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@@ -1,9 +1,27 @@ \documentclass[9pt]{beamer} -\renewcommand{\emph}[1]{\textcolor{blue}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} +% \newcommand{\red}[1]{\color{red}} \newif\iflong \longfalse +\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}} + \input{header} \input{macros} @@ -86,6 +104,13 @@ \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. + \end{itemize} + + \vspace{.3cm} \textbf{Challenges we do not address:} \begin{itemize} \item Choose thermodynamical ensemble; @@ -94,26 +119,20 @@ \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{center} + \Large + \color{blue} + Part I: Definition and examples of nonequilibrium systems + \end{center} - \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} + \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 + \end{itemize} + \end{minipage} \end{frame} \section{Equilibrium and nonequilibrium dynamics} @@ -140,24 +159,64 @@ \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 + {Example of nonequilibrium dynamics} + \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term} + \begin{equation} + \label{eq: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{\frac{2\gamma}{\beta}} \D W_t, + \end{aligned} + \right. + \end{equation} + \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} + {Worked example in dimension one} + Consider the perturbed overdamped Langevin dynamics with~$q_t \in \torus$ \[ - \frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0, + \d q_t = - V'(q_t) \, \d t + {\red \eta} \, \d t + \sqrt{2} \, \d W_t, \] - which can be solved as + The associated Fokker--Planck equation reads \[ - \rho_{\eta}(q) \propto \int_{\torus} \e^{V(q+y) - V(q) - \eta y} \, \d y. + \frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0. \] + \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} @@ -181,7 +240,6 @@ \begin{frame} {Existence of an invariant distribution} - \begin{theorem} Fix~$\eta_* > 0$ and $n \geq 2$, and let $\mathcal K_n(q, p) := 1 + \abs{p}^n$. @@ -206,16 +264,1016 @@ \end{frame} \begin{frame} - {Existence of an invariant measure} + {Existence of an invariant measure (1/2)} + For a Markov transition kernel~$\mathcal P\colon \mathcal E \times \mathcal B(\mathcal E) \to [0, 1]$, let + \[ + (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, \mathcal P(x, \d y), + \qquad + (\mathcal P^{\dagger} \mu)(A) := \int_{A} \mathcal P(x, A) \, \mu(\d x). + \] + \begin{theorem} + [Doeblin's theorem] + If there exists $\alpha \in (0, 1)$ and a probability measure $\eta$ such that + \[ + \mathcal P^{\dagger} \mu \geq \alpha \eta, + \] + then there exists $\mu_{\infty}$ such that $\mathcal P^{\dagger} \mu_{\infty} = \mu_{\infty}$. + Furthermore $d(\mathcal P^{\dagger^n} \mu, \mu_{\infty}) \leq \alpha^n d(\mu, \mu_{\infty})$. + \end{theorem} + + \emph{Sketch of proof.} + Use Banach's fixed point theorem. Define the Markov transition + \[ + \widetilde {\mathcal P}(x, \placeholder) := \frac{1}{1-\alpha} \mathcal P(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder). + \] + Let $V$ 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 V} \int_{\mathcal E} \phi(q) (\mathcal P^{\dagger} \mu - \mathcal P^{\dagger} \nu) (\d q) + = \sup_{\phi \in V} \int_{\mathcal E} \mathcal P \phi(q) \bigl(\mu - \nu\bigr) (\d q) \\ + &= (1 - \alpha) \sup_{\phi \in V} \int_{\mathcal E} \widetilde {\mathcal P} \phi(q) (\mu - \nu) (\d q) + \leq (1 - \alpha) \, d(\mu, \nu). + \end{align*} +\end{frame} + +\begin{frame} + {Existence of an invariant measure (2/2)} + \begin{itemize} + \item + Suppose that $\phi$ is uniformly bounded and let $\overline \phi = \int_{\mathcal E} \phi \, \d \mu_{\infty}$. Then + \[ + \Bigl\lVert \mathcal P \left(\phi - \overline \phi\right) \Bigr\rVert_{L^\infty} + = (1 - \alpha) \Bigl\lVert \widetilde {\mathcal P} (\phi - \overline \phi) \Bigr\rVert_{L^{\infty}} + \leq (1 - \alpha) \Bigl\lVert \phi - \overline \phi \Bigr\rVert_{L^{\infty}}, + \] + + + \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} + \end{itemize} +\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 $\dps R(q,p) = F^\t M^{-1}p$ + + \begin{block} + {Definition of the mobility} + \[ + \rho_F + = \lim_{\eta \to 0} \frac{\expect_\eta (R)-\expect_0 (R)}{\eta} + = \lim_{\eta \to 0} \frac{\expect_\eta (R)}{\eta} + \] + \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 + + \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} +\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 +\[ +\Pi_0 f = f - \int_{\mathcal E} f \, \psi_0 +\] + +\end{frame} + + +\begin{frame}\frametitle{Linear response of nonequilibrium dynamics (2)} + +\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} + +\medskip + +\bu Note that $\dps \int_{\mathcal E} \psi_\eta = 1$ + +\medskip + +\bu Linear response result: $\dps \rho_F = -\int_{\mathcal E} R \left[(\wcL \mathcal L_0^{-1})^* \mathbf{1}\right] \psi_0 $ + + +\end{frame} + + +\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) +\[ +\| \wcL \mathcal L_0^{-1} \varphi \|^2_{L^2(\psi_0)}\leq \frac{\beta}{\gamma} \| \varphi \|_{L^2(\psi_0)} \| \mathcal L_0^{-1} \varphi \|_{L^2(\psi_0)}. +\] + +\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 $L^2_0(\psi_0)$ + +\medskip + +\bu Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L^* + \eta \wcL^*$ +\vspace{-0.2cm} +\[ +\mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\wcL \mathcal L_0^{-1})^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0 +\] + +\bu {\red Prove that $f_\eta \geq 0$} (use some ergodicity result to show that $\psi_\eta = f_\eta \psi_0$) + +\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} + +\medskip + +\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 +\[ + \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p )}{\eta} + = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0) \Big) d t +\] + +\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}\frametitle{Generalization to other dynamics} + +\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} + +\bigskip + +\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} + +\end{frame} + + + +\begin{frame} + \begin{center} +\Huge{Other examples} +\end{center} +\end{frame} + + +\begin{frame}\frametitle{Shear viscosity in fluids (1)} + +\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} + +\end{frame} + + +\begin{frame}\frametitle{Shear viscosity in fluids (2)} + +\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, +\end{aligned} \right. +} +\end{block} + +\smallskip + +\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma_x,\gamma_y>0$ + +\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} + $u_x(Y) = \dps \lim_{\varepsilon \to 0} + \lim_{\eta \to 0} \frac{\left\langle U_x^\varepsilon(Y,\cdot)\right\rangle_\eta}{\eta}$ + where + \vspace{-0.3cm} \[ - d(P \mu, P \nu) - \leq + U_x^\varepsilon(Y,q,p) = \frac{L_y}{Nm}\sum_{i=1}^N p_{xi} + \chi_{\varepsilon}\left(q_{yi}-Y\right) \] + \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} + \[ + \hspace{-0.1cm} + \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right) + - \! \! \! \! \! \! \! \! + \sum_{1 \leq i < j \leq N} \! \! \! \! + \mathcal{V}'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|} + \!\int_{q_{yj}}^{q_{yi}} \!\chi_{\varepsilon}(s-Y) \, ds \right) + \] + +\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_x \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}\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} +\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} + +\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} |