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diff --git a/figures/mobilityFctGamma.pdf b/figures/mobilityFctGamma.pdf Binary files differnew file mode 100644 index 0000000..f0b1fdf --- /dev/null +++ b/figures/mobilityFctGamma.pdf diff --git a/figures/nanotube.jpg b/figures/nanotube.jpg Binary files differnew file mode 100644 index 0000000..e4784ff --- /dev/null +++ b/figures/nanotube.jpg diff --git a/figures/nanotube.png b/figures/nanotube.png Binary files differnew file mode 100644 index 0000000..bfbb3ed --- /dev/null +++ b/figures/nanotube.png diff --git a/figures/shear1.png b/figures/shear1.png Binary files differnew file mode 100644 index 0000000..adb542b --- /dev/null +++ b/figures/shear1.png diff --git a/figures/shear2.png b/figures/shear2.png Binary files differnew file mode 100644 index 0000000..9d81c12 --- /dev/null +++ b/figures/shear2.png @@ -88,6 +88,43 @@ \end{frame} \section{Introduction} + +\begin{frame} + \frametitle{Introduction} + {\bf Aims of computational statistical physics} + \begin{itemize} + \item {\red numerical microscope} + \item computation of {\blue average properties}, static or dynamic + \end{itemize} + \begin{center} + \begin{minipage}[t]{.6\textwidth} + \begin{figure}[ht] + \centering + \resizebox{\textwidth}{!}{% + \begin{tikzpicture} + \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[width=\textwidth]{figures/nanotube.png}}; + \node [draw, color=red!60, fill=red!5, very thick, rectangle, minimum height=1cm] (warm) at (0,1) {$T_+$}; + \node [draw, color=blue!60, fill=blue!5, very thick, rectangle, minimum height=1cm] (cold) at (7.3,4.1) {$T_-$}; + % \node [draw=none] (flux) at (5,2) {$J$}; + \draw[->, line width=1mm] (3.5,0.9) to node[below]{$J$} ++(2.5,2); + \node [draw=none, minimum height=1cm] (unknown) at (0,4) {% + \begin{minipage}{4cm} + \centering + \textbf{Fourier's law:}% + \[ + J = - \alert{\kappa} \grad T + \] + \end{minipage} + }; + \end{tikzpicture}} + \end{figure} + \end{minipage} + \end{center} + +``Given the structure and the {\red laws of interaction} of the particles, what are the {\blue macroscopic properties} of the systems composed of these particles?'' + +\end{frame} + \begin{frame} {Transport coefficients} At the \alert{macroscopic} level, @@ -113,9 +150,11 @@ \item Equilibrium vs nonequilibrium dynamics; \item Definition and computation of the mobility; \item Computation of other transport coefficients. + \item Error analysis \end{itemize} \end{frame} + \begin{frame} \begin{center} \Large @@ -161,7 +200,7 @@ \end{frame} \begin{frame} - {Examples of nonequilibrium dynamics} + {Paradigmatic examples of nonequilibrium dynamics} \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term} \begin{equation} \label{eq:overdamped_Langevin_F} @@ -182,7 +221,7 @@ \end{aligned} \right. \end{equation} - In the rest of the presentation we take {\blue $ M = \I$} for simplicity. + In the rest of this section, we take {\blue $ M = \I$} for simplicity. \end{block} where \begin{itemize} @@ -194,41 +233,14 @@ \end{frame} \begin{frame} - {Another example useful for thermal transport} - \begin{block}{Langevin dynamics with modified fluctuation} - \[ - \left\{ - \begin{aligned} - \d q_t & = M^{-1} p_t \, \d t, \\* - \d p_t & = -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t \, \d t - + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t, - \end{aligned} - \right. - \] - \end{block} - with non-negative temperature - \[ - T_\eta(q) = T_{\rm ref} + \eta \widetilde{T}(q) - \] - Typically, $\widetilde{T}$ constant and positive on $\mathcal D_+ \subset \mathcal C$, - and constant and negative on $\mathcal D_- \subset \mathcal D$. - \begin{itemize} - \item - Non-zero energy flux from $\mathcal D_+$ to $\mathcal D_-$ expected in the steady-state - - \item - - Simplified model of thermal transport (in 3D materials or atom chains) - \end{itemize} -\end{frame} - -\begin{frame} {When {\yellow $\eta = 0$}, these dynamics are reversible} \begin{itemize} \item For overdamped Langevin dynamics \[ \mathcal L_{\rm ovd} \Big\vert_{\red \eta = 0} = - \grad V \cdot \grad + \laplacian = - \grad^* \grad, + \qquad + \mu(\d q) = \frac{1}{Z} \e^{-V(q)} \, \d q. \] where $\grad^* := (\grad V - \grad) \cdot $. For any $f, g \in C^{\infty}_{\rm c}(\mathcal E)$, we have @@ -256,6 +268,35 @@ \end{frame} \begin{frame} + {Another example useful for thermal transport} + \begin{block}{Langevin dynamics with modified fluctuation} + \[ + \left\{ + \begin{aligned} + \d q_t & = M^{-1} p_t \, \d t, \\* + \d p_t & = -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t \, \d t + + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t, + \end{aligned} + \right. + \] + \end{block} + with non-negative temperature + \[ + T_\eta(q) = T_{\rm ref} + \eta \widetilde{T}(q) + \] + Typically, $\widetilde{T}$ constant and positive on $\mathcal D_+ \subset \mathcal C$, + and constant and negative on $\mathcal D_- \subset \mathcal D$. + \begin{itemize} + \item + Non-zero energy flux from $\mathcal D_+$ to $\mathcal D_-$ expected in the steady-state + + \item + + Simplified model of thermal transport (in 3D materials or atom chains) + \end{itemize} +\end{frame} + +\begin{frame} {Worked example in dimension one} Consider the perturbed overdamped Langevin dynamics with~$q_t \in \torus$ \[ @@ -335,7 +376,7 @@ \emph{Sketch of proof.} Define the Markov transition kernel \[ - \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder), + \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \pi(\placeholder), \] Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$. We have @@ -373,7 +414,7 @@ \[ L^{\infty}_{*} := \left\{ \phi \in L^{\infty}(\mathcal E) : \int_{\mathcal E} \phi \, \d \mu_{*} = 0 \right\}. \] - Thus $\I - \mathcal P$ is invertible and + Thus $\I - \mathcal P$ is invertible on~$L^{\infty}_{*}$ and \[ (\I - \mathcal P)^{-1} = \I + \mathcal P + \mathcal P^2 + \dotsb \] @@ -404,7 +445,7 @@ and additionally ${\rm Law} (q_t)$ is smooth by parabolic regularity. \item \textbf{Decay of the semigroup}: - For $t \in [0, \infty)$ and bounded $\varphi$, it holds that + For $t \in [0, \infty)$ and $\varphi \in L^{\infty}_*$, it holds that \begin{align*} \lVert \e^{t \mathcal L_{\rm ovd}} \varphi \rVert_{L^{\infty}} &= \left\lVert \e^{(t- \lfloor t \rfloor) \mathcal L_{\rm ovd}} \left( \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi \right) \right\rVert_{L^{\infty}} \\ @@ -413,7 +454,7 @@ \end{align*} \item - \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{\eta}$, + \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{*}$, and \[ \mathcal L_{\rm ovd}^{-1} @@ -527,7 +568,7 @@ \[ \mathcal L_\eta^* f_\eta = 0 \] - Observe that + Observe that $\mathcal L_{\eta}^* = \mathcal L_0^* + \widetilde {\mathcal L}^*$ with \[ \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p - \gamma \grad_p^* \grad_p^*, \qquad \widetilde {\mathcal L}^* \placeholder = \grad_p^* (F \placeholder) @@ -549,8 +590,8 @@ \mathcal L_{\eta}^* f_{\eta} &= \mathcal L_0^* \mathfrak f_0 \\ &\quad + \eta \left(\widetilde {\mathcal L}^* \mathfrak f_0 + \mathcal L_0^* \mathfrak f_1\right) \\ - &\quad + \eta^2 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_2\right) \\ - &\quad + \eta^3 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_2\right) + \dotsb + &\quad + \eta^2 \left(\widetilde {\mathcal L}^* \mathfrak f_1 + \mathcal L_0^* \mathfrak f_2\right) \\ + &\quad + \eta^3 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_3\right) + \dotsb \end{align*} This suggests that $\mathfrak f_{i+1} = -(\mathcal L_0^*)^{-1} (\widetilde {\mathcal L}^* \mathfrak f_i)$ and so \[ @@ -640,11 +681,11 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \end{center} \centering - \begin{minipage}{.8\textwidth} + \begin{minipage}{.6\textwidth} \begin{itemize} \item Definition through linear response \item Green--Kubo reformulation - \item Numerical approximation + \item Link with effective diffusion \end{itemize} \end{minipage} \end{frame} @@ -709,10 +750,26 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. Therefore \[ \rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0 - = \int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L} \mathbf 1) \, \psi_0 + = -\int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L}^* \mathbf 1) \, \psi_0 \] \end{frame} +\begin{frame} + {Numerical results (1)} + \begin{figure} + \centering + \includegraphics[width=.75\textwidth]{figures/LR.eps} + \end{figure} +\end{frame} + +\begin{frame} + {Numerical results (2)} + \begin{figure} + \centering + \includegraphics[width=.75\textwidth]{figures/mobilityFctGamma.pdf} + \caption{Mobility as a function of~$\gamma$~\footnote{See J.~Roussel and G.~Stoltz, \emph{ESAIM: M2AN} (2018)}} + \end{figure} +\end{frame} \begin{frame} {Reformulation as integrated correlation functions} @@ -734,37 +791,120 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. \end{block} For the mobility, - it holds $S(q,p) = \beta R(q,p) = F^T p$ and so + it holds $S(q,p) = R(q,p) = F^\t p$ and so \[ - \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p )}{\eta} - = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0) \Big) d t + \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta \bigl(F^\t p \bigr)}{\eta} + = \int_0^{+\infty} \expect_0 \Big( \bigl(F^\t p_t\bigr) \bigl(F^\t p_0\bigr) \Big) \, \d t \] - \end{frame} +\begin{frame} + {Elements of proof} -\begin{frame}\frametitle{Elements of proof} - -\bu Proof based on the following equality on $\mathcal{B}(L_0^2(\psi_0))$ +\bu Proof based on the following equality on $\mathcal{B}\bigl(L_0^2(\psi_0)\bigr)$ \[ --\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, d t +-\mathcal L_0^{-1} = \int_0^{+\infty} \mathrm{e}^{t \mathcal L_0} \, \d t. \] \bu Then, \begin{align*} \lim_{\eta \to 0} \frac{\expect_\eta(R)}{\eta} & = -\int_{\mathcal E} R \left[(\wcL \mathcal L_0^{-1})^* \mathbf{1}\right] \psi_0 -= -\int_{\mathcal{E}} [\mathcal L_0^{-1}R ] [\wcL^* \mathbf{1} ] \, \psi_0 \notag \\* -& = \int_0^{+\infty} \left( \int_{\mathcal{E}} \left(\mathrm{e}^{t \mathcal L_0} R\right) \, S \, \psi_0\right)dt \notag \\ -& = \int_0^{+\infty} \expect \Big( R(q_t,p_t)S(q_0,p_0) \Big) d t += -\int_{\mathcal{E}} [\mathcal L_0^{-1}R ] [\wcL^* \mathbf{1} ] \, \psi_0 \\ +& = \int_0^{+\infty} \left( \int_{\mathcal{E}} \left(\mathrm{e}^{t \mathcal L_0} R\right) \, S \, \psi_0\right) \d t \\ +& = \int_0^{+\infty} \expect \Big( R(q_t,p_t)S(q_0,p_0) \Big) \, \d t \end{align*} -\bu Note also that $S$ has average 0 w.r.t. invariant measure since +\bu Note also that $S$ has average 0 w.r.t.\ invariant measure since \[ -\int_\cX S \, d\pi = \int_\cX \wcL^* \mathbf{1} \, d\pi = \int_\cX \wcL\mathbf{1} \, d\pi = 0 +\int_\cX S \, \d\pi = \int_\cX \wcL^* \mathbf{1} \, \d\pi = \int_\cX \wcL\mathbf{1} \, \d\pi = 0 \] \end{frame} +\begin{frame} + {Connection with effective diffusion} + It is possible to show a {\blue functional central limit theorem} for the Langevin dynamics: + \begin{equation*} + \varepsilon \widetilde {q}_{s/\varepsilon^2} \xrightarrow[\varepsilon \to 0]{} \sqrt{2 \mat D} \, W_s + \qquad \text{weakly on } C([0, \infty)), \qquad \widetilde {q}_t := q_0 + \int_{0}^{t} p_s \, \d s \in {\blue \real^{d}}. + \end{equation*} + In particular, $\widetilde {q}_t /\sqrt{t} \xrightarrow[t \to \infty]{} \mathcal N(0, 2 \mat D)$ weakly. + + \vspace{-.25cm} + \begin{figure}[ht] + \centering + \href{run:videos/gle/effective-diffusion.webm?autostart&loop}% + {\includegraphics[width=0.75\textwidth]{videos/gle/effective-diffusion.png}}% + \caption{Histogram of $q_t/\sqrt{t}$. The potential $V(q) = - \cos(q) / 2$ is illustrated in the background.} + \end{figure} +\end{frame} + +\begin{frame} + {Mathematical expression for the effective diffusion (dimension 1)} + \vspace{.2cm} + \begin{block}{Expression of $D$ in terms of the solution to a Poisson equation} + Effective diffusion tensor given by $D = \ip{\phi}{p}_{L^2(\mu)}$ and $\phi$ is the solution to + \[ + - \mathcal L \phi = p, + \qquad \phi \in L^2_0(\mu). + \] + \end{block} + \textbf{Key idea of the proof:} Apply It\^o's formula to $\phi$ + \begin{align*} + \d \phi(q_s, p_s) + % &= \frac{1}{\varepsilon^2} \mathcal L_{L} \phi (q_t, p_t) + \frac{1}{\varepsilon} \, \sqrt{2 \gamma \beta^{-1}} \, \derivative{1}[\phi]{p}(q_t, p_t) \, \d W_t, \\ + &= - p_s \, \d s + \sqrt{2} \, \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s + \end{align*} + and then rearrange: + \begin{align*} + \alert\varepsilon (\widetilde q_{t/\alert\varepsilon^2} - \widetilde q_{0}) &= \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} p_s \, \d s \\ + &= \underbrace{\alert\varepsilon \bigl(\phi(q_0, p_0) - \phi(q_{t/\alert\varepsilon^2}, p_{t/\alert\varepsilon^2})\bigr)}_{\to 0 + % ~\text{in $L^p(\Omega, C([0, T], \real))$} + } + + \underbrace{\sqrt{2 \gamma} \alert\varepsilon \int_{0}^{t/\alert\varepsilon^2} \derivative{1}[\phi]{p}(q_s, p_s) \, \d W_s}_{\to \sqrt{2 D} W_t~\text{weakly by MCLT}}. + \end{align*} + % where + % \begin{align*} + % D &= \gamma \beta^{-1} \, \int \abs{\textstyle \derivative{1}[\phi]{p}(q, p)}^2 \, \mu(\d q \, \d p) + % = - \int \phi (\mathcal L \phi) \, \d \mu + % = \ip{\phi}{p}. + % \end{align*} + + \textbf{In the multidimensional setting}, $D_{F} = \ip{\phi_{F}}{F^\t p}$ with $- \mathcal L \phi_{F} = F^\t p$. + + \textbf{Einstein's relation:} we just showed + \( + D_F = \beta^{-1} \rho_F. + \) +\end{frame} + + +\begin{frame} + {Summary: numerical approaches for calculating the mobility} + \begin{itemize} + \itemsep.5cm + \item {\blue Linear response approach}: + \begin{equation*} + \rho_F = \lim_{\eta \to 0} \frac{1}{\alert{\eta}} \expect_{\alert{\eta}} \, \bigl[F^\t p\bigr]. + \end{equation*} + where $\mu_{\eta}$ is the invariant distribution of the system with external forcing. + + \item {\blue Einstein's relation}: + \[ + \rho_F = \lim_{t \to \infty} \frac{1}{2t} \expect_{\mu} \Bigl[ \bigl| F^\t (\widetilde {q}_t - q_0)\bigr|^2 \Bigr]. + \] + + \item Deterministic method, e.g. {\blue Fourier/Hermite Galerkin}, for the Poisson equation + \[ + - \mathcal L_0 \phi_{F} = F^\t p, \qquad \rho_F = \ip{\phi_F}{F^\t p}. + \] + + \item {\blue Green--Kubo formula}: + \begin{align*} + \rho_F &= \int_{0}^{\infty} \expect_{\blue 0}\bigl((F^\t p_0) (F^\t p_t)\bigr) \, \d t. + \end{align*} + \end{itemize} +\end{frame} \begin{frame} \begin{center} @@ -820,7 +960,7 @@ Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. The Hamiltonian of the system is the sum of the potential and kinetic energies: \begin{equation*} - H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2}, + H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2}, \quad V(r) = \sum_{n=1}^{N-1} v(r_n). \end{equation*} \end{frame} @@ -904,57 +1044,62 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: {Shear viscosity in fluids (2)} Macroscopic description by Navier--Stokes equation \[ - \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \eta \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x} + \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \nu \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x} \] Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives \[ - - \eta U_x''(y) = \overline{\rho} F(y) + - \nu U_x''(y) = \overline{\rho} F(y), \qquad \overline \rho := \frac{\rho}{m} = \frac{N}{|\mathcal D|} + \] + Therefore, for the test function~$g(y) = \e^{2i\pi \frac{y}{L_y}}$ + \[ + \nu \int_0^{L_y} U_x(y) g''(y) \, \d y = \overline{\rho} \int_{0}^{L_y} F(y) g(y) \, \d y + \] + + $\rightarrow$ Suggests estimating the shear viscosity from molecular dynamics as + \[ + \nu = \frac{\dps \frac{\overline{\rho}}{L_y}\int_{0}^{L_y} F(y) g(y) \, \d y} + {\dps \expect_{F} \left[ \frac{1}{N}\sum_{n=1}^{N} \frac{p_{xi}}{m} g''(q_{yi}) \right]}. \] \end{frame} \begin{frame} {Shear viscosity in fluids (2)} - - pairwise interactions + Assume pairwise interactions \[ V(q) = \sum_{1 \leq i < j \leq N} \mathcal V(\abs{q_i - q_j}). \] \bu Add a smooth {\blue nongradient force} in the $x$ direction, depending on~$y$ \begin{block}{Langevin dynamics under flow} \centerequation{\left \{ \begin{aligned} - d q_{i,t} &= \frac{p_{i,t}}{m} \, dt,\\ - d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, dt + {\red \eta F(q_{yi,t}) \, dt} - - \gamma \frac{p_{xi,t}}{m} \, dt + \sqrt{\frac{2\gamma}{\beta}} \, dW^{xi}_t, \\ - d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, dt - \gamma \frac{p_{yi,t}}{m} \, dt - + \sqrt{\frac{2\gamma}{\beta}} \, dW^{yi}_t, + \d q_{i,t} &= \frac{p_{i,t}}{m} \, \d t,\\ + \d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, \d t + {\red \eta F(q_{yi,t}) \, \d t} + - \gamma \frac{p_{xi,t}}{m} \, \d t + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{xi}_t, \\ + \d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, \d t - \gamma \frac{p_{yi,t}}{m} \, \d t + + \sqrt{\frac{2\gamma}{\beta}} \, \d W^{yi}_t. \end{aligned} \right. } \end{block} \smallskip -\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma,\gamma>0$ +\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma>0$ \smallskip \bu The perturbation $\dps \wcL = \sum_{i=1}^N \! F(q_{y,i}) \partial_{p_{x,i}}$ is $\mathcal{L}_0$-bounded - -\smallskip - -\bu {\blue Linear response}: - $\dps - \lim_{\eta \rightarrow 0} \frac{\left\langle \mathcal L_0 h \right\rangle_\eta}{\eta} - = - \frac{\beta}{m} \! - \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)} - $ -\medskip - \end{frame} -\begin{frame}\frametitle{Shear viscosity in fluids (3)} +\begin{frame} + {Shear viscosity in fluids (3)} +\bu {\blue Linear response}: +\[ + \lim_{\eta \rightarrow 0} \frac{\expect_{\eta} [\mathcal L_0 h]}{\eta} + = - \frac{\beta}{m} \! + \left\langle \!h, \sum_{i=1}^N p_{xi} F(q_{yi}) \!\right\rangle_{L^2(\psi_0)}. +\] \bu Average {\red longitudinal velocity} @@ -970,11 +1115,12 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: \bu Average {\red off-diagonal stress} $\dps \sigma_{xy}(Y) = \lim_{\varepsilon \to 0} - \lim_{\eta \to 0} \frac{\left\langle ... \right\rangle_\eta}{\eta}$, - where $... =$ + \lim_{\eta \to 0} \frac{\expect_{\eta} [...]}{\eta}$, + where \vspace{-0.4cm} \[ \hspace{-0.1cm} + ... = \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right) - \! \! \! \! \! \! \! \! \sum_{1 \leq i < j \leq N} \! \! \! \! @@ -984,15 +1130,12 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: \bu {\blue Local conservation} of momentum\footnote{Irving and Kirkwood, {\it J. Chem. Phys.} {\bf 18} (1950)}: replace $h$ by $U_x^\varepsilon$ (with $\overline{\rho} = N/|\mathcal{D}|$) \[ -\frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y) + \frac{d\sigma_{xy}(Y)}{dY} + \gamma_{x} \overline{\rho} u_x(Y) = \overline{\rho} F(Y) \] - \end{frame} - - \begin{frame} -\frametitle{Shear viscosity in fluids (4)} + {Shear viscosity in fluids (4)} \bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$ @@ -1000,62 +1143,61 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies: \centerequation{-\eta u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)} \end{block} -\bigskip - -\hspace{-0.5cm} -\begin{minipage}{6cm} -\psfrag{F}{{\scriptsize $F$}} -\psfrag{U}{{\scriptsize $u$}} -\psfrag{Y}{{\scriptsize $\ \ Y$}} -\psfrag{v}{{\scriptsize value}} -\includegraphics[width=6cm]{figures/ux5.eps} -\end{minipage} -\hspace{-0.5cm} -\begin{minipage}{6cm} -\psfrag{Y}{} -\psfrag{v}{{\scriptsize value}} -\psfrag{S}{{\scriptsize $\sigma_{xy}$}} -\psfrag{D}{{\scriptsize $-\nu u'$}} -\includegraphics[width=6cm]{figures/dux5.eps} -\end{minipage} - +\begin{figure}[ht] + \centering + \includegraphics[width=\linewidth]{figures/shear1.png} +\end{figure} \end{frame} - +% \begin{frame} +% \end{frame} \begin{frame} \begin{center} -\Huge{Error estimates on} \\ -\bigskip -\Huge{the computation of} \\ -\bigskip -\Huge{transport coefficients} -\end{center} + \Large + \color{blue} + Part IV: Error estimates on the estimation of transport coefficients + \end{center} + + \centering + \begin{minipage}{.8\textwidth} + \begin{itemize} + \item Reminders: strong order, weak order + \item Error analysis for the linear response method + \item Error analysis for Green--Kubo method + \end{itemize} + \end{minipage} \end{frame} -\begin{frame}\frametitle{Reminder: Error estimates in Monte Carlo simulations} +\begin{frame} + {Reminder: Error estimates in Monte Carlo simulations} -\bu General SDE $dx_t = b(x_t)\,dt + \sigma(x_t) \, dW_t$, invariant measure $\pi$ +Consider the general SDE +\[ + \d x_t = b(x_t)\,\d t + \sigma(x_t) \, \d W_t +\] +with invariant measure $\pi$. \bigskip \bu {\red Discretization} $x^{n} \simeq x_{n\dt}$, {\blue invariant measure $\pi_\dt$}. For instance, \[ -x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \sim \mathcal{G}(0,{\rm Id}) \ \mathrm{i.i.d.} +x^{n+1} = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n, \qquad G^n \stackrel{\rm{i.i.d.}}{\sim} \mathcal N(0,{\rm Id}) \] \medskip \bu {\blue Ergodicity} of the numerical scheme with invariant measure~$\pi_\dt$ \[ -\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(dx) +\frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) \xrightarrow[N_{\rm iter}\to+\infty]{} \int_\cX A(x) \, \pi_\dt(\d x) \] \begin{block}{Error estimates for {\red finite} trajectory averages} \[ -\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) = \expect_\pi(A) + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error} +\widehat{A}_{N_{\rm iter}} = \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}} A(x^n) += \expect_\pi(A) + \underbrace{\frac{C}{N_{\rm iter} \dt}}_{\rm bias} + \underbrace{C\dt^\alpha}_{\rm bias} + \underbrace{\frac{\sigma_{A,\dt}}{\sqrt{N_{\rm iter}\dt}} \mathscr{G}}_\mathrm{statistical~error} \] \end{block} @@ -1076,105 +1218,127 @@ P_\dt \varphi(x) = \expect\Big( \varphi\left(x^{n+1}\right)\Big| x^n = x\Big) \] where $(x^n)$ is an approximation of $(x_{n \dt})$ -\bigskip - -\bu (Infinitely) Many possibilities! Numerical analysis allows to {\blue discriminate} - \medskip \bu Standard notions of error: {\red fixed integration time $T < +\infty$} \begin{itemize} -\item {\blue Strong error} $\dps \sup_{0 \leq n \leq T/\dt} \expect | X^n - X_{n\dt} | \leq C \dt^p$ -\item {\blue Weak error}: $\dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(X^n\right)\right] - \expect\left[\varphi\left(X_{n\dt}\right)\right] \Big| \leq C \dt^p$ (for any $\varphi$) +\item {\blue Strong error}: + \[ + \dps \sup_{0 \leq n \leq T/\dt} \expect | x^n - x_{n\dt} | \leq C \dt^p + \] +\item {\blue Weak error}: for any $\varphi$, + \[ + \dps \!\!\!\! \sup_{0 \leq n \leq T/\dt} \Big| \expect\left[\varphi\left(x^n\right)\right] - \expect\left[\varphi\left(x_{n\dt}\right)\right] \Big| \leq C \dt^p + \] %\item ``mean error'' \emph{vs.} ``error of the mean'' \end{itemize} -%\medskip -%\bu Example: for Euler-Maruyama, weak order~1, strong order $1/2$ (1 when $\sigma$ constant) -%\medskip - \begin{block}{$\dt$-expansion of the evolution operator} -\centerequation{P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt}} + \[ + P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt} + \] \end{block} \smallskip -\bu {\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$ +{\red Weak order}~$p$ when $\mathcal A_k = \mathcal L^k/k!$ for $1 \leq k \leq p$. \end{frame} +\begin{frame} + {Elements of proof} + \begin{itemize} + \item + Since $u(t, x) := \e^{t \mathcal L} \varphi(x)$ solves the backward Kolmogorov equation + \begin{align*} + \partial_t u = \mathcal L u, + \qquad u(0, x) = \varphi. + \end{align*} + we can write formally + \[ + \e^{\dt \mathcal L} \varphi = \I + \dt \mathcal L \varphi + \frac{\dt^2}{2} \mathcal L^2\varphi + \dotsb + \] + \item + Introduce a telescopic sum + \begin{align*} + \expect \bigl[\varphi(x^N)\bigr] - \expect \bigl[\varphi(x_{N \dt})\bigr] + &= P_{\dt}^N \varphi (x_0) - \e^{N \dt \mathcal L} \varphi(x_0) \\ + &= \sum_{n=0}^{N-1} \left( P_{\dt}^{N-n} \e^{n \dt \mathcal L} \varphi(x_0) - P_{\dt}^{N-(n+1)} \e^{(n+1) \dt \mathcal L} \varphi (x_0) \right) \\ + &= \sum_{n=0}^{N-1} P_{\dt}^{N-(n+1)} \left( P_{\dt} - \e^{\dt \mathcal L} \right) \e^{n \dt \mathcal L} \varphi (x_0) + \end{align*} + \end{itemize} +\end{frame} -\begin{frame}\frametitle{Example: Euler-Maruyama, weak order~1} - -\medskip -\bu Scheme $x^{n+1} = \Phi_\dt(x^n,G^n) = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n$ - -\bigskip +\begin{frame} + {Example: Euler-Maruyama, weak order~1} + Consider the scheme + \[ + x^{n+1} = \Phi_\dt(x^n,G^n) = x^n + \dt\,b(x^n)+\sqrt{\dt}\,\sigma(x^n)\,G^n + \] -\bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$ + \bigskip -\bigskip + \bu Note that $P_\dt \varphi(x) = \expect_G\left[ \varphi\big(\Phi_\dt(x,G)\big) \right]$ -\bu Technical tool: {\blue Taylor expansion} -\vspace{-0.2cm} -\[ -\varphi(x + \delta) = \varphi(x) + \delta^T \nabla \varphi(x) + \frac12 \delta^T \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots -\] + \bigskip -\medskip + \bu Technical tool: {\blue Taylor expansion} + \vspace{-0.2cm} + \[ + \varphi(x + \delta) = \varphi(x) + \delta^\t \nabla \varphi(x) + \frac12 \delta^\t \nabla^2\varphi(x) \delta + \frac16 D^3\varphi(x):\delta^{\otimes 3} + \dots + \] -\bu Replace $\delta$ with $\sqrt{\dt}\, \sigma(x)\,G + \dt\,b(x)$ and {\blue gather in powers of $\dt$} -\[ -\begin{aligned} -\varphi\big(\Phi_\dt(x,G)\big) & = \varphi(x) + \sqrt{\dt}\, \sigma(x)\,G \cdot \nabla \varphi(x) \\ -& \ \ \ + \dt \left(\frac{\sigma(x)^2}{2} G^T \left[\nabla^2\varphi(x)\right]G + b(x)\cdot\nabla \varphi(x) \right) + \dots -\end{aligned} -\] + \medskip -\medskip + \bu Replace $\delta$ with $\sqrt{\dt}\, \sigma(x)\,G + \dt\,b(x)$ and {\blue gather in powers of $\dt$} + \[ + \begin{aligned} + \varphi\big(\Phi_\dt(x,G)\big) & = \varphi(x) + \sqrt{\dt}\, \sigma(x)\,G \cdot \nabla \varphi(x) \\ + & \ \ \ + \dt \left(\frac{\sigma(x)^2}{2} G^\t \left[\nabla^2\varphi(x)\right]G + b(x)\cdot\nabla \varphi(x) \right) + \dots + \end{aligned} + \] -\bu Taking {\blue expectations w.r.t. $G$} leads to -\[ -P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathrm{O}(\dt^2) -\] + \medskip + \bu Taking {\blue expectations w.r.t. $G$} leads to + \[ + P_\dt\varphi(x) = \varphi(x) + \dt \underbrace{\left(\frac{\sigma(x)^2}{2} \Delta \varphi(x) + b(x)\cdot\nabla \varphi(x) \right)}_{= \mathcal{L}\varphi(x)} + \mathcal O(\dt^2) + \] \end{frame} - -\begin{frame}\frametitle{Error estimates on the invariant measure (equilibrium)} +\begin{frame} + {Error estimates on the invariant measure (equilibrium)} \bu {\red Assumptions} on the operators in the weak-type expansion -\begin{itemize} -\item invariance of $\pi$ by $\mathcal A_k$ for $1 \leq k \leq p$, namely -$\dps \int_\cX \mathcal A_k \varphi \, d\pi = 0$ -\item $\dps \int_\cX \mathcal A_{p+1}\varphi \, d\pi = \int_\cX g_{p+1} \varphi \, d\pi$ -(\textit{i.e.} $g_{p+1} = \mathcal A_{p+1}^* \mathbf{1}$) -\end{itemize} \begin{block}{Error estimates on $\pi_\dt$} -\centerequation{ -\int_\cX \varphi \, d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) d\pi + \dt^{p+1} R_{\varphi,\dt} -} + Suppose that + \begin{itemize} + \item + For all smooth $\varphi$, the following expansion holds + \[ + P_\dt \varphi = \varphi + \dt \, \mathcal A_1 \varphi + \dt^2 \mathcal A_2 \varphi + \dots + \dt^{p+1} \mathcal A_{p+1} \varphi + \dt^{p+2} r_{\varphi,\dt} + \] + \item The probability measure $\pi$ is invariant by $\mathcal A_k$ for $1 \leq k \leq p$, namely + \[ + \int_\cX \mathcal A_k \varphi \, d\pi = 0 + \] + \end{itemize} + Then + \[ + \int_\cX \varphi \, d\pi_\dt = \int_\cX \varphi \Big(1+\dt^{p}f_{p+1}\Big) d\pi + \dt^{p+1} R_{\varphi,\dt}, + \] + where $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$. \end{block} -\medskip - -\bu In fact, $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$ -\begin{itemize} -\item when $\mathcal A_1 = \mathcal L$, the first order correction can be {\red estimated} by some integrated correlation function as $\dps \int_0^{+\infty} \expect\Big(\varphi(x_t)g_{p+1}(x_0)\Big) \, dt$ -\item in general, first order term can be removed by Romberg extrapolation -\end{itemize} - -\medskip - -\bu Error on invariant measure can be {\blue (much) smaller} than the weak error +Error on invariant measure can be {\blue (much) smaller} than the weak error \end{frame} -%----------------------------------------------------------- -\begin{frame}\frametitle{Sketch of proof (1)} +\begin{frame} + {Motivation of the result} -{\bf Step~1: Establish the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$} +We verify the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$. \medskip @@ -1190,8 +1354,8 @@ $\dps \int_\cX \mathcal A_k \varphi \, d\pi = 0$ \[ \begin{aligned} & \int_\cX \left[ \left(\frac{P_\dt-\I}{\dt}\right)\psi\right] (1+\dt^{p}f_{p+1})\, d\pi \\ -& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathrm{O}\left(\dt^{p+1}\right) \\ -& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathrm{O}\left(\dt^{p+1}\right) +& \qquad = \dt^{p} \int_\cX \Big( \mathcal A_{p+1}\psi + (\mathcal A_1 \psi) f_{p+1} \Big) d\pi + \mathcal O\left(\dt^{p+1}\right) \\ +& \qquad = \dt^p \int_\cX \Big( g_{p+1} + \mathcal A_1^* f_{p+1} \Big) \psi \, d\pi + \mathcal O\left(\dt^{p+1}\right) \end{aligned} \] @@ -1201,40 +1365,50 @@ Suggests $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$ \end{frame} -%----------------------------------------------------------- -\begin{frame}\frametitle{Sketch of proof (2)} +\begin{frame} + {Numerical estimators and associated challenges} + \begin{itemize} + \item + Estimator of linear response (observable~$R$ with equilibrium average~0) + \[ + \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, ds \xrightarrow[t\to+\infty]{\mathrm{a.s.}} + \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, d\mu = \alpha + \mathcal O(\eta) + \] + {\bf Issues with linear response methods:} + \begin{itemize} + \item Statistical error with {\red asymptotic variance $\mathcal O(\eta^{-2})$} + \item Bias $\mathcal O(\eta)$ due to $\eta \neq 0$ + \item Bias from finite integration time + \end{itemize} -{\bf Step~2: Define an approximate inverse} + \end{itemize} +\end{frame} -\medskip +\begin{frame}\frametitle{Analysis of variance / finite integration time bias} -\bu Issue: derivatives of $(\I-P_\dt)^{-1}\varphi$ are not controlled + \bu {\bf Statistical error} dictated by {\blue Central Limit Theorem}: + \[ + \sqrt{t} \left(\widehat{A}_{\eta,t} - \alpha_\eta \right) \xrightarrow[t \to +\infty]{\mathrm{law}} \mathcal{N}\left(0,\frac{\sigma_{R,\eta}^2}{\eta^2}\right), + \qquad + \sigma_{R,\eta}^2 = \sigma_{R,0}^2 + \mathcal O(\eta) + \] + so $\dps \widehat{A}_{\eta,t} = \alpha_\eta + \mathcal O_{\rm P}\left(\frac{1}{\eta \sqrt{t}}\right)$ $\to$ requires {\red long simulation times} $t \sim \eta^{-2}$ + + \bigskip + + \bu {\bf Finite time integration bias}: $\dps \left| \mathbb{E}\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\ + Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error} -\bigskip +%\bigskip + %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$ -\bu Consider $\dps \left(\Pi \frac{P_\dt-\I}{\dt} \Pi\right) Q_\dt\psi = \psi + \dt^{p+1} \widetilde{r}_{\psi,\dt}$ where -\vspace{-0.2cm} -\[ -\Pi \varphi = \varphi - \int_\cX \varphi \, d\pi -\] +\bigskip -\bu Idea of the construction: truncate the formal series expression +\bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, d\mu$ \[ -(A + \dt \, B)^{-1} = A^{-1} - \dt \, A^{-1}B A^{-1} + \dt^{2} \, A^{-1}B A^{-1}B A^{-1} + \dots +\widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T dW_s \] -\bigskip - -{\bf Step~3: Conclusion} - -\medskip - -\bu Write the invariances with $\dps \Pi \left(\frac{P_\dt-\I}{\dt}\right) \Pi \psi$ instead of $\dps \left(\frac{P_\dt-\I}{\dt}\right) \psi$ - -\medskip - -\bu Replace $\psi$ by $Q_\dt \varphi$, and gather in~$R_{\varphi,\dt}$ all the higher order terms - \end{frame} @@ -1256,12 +1430,12 @@ C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p \bu Trotter splitting $\to$ weak order 1 \[ -P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^2) +P^{ZYX}_\dt = \e^{\dt Z} \e^{\dt Y} \e^{\dt X} = \e^{\dt \mathcal L} + \mathcal O(\dt^2) \] \bu Strang splitting $\to$ {\blue weak order 2} \[ -P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathrm{O}(\dt^3) +P^{ZYXYZ}_\dt = \e^{\dt Z/2} \e^{\dt Y/2} \e^{\dt X} \e^{\dt Y/2} \e^{\dt Z/2} = \e^{\dt \mathcal L} + \mathcal O(\dt^3) \] \bu Other category: {\red Geometric Langevin}\footnote{N.~Bou-Rabee and H.~Owhadi, {\em SIAM J. Numer. Anal.} (2010)} algorithms, \textit{e.g.} $P_\dt^{\gamma C,A,B_\eta,A}$ \\ @@ -1323,8 +1497,8 @@ where the remainder is compatible with linear response \bu Corollary: error estimates on the {\blue numerically computed mobility} \[ \begin{aligned} -\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E} F^T M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\ -& = \rho_{F} + \dt^\alpha \int_{\mathcal E} F^T M^{-1} p \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt} +\rho_{F,\dt} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,\eta,\dt}(d{q}\,d{p}) - \int_{\mathcal E} F^\t M^{-1} p \, \mu_{\gamma,0,\dt}(d{q}\,d{p}) \right) \\ +& = \rho_{F} + \dt^\alpha \int_{\mathcal E} F^\t M^{-1} p \, f_{\alpha,1,\gamma} \, d{\mu} + \dt^{\alpha+1} r_{\gamma,\dt} \end{aligned} \] @@ -1339,17 +1513,11 @@ where the remainder is compatible with linear response \begin{figure} \begin{center} -\includegraphics[width=6.2cm]{figures/LR.eps} -\includegraphics[width=6.2cm]{figures/mobility_Langevin.eps} +\includegraphics[width=.8\textwidth]{figures/mobility_Langevin.eps} \end{center} \end{figure} -\small -{\bf Left:} Linear response of the average velocity as a function of $\eta$ for the scheme associated with $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$ and $\dt = 0.01, \gamma = 1$. \\ - -\smallskip - -{\bf Right:} Scaling of the mobility $\nu_{F,\gamma,\dt}$ for the first order scheme $P_\dt^{A,B_\eta,\gamma C}$ and the second order scheme $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$. +Scaling of the mobility for the first order scheme $P_\dt^{A,B_\eta,\gamma C}$ and the second order scheme $P_\dt^{\gamma C, B_\eta,A,B_\eta, \gamma C}$. \end{frame} @@ -1490,20 +1658,6 @@ with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \d \end{frame} -\begin{frame}\frametitle{Numerical results} - -\vspace{-0.5cm} -\begin{figure} -\begin{center} -\includegraphics[width=11.8cm]{figures/error_diffusion.eps} -%\includegraphics[width=8.2cm]{figures/error_diffusion_zoom.eps} -\end{center} -\end{figure} - -\end{frame} - - - \begin{frame} \begin{center} \Huge{Conclusion and perspectives} diff --git a/videos/gle/effective-diffusion.png b/videos/gle/effective-diffusion.png Binary files differindex 349184d..85b0c95 100644 --- a/videos/gle/effective-diffusion.png +++ b/videos/gle/effective-diffusion.png |
