, d\pi = 0 \] \end{frame} \begin{frame} \begin{center} \Large \color{blue} Part III: Computation of other transport coefficients \end{center} \centering \begin{minipage}{.6\textwidth} \begin{itemize} \item Thermal conductivity \item Shear viscosity \end{itemize} \end{minipage} \end{frame} \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} \begin{frame} {Thermal transport in one-dimensional chains (2)} \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} {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)} 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} 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} {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 \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 $\gamma,\gamma>0$ \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)} \bu Average {\red longitudinal velocity} $u_x(Y) = \dps \lim_{\varepsilon \to 0} \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\cdot) \right]}{\eta}$ where \vspace{-0.3cm} \[ U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{i=1}^N p_{xi} \chi_{\varepsilon}\left(q_{yi}-Y\right) \] \vspace{-0.5cm} \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 $... =$ \vspace{-0.4cm} \[ \hspace{-0.1cm} \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right) - \! \! \! \! \! \! \! \! \sum_{1 \leq i < j \leq N} \! \! \! \! v'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|} \!\int_{q_{yj}}^{q_{yi}} \!\chi_{\varepsilon}(s-Y) \, ds \right) \] \bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ (with $\overline{\rho} = N/|\mathcal{D}|$) \[ \frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y) \] \end{frame} \begin{frame} \frametitle{Shear viscosity in fluids (4)} \bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$ \begin{block}{Velocity profile in Langevin dynamics under flow} \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} \end{frame} \begin{frame} \begin{center} \Huge{Error estimates on} \\ \bigskip \Huge{the computation of} \\ \bigskip \Huge{transport coefficients} \end{center} \end{frame} \begin{frame}\frametitle{Reminder: Error estimates in Monte Carlo simulations} \bu General SDE $dx_t = b(x_t)\,dt + \sigma(x_t) \, dW_t$, invariant measure $\pi$ \bigskip \bu {\red Discretization} $x^{n} \simeq x_{n\dt}$, {\blue invariant measure $\pi_\dt$}. For instance, \[ x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \sim \mathcal{G}(0,{\rm Id}) \ \mathrm{i.i.d.} \] \medskip \bu {\blue Ergodicity} of the numerical scheme with invariant measure~$\pi_\dt$ \[ \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(dx) \] \begin{block}{Error estimates for {\red finite} trajectory averages} \[ \widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) = \expect_\pi(A) + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error} \] \end{block} \smallskip \bu Bias $\expect_{\pi_\dt}(A)-\expect_\pi(A) \longrightarrow$ {\bf Focus today} \medskip \end{frame} \begin{frame}\frametitle{Weak type expansions} \bu Numerical scheme = {\red Markov chain} characterized by {\blue evolution operator} \[ P_\dt \varphi(x) = \expect\Big( \varphi\left(x^{n+1}\right)\Big| x^n = x\Big) \] 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 ``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}} \end{block} \smallskip \bu {\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$ \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 \bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$ \bigskip \bu Technical tool: {\blue Taylor expansion} \vspace{-0.2cm} \[ \varphi(x + \delta) = \varphi(x) + \delta^T \nabla \varphi(x) + \frac12 \delta^T \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots \] \medskip \bu Replace $\delta$ with $\sqrt{\dt}\, \sigma(x)\,G + \dt\,b(x)$ and {\blue gather in powers of $\dt$} \[ \begin{aligned} \varphi\big(\Phi_\dt(x,G)\big) & = \varphi(x) + \sqrt{\dt}\, \sigma(x)\,G \cdot \nabla \varphi(x) \\ & \ \ \ + \dt \left(\frac{\sigma(x)^2}{2} G^T \left[\nabla^2\varphi(x)\right]G + b(x)\cdot\nabla \varphi(x) \right) + \dots \end{aligned} \] \medskip \bu Taking {\blue expectations w.r.t. $G$} leads to \[ P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathrm{O}(\dt^2) \] \end{frame} \begin{frame}\frametitle{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} } \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 \end{frame} %----------------------------------------------------------- \begin{frame}\frametitle{Sketch of proof (1)} {\bf Step~1: Establish the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$} \medskip \bu Idea: $\pi_\dt = \pi (1 + \dt^p f_{p+1} + \dots)$ \medskip \bu by definition of $\pi_\dt$ \[ \int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right) \psi \right] d\pi_\dt = 0 \] \bu compare to first order correction to the invariant measure \[ \begin{aligned} & \int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right)\psi\right] (1+\dt^{p}f_{p+1})\, d\pi \\ & \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathrm{O}\left(\dt^{p+1}\right) \\ & \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathrm{O}\left(\dt^{p+1}\right) \end{aligned} \] \begin{block}{} Suggests $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$ \end{block} \end{frame} %----------------------------------------------------------- \begin{frame}\frametitle{Sketch of proof (2)} {\bf Step~2: Define an approximate inverse} \medskip \bu Issue: derivatives of $(\I-P_\dt)^{-1}\varphi$ are not controlled \bigskip \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} \[ \Pi \varphi = \varphi - \int_\cX \varphi \, d\pi \] \bu Idea of the construction: truncate the formal series expression \[ (A + \dt \, B)^{-1} = A^{-1} - \dt \, A^{-1}B A^{-1} + \dt^{2} \, A^{-1}B A^{-1}B A^{-1} + \dots \] \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} \begin{frame}\frametitle{Examples of splitting schemes for Langevin dynamics (1)} \bu Example: Langevin dynamics, discretized using a {\blue splitting} strategy \[ A = M^{-1} p \cdot \nabla_q, \quad B_\eta = \Big(-\nabla V(q) + \eta\,F\Big)\cdot \nabla_p, \quad C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p \] \bu Note that $\mathcal L_\eta = A + B_\eta + \gamma C$ \medskip \bu Trotter splitting $\to$ weak order 1 \[ P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^2) \] \bu Strang splitting $\to$ {\blue weak order 2} \[ P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^3) \] \bu Other category: {\red Geometric Langevin}\footnote{N.~Bou-Rabee and H.~Owhadi, {\em SIAM J. Numer. Anal.} (2010)} algorithms, \textit{e.g.} $P_\dt^{\gamma C,A,B_\eta,A}$ \\ $\to$ weak order 1 but measure preserved at order 2 in $\dt$ \end{frame} \begin{frame}\frametitle{Examples of splitting schemes for Langevin dynamics (2)} \small \bu $P_\dt^{B_\eta,A,\gamma C}$ corresponds to %\begin{equation} %\label{eq:Langevin_splitting} $\dps \left\{ \begin{aligned} \widetilde{p}^{n+1} & = p^n + \Big(-\nabla V(q^{n}) + \eta F\Big)\dt, \\ q^{n+1} & = q^n + \dt \, M^{-1} \widetilde{p}^{n+1}, \\ p^{n+1} & = \alpha_\dt \widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha^2_\dt}{\beta}M} \, G^n \end{aligned} \right.$ \\[5pt] %\end{equation} where $G^n$ are i.i.d. Gaussian and $\alpha_\dt = \exp(-\gamma M^{-1} \dt)$ \bigskip \bu $P^{\gamma C,B_\eta,A,B_\eta,\gamma C}_\dt$ for %\[ $\dps \left\{ \begin{aligned} \widetilde{p}^{n+1/2} & = \alpha_{\dt/2} p^{n} + \sqrt{\frac{1-\alpha_{\dt}}{\beta}M} \, G^{n}, \\ p^{n+1/2} & = \widetilde{p}^{n+1/2} + \frac{\dt}{2} \Big( -\nabla V(q^{n})+\eta F\Big), \\ q^{n+1} & = q^n + \dt \, M^{-1} p^{n+1/2}, \\ \widetilde{p}^{n+1} & = p^{n+1/2} + \frac{\dt}{2} \Big(- \nabla V(q^{n+1}) +\eta F\Big), \\ p^{n+1} & = \alpha_{\dt/2} \widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha_{\dt}}{\beta}M} \, G^{n+1/2} \end{aligned} \right.$ %\] \end{frame} \begin{frame}\frametitle{Error estimates on linear response} \begin{block}{Error estimates for nonequilibrium dynamics} There exists a function $f_{\alpha,1,\gamma} \in H^1(\mu)$ such that \vspace{-0.3cm} \[ \int_{\mathcal E} \psi \, d{\mu}_{\gamma,\eta,\dt} = \int_{\mathcal E} \psi \Big(1+ \eta f_{0,1,\gamma} + \dt^\alpha f_{\alpha,0,\gamma} + \eta \dt^\alpha f_{\alpha,1,\gamma} \Big) d{\mu} + r_{\psi,\gamma,\eta,\dt}, \] where the remainder is compatible with linear response \vspace{-0.1cm} \[ \left|r_{\psi,\gamma,\eta,\dt}\right| \leq K(\eta^2 + \dt^{\alpha+1}), \qquad \left|r_{\psi,\gamma,\eta,\dt} - r_{\psi,\gamma,0,\dt}\right| \leq K \eta (\eta + \dt^{\alpha+1}) \] \end{block} \medskip \bu Corollary: error estimates on the {\blue numerically computed mobility} \[ \begin{aligned} \rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\ & = \rho_{F} + \dt^\alpha \int_{\mathcal E} F^T M^{-1} p \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt} \end{aligned} \] \bu Results in the {\red overdamped} limit\footnote{B.~Leimkuhler, C.~Matthews and G.~Stoltz, {\em IMA J. Numer. Anal.} (2015)} \bigskip \end{frame} \begin{frame}\frametitle{Numerical results} \begin{figure} \begin{center} \includegraphics[width=6.2cm]{figures/LR.eps} \includegraphics[width=6.2cm]{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 $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$. \end{frame} %----------------------------------------------------------- \begin{frame}\frametitle{Error estimates on Green-Kubo formulas (1)} \bu For methods of {\bf weak order}~1, {\red Riemman sum} ($\phi,\varphi$ average 0 w.r.t. $\pi$) \vspace{-0.2cm} \[ \begin{aligned} & \int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt) \\[-7pt] & \mathrm{where} \ \Pi_\dt \phi = \phi - \int_\cX \phi \, d\pi_\dt \end{aligned} \] \bu Correlation approximated in practice using $K$ independent realizations %\bi %\item truncating the integration (decay estimates) %\item using empirical averages ($K$ independent realizations) \vspace{-0.2cm} \[ \begin{aligned} & \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) \simeq \frac{1}{K}\sum_{m=1}^{K} \left( \phi(x^{n,k}) - \overline{\phi}^{n,K} \right)\left( \varphi(x^{n,k}) - \overline{\varphi}^{n,K} \right) \\[-10pt] & \mathrm{where} \ \overline{\phi}^{n,K} = \frac1K \sum_{m=1}^{K} \phi(x^{n,k}) \end{aligned} \] \bu For methods of {\bf weak order} 2, {\blue trapezoidal rule} \vspace{-0.1cm} \[ \begin{aligned} \int_0^{+\infty} \expect \Big( \phi(x_t) \varphi(x_0) \Big) d{t} & = \frac{\dt}{2} \expect_\dt \left(\Pi_\dt \phi\left(x^{0}\right)\varphi\left(x^0\right)\right) \\ & \ \ + \dt \sum_{n=1}^{+\infty} \expect_\dt \left(\Pi_\dt \phi\left(x^{n}\right)\varphi\left(x^0\right)\right) + \mathrm{O}(\dt^2) \end{aligned} \] %\bu Allows to quantify the variance $\dps \frac{\sigma^2_{A,\dt}}{N_{\rm iter}\dt} \simeq \frac{\dps 2 \int_0^{+\infty} \expect\left[\delta A(x_t)\delta A(x_0)\right] \, dt}{T}$ where $T = N_{\rm iter}\dt$ \end{frame} %----------------------------------------------------------- \begin{frame}\frametitle{Error estimates on Green-Kubo formulas (2)} \bu Error of {\red order~$\alpha$ on invariant measure}: $\dps \int_\cX \psi \, d{\pi}_\dt = \int_\cX \psi \, d{\pi} + \mathrm{O}(\dt^\alpha)$ \medskip \bu Expansion of the evolution operator ($p+1 \geq \alpha$ and $\mathcal A_1 = \mathcal L$) \[ P_\dt \varphi = \varphi + \dt \, \mathcal L \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt} \] \begin{block}{Ergodicity of the numerical scheme} \centerequation{ \forall n \in \mathbb{N}, \qquad \left\| P_\dt^n \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq C_s \e^{-\lambda_s n\dt} } where $\mathcal{K}_s$ is a Lyapunov function ($1+|p|^{2s}$ for Langevin) and \[ L^\infty_{\Li_s,\dt} = \left\{ \frac{\varphi}{\mathcal{K}_s} \in L^\infty(\cX), \ \int_\cX \varphi \, d\pi_\dt = 0\right\} \] \end{block} \bu Proof: Lyapunov condition + uniform-in-$\dt$ minorization condition\footnote{M. Hairer and J. Mattingly, \emph{Progr. Probab.} (2011)} \end{frame} %----------------------------------------------------------- \begin{frame}\frametitle{Error estimates on Green-Kubo formulas (3)} \begin{block}{Error estimates on integrated correlation functions} Observables $\varphi,\psi$ with average~0 w.r.t. invariant measure~$\pi$ \[ \int_0^{+\infty} \expect \Big( \psi(x_t) \varphi(x_0) \Big) d{t} = \dt \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(x^{n}\right)\varphi\left(x^0\right)\right) + \dt^\alpha r^{\psi,\varphi}_\dt, \] where $\expect_\dt$ denotes expectations w.r.t. initial conditions $x_0 \sim \pi_\dt$ and over all realizations of the Markov chain $(x^n)$, and \[ \widetilde{\psi}_{\dt,\alpha} = \psi_{\dt,\alpha} - \int_\cX \psi_{\dt,\alpha} \, d\pi_\dt\] with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \dots + \dt^{\alpha-1} \mathcal A_{\alpha}\mathcal L^{-1} \Big)\psi$ \end{block} \bu Useful when $\mathcal A_k \mathcal L^{-1}$ can be computed, \emph{e.g.} $\mathcal A_k = a_k \mathcal L^{k}$ \medskip \bu Reduces to trapezoidal rule for second order schemes \end{frame} %----------------------------------------------------------- \begin{frame}\frametitle{Sketch of proof (1)} \bu Define $\dps \Pi_\dt \varphi = \varphi - \int_\cX \varphi \, d\pi_\dt$ \smallskip \bu Since $\mathcal L^{-1}\psi$ has average~0 w.r.t.~$\pi$, introduce $\pi_\dt$ as \begin{align*} \int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} & = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi} \nonumber \\ %& = \int_\cX \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt, \nonumber \\ & = \int_\cX \Pi_\dt \left(-\mathcal L^{-1} \psi\right) \Pi_\dt \varphi \, d{\pi}_\dt + \dt^\alpha r^{\psi,\varphi}_\dt, \end{align*} \bu Rewrite $-\Pi_\dt \mathcal L^{-1}$ in terms of $P_\dt$ as \[ \begin{aligned} & -\Pi_\dt \mathcal L^{-1} \psi = -\Pi_\dt \left(\dt\sum_{n=0}^{+\infty} P_\dt^n \right) \Pi_\dt \left(\frac{\I - P_\dt}{\dt}\right) \mathcal L^{-1} \psi \\ & \ \ = \dt \left(\sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \right) \left(\mathcal L + \dots + \dt^{\alpha-1} S_{\alpha-1} + \dt^\alpha \widetilde{R}_{\alpha,\dt}\right) \mathcal L^{-1} \psi, \\ & \ \ = \dt \sum_{n=0}^{+\infty} \left[ \Pi_\dt P_\dt \Pi_\dt\right]^n \widetilde{\psi}_{\dt,\alpha} + \dt^\alpha \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \Pi_\dt \widetilde{R}_{\alpha,\dt} \mathcal L^{-1} \psi. \end{aligned} \] \end{frame} %----------------------------------------------------------- \begin{frame}\frametitle{Sketch of proof (2)} \bu Uniform resolvent bounds $\dps \left\| \left(\frac{\I - P_\dt}{\dt}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{\Li_s,\dt})} \leq \frac{C_s}{\lambda_s}$ \medskip \bu Coming back to the initial equality, \[ \int_\cX \left(-\mathcal L^{-1} \psi\right) \varphi \, d{\pi} = \dt \int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \left( \Pi_\dt \varphi \right) d{\pi}_\dt + \mathrm{O}\left(\dt^\alpha\right) \] \bu Rewrite finally \[ \begin{aligned} \int_\cX \sum_{n=0}^{+\infty} \left( \Pi_\dt P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right)\left( \Pi_\dt \varphi \right) d{\pi}_\dt & = \int_\cX \sum_{n=0}^{+\infty} \left(P_\dt^n \widetilde{\psi}_{\dt,\alpha} \right) \varphi \, d{\pi}_\dt \\ & = \sum_{n=0}^{+\infty} \expect_\dt \left(\widetilde{\psi}_{\dt,\alpha}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\right) \end{aligned} \] \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} \end{center} \end{frame} \begin{frame} {Main points: recall the outline!} \bu {\bf Definition and examples of nonequilibrium systems} \bigskip \bu {\bf Computation of transport coefficients} \begin{itemize} \item a survey of computational techniques \item linear response theory \item relationship with Green-Kubo formulas \end{itemize} \bigskip \bu {\bf Elements of numerical analysis} \begin{itemize} \item estimation of biases due to timestep discretization \item {\blue (largely) open issue: variance reduction} \item {\red (not discussed) use of non-reversible dynamics to enhance sampling} \end{itemize} \end{frame} \end{document} % vim: ts=4 sw=4