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| -rw-r--r-- | figures.jl | 12 | ||||
| -rwxr-xr-x | header.tex | 9 | ||||
| -rwxr-xr-x | main.bib | 12 | ||||
| -rwxr-xr-x | main.tex | 839 |
4 files changed, 561 insertions, 311 deletions
@@ -2,16 +2,22 @@ using Plots using QuadGK using LaTeXStrings -V(x) = 1 - cos(x) +Plots.default(fontfamily="Computer Modern", + titlefontsize=20, + xtickfontsize=12, + ytickfontsize=12, + legendfontsize=12) + +V(x) = 1/2 * (1 - cos(x)) η = .1 -# plot(title="Invariant distribution of perturbed OL") +plot(title=L"Steady state $\rho_{\eta}$ with forcing") plot(title="") for η ∈ [0, .5, 1, 2] ρ(q) = exp(-V(q)) * quadgk(y -> exp(V(q + y) - η*y), 0, 2π)[1] Z = quadgk(ρ, - π, π)[1] plot!(q -> ρ(q) / Z, -π, π, label=L"\eta = " * string(η), lw=2) end -plot!(xlims = (-π, π), grid=false, ylims=(0, .4), framestyle=:box) +plot!(xlims = (-π, π), grid=false, framestyle=:box) savefig("figures/invariant_perturbed_ol.pdf") @@ -23,6 +23,7 @@ \usepackage{subcaption} \usepackage{appendixnumberbeamer} \usepackage{xparse} +\usepackage{dsfont} \usepackage[makeroom]{cancel} \renewcommand{\CancelColor}{\color{red}} \usepackage[% @@ -39,9 +40,11 @@ \usetikzlibrary{quotes} \pgfplotsset{compat=1.17} -\definecolor{darkred}{rgb}{.5,0,0} -\definecolor{darkgreen}{rgb}{0,.5,0} -\definecolor{darkblue}{rgb}{0,0,.5} +\definecolor{linea}{RGB}{211, 94, 60} +\definecolor{lineb}{RGB}{ 78, 144, 204} +\colorlet{dark_color}{black!67} +\tikzstyle{dark} =[very thick, color=dark_color] +\tikzstyle{green} =[very thick, color=darkgreen] \newtheorem{proposition}{Proposition} @@ -451,3 +451,15 @@ MRREVIEWER = {Wojciech\ Bartoszek}, DOI = {10.1007/978-3-0348-0021-1\_7}, URL = {https://doi.org/10.1007/978-3-0348-0021-1_7}, } + +@book {MR2723222, + AUTHOR = {Tuckerman, Mark E.}, + TITLE = {Statistical Mechanics: Theory and Molecular Simulation}, + SERIES = {Oxford Graduate Texts}, + PUBLISHER = {Oxford University Press}, + YEAR = {2010}, + ISBN = {978-0-19-852526-4}, + MRCLASS = {82-01 (81V55 82-08 82C80)}, + MRNUMBER = {2723222}, + MRREVIEWER = {Gabriel Stoltz}, +} @@ -1,9 +1,8 @@ \documentclass[9pt]{beamer} -\newcommand{\blue}[1]{\textcolor{blue}{#1}} -% \newcommand{\red}[1]{\color{red}} \newif\iflong \longfalse +\usepackage{psfrag} \newcommand{\placeholder}{\mathord{\color{black!33}\bullet}}% \newcommand{\bu}{$\bullet \ $} \newcommand{\bi}{\begin{itemize}} @@ -21,6 +20,8 @@ \newcommand{\I}{\mathrm{Id}} \newcommand{\dps}{\displaystyle} \newcommand{\red}{\color{red}} +\newcommand{\blue}{\color{blue}} +\newcommand{\yellow}{\color{yellow}} \input{header} \input{macros} @@ -80,16 +81,12 @@ \itemsep.2cm \item \fullcite{MR3509213} \item \fullcite{pavliotis2011applied} + \item \fullcite{MR2723222} \item Lecture notes by Gabriel Stoltz on computational statistical physics: \url{http://cermics.enpc.fr/~stoltz/Cours/intro_phys_stat.pdf} \end{itemize} \end{frame} -\begin{frame} - {Outline} - \tableofcontents -\end{frame} - \section{Introduction} \begin{frame} {Transport coefficients} @@ -100,21 +97,22 @@ \begin{itemize} \item The \emph{mobility} relates an external force to a velocity; \item The \emph{heat conductivity} relates a temperature difference to a heat flux; - \item The \emph{shear viscosity} relates a shear velocity to a shear stress; + \item The \emph{shear viscosity} relates a shear velocity to a shear stress. \end{itemize} \vspace{.3cm} They can be estimated from molecular simulation at the \blue{microscopic level}. \begin{itemize} - \item They are defined from \emph{nonequilibrium} dynamics; - \item There are three main classes of methods to calculate them. + \item Defined from \emph{nonequilibrium} dynamics; + \item Three main classes of methods to calculate them. \end{itemize} \vspace{.3cm} - \textbf{Challenges we do not address:} + \textbf{\blue Outline of this talk} \begin{itemize} - \item Choose thermodynamical ensemble; - \item Prescribe microscopic dynamics; + \item Equilibrium vs nonequilibrium dynamics; + \item Definition and computation of the mobility; + \item Computation of other transport coefficients. \end{itemize} \end{frame} @@ -131,6 +129,7 @@ \item Equilibrium vs nonequilibrium dynamics \item Existence of an invariant measure for nonequilibrium dynamics \item Convergence to the invariant measure + \item Perturbation expansion of the invariant measure \end{itemize} \end{minipage} \end{frame} @@ -156,13 +155,16 @@ \begin{theorem} A stationary diffusion processes $x_t$ in $\real^d$ with generator $\mathcal L$ and invariant measure~$\mu$ is reversible if and only if $\mathcal L$ is self-adjoint in~$L^2(\mu)$. \end{theorem} + + In this course, equilibrium = reversible, + possibly up to a one-to-one transformation preserving the invariant measure. \end{frame} \begin{frame} - {Example of nonequilibrium dynamics} + {Examples of nonequilibrium dynamics} \begin{block}{Overdamped Langevin dynamics perturbed by a constant force term} \begin{equation} - \label{eq:Langevin_F} + \label{eq:overdamped_Langevin_F} \tag{NO} \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F} + \sqrt{2} \, \d W_t \end{equation} @@ -176,10 +178,11 @@ \begin{aligned} \d q_t & = M^{-1} p_t \D t, \\* \d p_t & = \bigl( -\nabla V(q_t) + {\red \eta F} \bigr) \D t - \gamma M^{-1} p_t \D t - + \sqrt{\frac{2\gamma}{\beta}} \D W_t, + + \sqrt{2\gamma} \D W_t, \end{aligned} \right. \end{equation} + In the rest of the presentation we take {\blue $ M = \I$} for simplicity. \end{block} where \begin{itemize} @@ -191,6 +194,68 @@ \end{frame} \begin{frame} + {Another example useful for thermal transport} + \begin{block}{Langevin dynamics with modified fluctuation} + \[ + \left\{ + \begin{aligned} + \d q_t & = M^{-1} p_t \, \d t, \\* + \d p_t & = -\nabla V(q_t) \, \d t - \gamma M^{-1} p_t \, \d t + + \sqrt{2\gamma {\red T_\eta(q)}} \, \d W_t, + \end{aligned} + \right. + \] + \end{block} + with non-negative temperature + \[ + T_\eta(q) = T_{\rm ref} + \eta \widetilde{T}(q) + \] + Typically, $\widetilde{T}$ constant and positive on $\mathcal D_+ \subset \mathcal C$, + and constant and negative on $\mathcal D_- \subset \mathcal D$. + \begin{itemize} + \item + Non-zero energy flux from $\mathcal D_+$ to $\mathcal D_-$ expected in the steady-state + + \item + + Simplified model of thermal transport (in 3D materials or atom chains) + \end{itemize} +\end{frame} + +\begin{frame} + {When {\yellow $\eta = 0$}, these dynamics are reversible} + \begin{itemize} + \item For overdamped Langevin dynamics + \[ + \mathcal L_{\rm ovd} \Big\vert_{\red \eta = 0} = - \grad V \cdot \grad + \laplacian + = - \grad^* \grad, + \] + where $\grad^* := (\grad V - \grad) \cdot $. + For any $f, g \in C^{\infty}_{\rm c}(\mathcal E)$, we have + \[ + \int_{\mathcal E} (\mathcal L_{\rm ovd} f ) g \, \d \mu + = - \int_{\mathcal E} \nabla f \cdot \nabla g \, \d \mu + = \int_{\mathcal E} (\mathcal L_{\rm ovd} g ) f \, \d \mu. + \] + \item For Langevin dynamics + \begin{align*} + \mathcal L\Big\vert_{\red \eta = 0} + = p \cdot \grad_q - \grad V \cdot \grad_p + \gamma \left( - p \cdot \grad_p + \laplacian_p \right) + = \grad_p^* \grad_q - \grad_q^* \grad_p - \gamma \grad_p^* \grad_p^*, + \end{align*} + where $\grad_q^* := (\grad V - \grad_q) \cdot $ and $\grad_p^* = (p -\grad_p) \cdot$ are the formal adjoints. + We have + \begin{align*} + \int_{\mathcal E} (\mathcal Lf ) g \, \d \mu + &= \int_{\mathcal E} g \left(\grad_p^* \grad_q - \grad_q^* \grad_p\right) f - \gamma \grad_p f \cdot \grad_p g \, \d \mu \\ + &= \int_{\mathcal E} {\red -} f \left(\grad_p^* \grad_q - \grad_q^* \grad_p\right) g - \gamma \grad_p f \cdot \grad_p g \, \d \mu \\ + &= \int_{\mathcal E} (f \circ S) \bigl(\mathcal L (g \circ S)\bigr) \, \d \mu + \qquad S f(q, p) := f(q, -p). + \end{align*} + \end{itemize} +\end{frame} + +\begin{frame} {Worked example in dimension one} Consider the perturbed overdamped Langevin dynamics with~$q_t \in \torus$ \[ @@ -221,22 +286,22 @@ \begin{frame} {Nonequilibrium overdamped Langevin dynamics} - In general, can we prove existence of and convergence to an invariant measure for + In general, how can we prove existence of an invariant measure for \[ \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F} + \sqrt{2} \, \d W_t \, ? \] \begin{itemize} - \item + \item If the state space is compact (e.g. $\torus^d$), - then we can apply Doeblin's theorem. + apply Doeblin's theorem. \item - If not, we can apply its generization, Harris' theorem. + If not, use its generization, Harris' theorem. \end{itemize} \medskip - Fix $t > 0$ and denote by $p\colon \mathcal E \times \mathcal B(\mathcal E)$ the Markov transition kernel + Fix ${\blue t = 1}$ and denote by $p\colon \mathcal E \times \mathcal B(\mathcal E)$ the Markov transition kernel \[ p(x, A) := \proba \left[ q_t \in A \, \middle| \, q_0 = x \right]. \] @@ -254,21 +319,23 @@ \end{frame} \begin{frame} - {Existence of an invariant measure (1/2)} - Remember that the set of probability measures with TV distance $d(\placeholder, \placeholder)$ is complete. + {Existence of an invariant measure for compact state space (1/2)} + Let $d(\placeholder, \placeholder)$ denote the total variation metric. \begin{theorem} [Doeblin's theorem] - If there exists $\alpha \in (0, 1)$ and a probability measure $\eta$ such that + If there exists $\alpha \in (0, 1)$ and a probability measure $\pi$ such that \[ - \mathcal P^{\dagger} \mu \geq \alpha \eta, + \forall \mu, \qquad + \mathcal P^{\dagger} \mu \geq \alpha \pi, + \qquad \text{ (Minorization condition) } \] - then there exists $\mu_{\infty}$ such that $\mathcal P^{\dagger} \mu_{\infty} = \mu_{\infty}$. - Furthermore $d(\mathcal P^{\dagger^n} \mu, \mu_{\infty}) \leq \alpha^n d(\mu, \mu_{\infty})$. + then there exists $\mu_*$ such that $\mathcal P^{\dagger} \mu_* = \mu_*$. + Furthermore $d(\mathcal P^{\dagger^n} \mu, \mu_*) \leq \alpha^n d(\mu, \mu_*)$. \end{theorem} \emph{Sketch of proof.} Define the Markov transition kernel \[ - \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder), + \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder), \] Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$. We have @@ -283,34 +350,84 @@ \end{frame} \begin{frame} - {Existence of an invariant measure (2/2)} + {Existence of an invariant measure for compact state space (2/2)} + Two simple corollaries: \begin{itemize} \item Suppose that $\phi$ is uniformly bounded. Then \begin{align*} \left\lvert \mathcal P^n \phi(x) - \overline \phi \right\rvert - &= \int_{\mathcal E} \mathcal P^n (\phi - \overline \phi) \, \d(\delta_x - \mu_{\infty}) - = \int_{\mathcal E} (\phi - \overline \phi) \, (\mathcal P^{\dagger n} \delta_x - \mathcal P^{\dagger n} \mu_{\infty}) (\d q) \\ - &\leq \norm{\phi - \overline \phi}_{L^{\infty}} (1-\alpha)^n d(\delta_x, \mu_{\infty}) + &= \int_{\mathcal E} \mathcal P^n (\phi - \overline \phi) \, \d(\delta_x - \mu_{*}) + = \int_{\mathcal E} (\phi - \overline \phi) \, (\mathcal P^{\dagger n} \delta_x - \mathcal P^{\dagger n} \mu_{*}) (\d q) \\ + &\leq \norm{\phi - \overline \phi}_{L^{\infty}} (1-\alpha)^n d(\delta_x, \mu_{*}) \leq 2 \norm{\phi - \overline \phi}_{L^{\infty}} (1 - \alpha)^n. \end{align*} - + This shows that + \[ + \left\lVert \mathcal P^n \phi(x) - \overline \phi \right\rVert_{L^{\infty}} + \leq 2 (1 - \alpha)^n \norm{\phi - \overline \phi}_{L^{\infty}}. + \] \item - In molecular dynamics, this theorem can be employed for showing existence of and convergence to the invariant measure, - provided that the \blue{state space is compact}. - - \item - For \alert{noncompact state spaces}, an extension called \emph{Harris' theorem} + The Neumann series $\I + \mathcal P + \mathcal P^2 + \dotsb$ is convergent as a bounded operator on + \[ + L^{\infty}_{*} := \left\{ \phi \in L^{\infty}(\mathcal E) : \int_{\mathcal E} \phi \, \d \mu_{*} = 0 \right\}. + \] + Thus $\I - \mathcal P$ is invertible and + \[ + (\I - \mathcal P)^{-1} = \I + \mathcal P + \mathcal P^2 + \dotsb + \] \end{itemize} \end{frame} \begin{frame} - {Existence of an invariant measure for noneq.\ dynamics} + {Connection with the time-continuous setting} + Consider the overdamped Langevin dynamics on~$\torus^d$: + \[ + \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F \, \d t} + \sqrt{2} \, \d W_t, + \qquad q_t \in \torus^d. + \] + + \begin{itemize} + \itemsep.5cm + \item + The \textbf{minorization condition} is satisfied. + Indeed for $t > 0$ + \begin{align*} + p(x, A) + &= \expect \left[ q_t \in A \, \middle| \, q_0 = x \right] + = \expect \left[ \mathds 1_{A} \left(x + W_t \right) M_t \right] + && M_t = \text{Girsanov weight} \\ + &= \proba \left[ x + W_t \in A \right] \expect \left[ M_t \, | \, \{x + W_t \in A\} \right] \\ + &\geq C \proba \left[ x + W_t \in A \right] \geq C \lambda(A) && \lambda := \text{Lebesgue measure}. + \end{align*} + and additionally ${\rm Law} (q_t)$ is smooth by parabolic regularity. + \item + \textbf{Decay of the semigroup}: + For $t \in [0, \infty)$ and bounded $\varphi$, it holds that + \begin{align*} + \lVert \e^{t \mathcal L_{\rm ovd}} \varphi \rVert_{L^{\infty}} + &= \left\lVert \e^{(t- \lfloor t \rfloor) \mathcal L_{\rm ovd}} \left( \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi \right) \right\rVert_{L^{\infty}} \\ + &\leq \left\lVert \e^{\lfloor t \rfloor \mathcal L_{\rm ovd}} \varphi \right\rVert_{L^{\infty}} + \leq 2 \e^{\alpha} \e^{- \alpha t} \lVert \varphi \rVert_{L^{\infty}}. + \end{align*} + + \item + \textbf{Corollary}: $\mathcal L_{\rm ovd}$ is invertible on~$L^{\infty}_{\eta}$, + and + \[ + \mathcal L_{\rm ovd}^{-1} + = - \int_{0}^{\infty} \e^{t \mathcal L_{\rm ovd}} \, \d t. + \] + \end{itemize} +\end{frame} + +\begin{frame} + {Existence of an invariant measure for perturbed Langevin dynamics} Consider the paradigmatic dynamics \begin{align*} \d q_t &= M^{-1} p_t \, \d t, \\ - \d p_t &= - \bigl(\grad V(q_t) + \eta F\bigr) \, \d t - \gamma M^{-1} p_t \, \d t + \sqrt{\frac{2 \gamma}{\beta}} \, \d W_t, + \d p_t &= - \grad V(q_t) \, \d t + {\red \eta F \, \d t} - \gamma p_t \, \d t + \sqrt{2 \gamma} \, \d W_t, \end{align*} where $(q_t, p_t) = \torus^d \times \real^d$ and $F \in \real^d$ with $\abs{F} = 1$ is a given direction. @@ -325,14 +442,55 @@ \end{frame} \begin{frame} - {Existence of an invariant distribution} + {Harris' theorem} + Let $p(x, A)$ denote a Markov transition kernel and let + \[ + (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, p(x, \d y), + \qquad + (\mathcal P^{\dagger} \mu)(A) := \int_{A} p(x, A) \, \mu(\d x). + \] \begin{theorem} - Fix~$\eta_* > 0$ and $n \geq 2$, + [Harris's theorem] + Suppose that the following conditions are satisfied: + \begin{itemize} + \item + There exists $\mathcal K\colon \mathcal E \to [1, \infty)$ + and constants~$a > 0$ and $b \geq 0$ such that + \[ + \forall x \in \mathcal E, \qquad + \mathcal L \mathcal K(x) \leq - a \mathcal K(x) + b, + \] + \item + There exists a constant $\alpha \in (0, 1)$ and a probability measure~$\pi$ such that + \[ + \inf_{x \in \mathcal C} p(x, \d y) \geq \, \alpha \, \pi(\d y), + \] + where $\mathcal C = \{x \in \real \, | \, \mathcal K(x) \leq K_{\max} \}$ for some $K_{\max} \geq 1 + 2 \, \frac{b}{a}$. + \end{itemize} + Then there $\exists! \, \, \mu_{*}$ such that $\mathcal P^{\dagger} \mu_{*} = \mu_{*}$. + Furthermore there is $\gamma \in (0, 1)$ such that + \[ + \left\lVert \frac{\mathcal P^n \phi - \overline \phi}{\mathcal K} \right\rVert_{L^{\infty}} + \leq C \gamma^n \norm{ \frac{\mathcal P^n \phi - \overline \phi}{\mathcal K} }_{L^{\infty}}, + \qquad \overline \phi := \int_{\mathcal E} \phi \, \d \mu_*. + \] + \end{theorem} +\end{frame} + +\begin{frame} + {Application to perturbed Langevin dynamics} + For $\mathcal K \colon \mathcal E \to [1, \infty)$, let + \[ + L^{\infty}_{\mathcal K} + := \left\{ \varphi \text{~measureable } : \norm{\frac{\varphi}{\mathcal K}}_{L^{\infty}} < \infty \right\} + \] + + \begin{theorem} + Fix~$\eta > 0$ and $n \geq 2$, and let $\mathcal K_n(q, p) := 1 + \abs{p}^n$. - For any $\eta \in [- \eta_*, \eta_*]$, - there exists a unique invariant probability measure, + There exists a unique invariant probability measure, with a smooth density~$\psi_{\eta}(q, p)$ with respect to the Lebesgue measure. - Furthermore there exists $C = C(n, \eta_*) > 0$ and $\lambda = \lambda(n, \eta_*) > 0$ such that + Furthermore there exists $C = C(n, \eta) > 0$ and $\lambda = \lambda(n, \eta) > 0$ such that \[ \forall \phi \in L^{\infty}_{\mathcal K_n}(\mathcal E), \qquad \left\lVert \e^{t \mathcal L_n} \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}} @@ -341,102 +499,117 @@ \end{theorem} \textbf{Idea of the proof.} - Show that + Show that the assumptions of Harris' theorem are satisfied, + in particular that \begin{align*} - \mathcal L \mathcal K_n &\leq - c_1 \mathcal K_n(q, p) + c_2, + \mathcal L \mathcal K_n &\leq - a \mathcal K_n(q, p) + b, \end{align*} - for $c_1 > 0$ and $c_2 > 0$. - Then apply the main theorem from~\footfullcite{MR2857021}. + for $a > 0$ and $b \geq 0$. \end{frame} - \begin{frame} - {Linear response of nonequilibrium dynamics (1)} - \bu The force $\eta F$ induces a non-zero velocity in the direction $F$ - \medskip - - \bu Encoded by $\dps \expect_\eta(R) = \int_{\mathcal E} R \, \psi_\eta$ with $\dps R(q,p) = F^\t M^{-1}p$ +{Perturbation expansion for {\yellow $\eta$ sufficiently small} (1/2)} + Consider the perturbed Langevin dynamics and write + \[ + \mathcal L_{\eta} = \mathcal L_0 + {\red \eta \widetilde {\mathcal L}}, + \qquad \widetilde {\mathcal L} = F \cdot \grad_p + \] - \begin{block} - {Definition of the mobility} + It is {\red expected} that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and + \[ + f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathcal O(\eta^2) + \] + The invariance of $\psi_\eta$ can be written as + \[ + \int_{\mathcal E} (\mathcal L_\eta \varphi) \psi_\eta = 0 = \int_{\mathcal E} (\mathcal L_\eta \varphi) f_\eta \psi_0 + \] + \begin{block}{Fokker-Planck equation on $L^2(\psi_0)$} \[ - \rho_F - = \lim_{\eta \to 0} \frac{\expect_\eta (R)-\expect_0 (R)}{\eta} - = \lim_{\eta \to 0} \frac{\expect_\eta (R)}{\eta} + \mathcal L_\eta^* f_\eta = 0 + \] + Observe that + \[ + \mathcal L_0^* = - \grad_p^* \grad_q + \grad_q^* \grad_p - \gamma \grad_p^* \grad_p^*, + \qquad \widetilde {\mathcal L}^* \placeholder = \grad_p^* (F \placeholder) \] \end{block} - \medskip - - \bu It is {\red expected} that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and - \[ - f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathrm{O}(\eta^2) - \] - - \medskip - - \bu In this case, $\dps \rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0$ - \bigskip + {\bf Questions:} Can the expansion for $f_\eta$ be made rigorous? What is $\mathfrak{f}_1$? - \bu {\bf Questions:} Can the expansion for $f_\eta$ be made rigorous? What is $\mathfrak{f}_1$? \end{frame} \begin{frame} - {Computation of transport coefficients} - Three main classes of methods: - \begin{itemize} - \itemsep.2cm - \item - Non-equilibrium techniques - \begin{itemize} - \item Calculations from the steady state of a system out of equilibrium. - \item Comprises bulk-driven and boundary-driven approaches. - \end{itemize} - - \item - Equilibrium techniques based on the Green--Kubo formula - \[ - \rho = \int_{0}^{\infty} \expect_{\mu} \bigl[\varphi(x_t) \phi(x_0)\bigr] \, \d t. - \] - We will derive this formula from linear response. - \item - Transient techniques: - \end{itemize} + {Perturbation expansion for {\yellow $\eta$ sufficiently small} (2/3)} + \begin{block} + {Formal asymptotics} + Write $f_\eta = \mathfrak f_0 + \eta \mathfrak{f}_1 + \eta^2 \mathfrak{f}_2 + \dotsb$ and expand + \begin{align*} + \mathcal L_{\eta}^* f_{\eta} + &= \mathcal L_0^* \mathfrak f_0 \\ + &\quad + \eta \left(\widetilde {\mathcal L}^* \mathfrak f_0 + \mathcal L_0^* \mathfrak f_1\right) \\ + &\quad + \eta^2 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_2\right) \\ + &\quad + \eta^3 \left(\widetilde {\mathcal L}^* \mathfrak f_2 + \mathcal L_0^* \mathfrak f_2\right) + \dotsb + \end{align*} + This suggests that $\mathfrak f_{i+1} = -(\mathcal L_0^*)^{-1} (\widetilde {\mathcal L}^* \mathfrak f_i)$ and so + \[ + f_\eta = \sum_{i=0}^{\infty} (-\eta)^i \Bigl((\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^*\Bigr)^i \mathbf 1 + = \left(\I + \eta(\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right)^{-1} \mathbf 1. + \] + \end{block} \end{frame} -\iffalse -\begin{frame}\frametitle{Linear response of nonequilibrium dynamics (2)} - -\bu {\red Perturbative framework} where $\mathcal L_0$ considered on $L^2(\psi_0)$ is the reference - -\medskip - -\bu The invariance of $\psi_\eta$ can be written as -\[ -\int_{\mathcal E} (\mathcal L_\eta \varphi) \psi_\eta = 0 = \int_{\mathcal E} (\mathcal L_\eta \varphi) f_\eta \psi_0 -\] - -\begin{block}{Fokker-Planck equation on $L^2(\psi_0)$} -\centerequation{\mathcal L_\eta^* f_\eta = 0} -\end{block} - -\bigskip - -\bu Formally, $\mathcal L_\eta^* f_\eta = (\mathcal L_0)^* \underbrace{\left(\I + \wcL \mathcal L_0^{-1}\right)^*f_\eta}_{=1 ?} = 0$ - -\medskip - -\bu To make the result precise, introduce $L_0^2(\psi_0) = \Pi_0 L^2(\psi_0)$ with -\[ -\Pi_0 f = f - \int_{\mathcal E} f \, \psi_0 -\] +\begin{frame} + {Elements of proof} + Let us introduce + \[ + H^1_{p}(\psi_0) = + \Bigl\{ \varphi \in L^2(\psi_0) : \grad_p \varphi \in L^2(\psi) \Bigr\}, + \qquad \| \varphi \|_{H^1_{p}(\psi_0)}^2 = \| \varphi \|_{L^2(\psi_0)}^2 + \| \nabla_p \varphi \|_{L^2(\psi_0)}^2. + \] + \vspace{-.3cm} + \begin{itemize} + \itemsep.2cm + \item + The operator {\blue $\widetilde {\mathcal L}^*\colon H^1_p(\psi_0) \to L^2_0(\psi)$} is well-defined and bounded. + Indeed + \[ + \lVert \widetilde {\mathcal L}^* \varphi \rVert_{L^2_0(\psi_0)}^2 + = \ip{\nabla_p^* F \varphi}{\nabla_p^* F \varphi}_{L^2_0(\psi_0)} + \leq \lVert \varphi \rVert_{H^1_p(\psi_0)}^2 + \] + and + \[ + \int_{\mathcal E} \widetilde {\mathcal L}^* \phi \, \psi_0 + = \int_{\mathcal E} \nabla_p^* (F \phi) \, \psi_0 = 0. + \] + \item + The operator {\blue $(\mathcal L_0^*)^{-1} \colon L^2_0(\psi_0) \to H^1_p(\psi_0)$} is well-defined and bounded, + by {\red hypocoercivity} and {\red hypoelliptic regularization}. + % In particular, for $\phi = (\mathcal L_0^*)^{-1} \varphi$ + % \begin{align*} + % \| \phi \|_{L^2(\psi_0)}^2 + % + \| \nabla_p \phi \|_{L^2(\psi_0)}^2 + % &= \|(\mathcal L_0^*)^{-1} \varphi \|_{L^2(\psi_0)}^2 + % + \frac{1}{\gamma} \ip{-\mathcal L_0^* \phi}{\phi}_{L^2(\psi_0)} \\ + % &\leq \frac{1}{\gamma} \norm{(\mathcal L_0^*)^{-1}}_{\mathcal B\bigl(L^2(\psi_0)\bigr)}^2 + % \norm{\varphi}_{L^2(\psi_0)} + % \end{align*} + + \item Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L_0^* + \eta \wcL^*$ + \vspace{-0.2cm} + \[ + \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0. + \] + + \item {\red Prove that $f_\eta \geq 0$}. + \end{itemize} \end{frame} - -\begin{frame}\frametitle{Linear response of nonequilibrium dynamics (2)} +\begin{frame} + {Perturbation expansion for {\yellow $\eta$ sufficiently small} (3/3)} \begin{block}{Power expansion of the invariant measure} Spectral radius $r$ of the bounded operator @@ -445,7 +618,7 @@ Spectral radius $r$ of the bounded operator r = \lim_{n \to +\infty} \left\| \left[ \left(\wcL \mathcal L_0^{-1}\right)^* \right]^n \right\|^{1/n}. \] Then, for $|\eta| < r^{-1}$, the unique invariant measure can be written as $\psi_\eta = f_\eta\psi_0$, - where $f_\eta \in L^2(\psi_0)$ can be expanded as + where~$f_\eta \in L^2(\psi_0)$ can be expanded as \begin{equation} \label{eq:expansion_psi_xi_general} f_\eta = \left( 1+\eta (\wcL \mathcal L_0^{-1})^* \right)^{-1} \mathbf{1} @@ -454,67 +627,118 @@ Spectral radius $r$ of the bounded operator \end{equation} \end{block} -\medskip - -\bu Note that $\dps \int_{\mathcal E} \psi_\eta = 1$ - -\medskip +Note that $\dps \int_{\mathcal E} \psi_\eta = 1$. +\end{frame} -\bu Linear response result: $\dps \rho_F = -\int_{\mathcal E} R \left[(\wcL \mathcal L_0^{-1})^* \mathbf{1}\right] \psi_0 $ +\section{Computation of transport coefficients} +\begin{frame} + \begin{center} + \Large + \color{blue} + Part II: Definition and calculation of the mobility + \end{center} + \centering + \begin{minipage}{.8\textwidth} + \begin{itemize} + \item Definition through linear response + \item Green--Kubo reformulation + \item Numerical approximation + \end{itemize} + \end{minipage} \end{frame} +\begin{frame} + {Computation of transport coefficients} + Three main classes of methods: + \begin{itemize} + \itemsep.2cm + \item + Non-equilibrium techniques. + \begin{itemize} + \item Calculations from the steady state of a system out of equilibrium. + \item Comprises bulk-driven and boundary-driven approaches. + \end{itemize} -\begin{frame}\frametitle{Elements of proof} - -\bu Since $\dps \frac{\gamma}{\beta} \| \nabla_p \varphi \|^2_{L^2(\psi_0)} = -\langle \mathcal L_0 \varphi,\varphi \rangle_{L^2(\psi_0)}$, it follows that -\vspace{-0.2cm} -\[ -\| \wcL \varphi \|^2_{L^2(\psi_0)} \leq \| \nabla_p \varphi \|^2_{L^2(\psi_0)} \leq \frac{\beta}{\gamma} \| \mathcal L_0 \varphi \|_{L^2(\psi_0)} \| \varphi \|_{L^2(\psi_0)} -\] - -\bu {\red $\mathcal L_0^{-1}$ is a well defined bounded operator on $L_0^2(\psi_0)$} (hypocoercivity + hypoelliptic regularization) -\[ -\| \wcL \mathcal L_0^{-1} \varphi \|^2_{L^2(\psi_0)}\leq \frac{\beta}{\gamma} \| \varphi \|_{L^2(\psi_0)} \| \mathcal L_0^{-1} \varphi \|_{L^2(\psi_0)}. -\] + \item + Equilibrium techniques based on the Green--Kubo formula + \[ + \rho = \int_{0}^{\infty} \expect_{\mu} \bigl[\varphi(x_t) \phi(x_0)\bigr] \, \d t. + \] + We will derive this formula from linear response. + \item + Transient methods. + \begin{itemize} + \item System locally perturbed + \item Relaxation of this perturbation enables to calibrate macroscopic model. + \end{itemize} + \end{itemize} -\bu {\blue $\Pi_0 \wcL \mathcal L_0^{-1}$ is bounded on $L^2_0(\psi_0)$}, so $(\wcL \mathcal L_0^{-1})^* \Pi_0 = (\wcL \mathcal L_0^{-1})^*$ is also bounded on $L^2_0(\psi_0)$ + We illustrate the first two for the simplest transport coefficient: + the {\blue mobility}. +\end{frame} -\medskip +\begin{frame} + {Linear response of nonequilibrium dynamics} + Consider the nonequilibirium dynamics + \begin{align*} + \d q_t &= M^{-1} p_t \, \d t, \\ + \d p_t &= - \grad V(q_t) \, \d t + {\red \eta F \, \d t} - \gamma p_t \, \d t + \sqrt{2 \gamma} \, \d W_t, + \end{align*} -\bu Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L^* + \eta \wcL^*$ -\vspace{-0.2cm} -\[ -\mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\wcL \mathcal L_0^{-1})^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0 -\] + \begin{itemize} + \item The force {\red $\eta F$} induces a non-zero velocity in the direction $F$ + \item Encoded by $\dps \expect_\eta(R) = \int_{\mathcal E} R \, \psi_\eta$ with $\dps R(q,p) = F^\t p$ + \end{itemize} -\bu {\red Prove that $f_\eta \geq 0$} (use some ergodicity result to show that $\psi_\eta = f_\eta \psi_0$) + \begin{definition} + [Mobility] + The mobility in direction $F$ is defined mathematically as + \[ + \rho_{F} = + \lim_{\alert{\eta} \to 0} \frac{\expect_{\red \eta} [R] - \expect_{0} [R]}{\red \eta} + = \lim_{\eta \to 0} \frac{1}{\alert{\eta}}\expect_{\red \eta} [R] + \] + \end{definition} + We proved that $\psi_\eta = f_\eta\psi_0$ with $\psi_0(q,p) = Z^{-1} \e^{-\beta H(q,p)}$ and + \[ + f_\eta = \mathbf{1} + \eta \mathfrak{f}_1 + \mathrm{O}(\eta^2), \qquad \mathfrak f_1 = - (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \mathbf 1. + \] + Therefore + \[ + \rho_F = \int_{\mathcal E} R \mathfrak{f}_1 \psi_0 + = \int_{\mathcal E} \left(\mathcal L_0^{-1} R\right) (\widetilde {\mathcal L} \mathbf 1) \, \psi_0 + \] \end{frame} -\begin{frame}\frametitle{Reformulation as integrated correlation functions} - -\bu Conjugate response $S = \wcL^* \mathbf{1}$, equivalently $\dps \int_{\mathcal E} \left(\wcL \varphi\right) \psi_0 = \int_{\mathcal E} \varphi \, S\, \psi_0$ - -\medskip - -\begin{block}{Green--Kubo formula} - For any $R \in L^2_0(\psi_0)$, - \[ - \lim_{\eta \to 0} \frac{\expect_\eta(R)}{\eta} = \int_0^{+\infty} \expect_0 \Big(R(q_t,p_t)S(q_0,p_0) \Big) d t, - \] - where $\expect_\eta$ is w.r.t. to $\psi_\eta(q,p)\,d q\, p$, while $\expect_0$ is taken over initial conditions $(q_0,p_0) \sim \psi_0$ and over all realizations of the equilibrium dynamics. -\end{block} +\begin{frame} + {Reformulation as integrated correlation functions} + Define the conjugate response + \[ + S + = \wcL^* \mathbf{1} + = \nabla_p^* (F \mathbf 1) + = F^\t p. + \] -\medskip + \begin{block}{Green--Kubo formula} + For any $R \in L^2_0(\psi_0)$, + \[ + \lim_{\eta \to 0} \frac{\expect_\eta(R)}{\eta} = \int_0^{+\infty} \expect_0 \Big(R(q_t,p_t)S(q_0,p_0) \Big) d t, + \] + where $\expect_\eta$ is w.r.t.\ to $\psi_\eta(q,p)\, \d q\, \d p$, + while $\expect_0$ is w.r.t.\ initial conditions~$(q_0,p_0) \sim \psi_0$ and over all realizations of the equilibrium dynamics. + \end{block} -\bu For the dynamics \eqref{eq:Langevin_F}, it holds $S(q,p) = \beta R(q,p) = \beta F^T M^{-1} p$ so that -\[ - \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p )}{\eta} - = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0) \Big) d t -\] + For the mobility, + it holds $S(q,p) = \beta R(q,p) = F^T p$ and so + \[ + \rho_F = \lim_{\eta \to 0} \frac{\dps \expect_\eta (F\cdot M^{-1}p )}{\eta} + = \beta \int_0^{+\infty} \expect_0 \Big( (F\cdot M^{-1}p_t) (F\cdot M^{-1}p_0) \Big) d t + \] \end{frame} @@ -542,71 +766,175 @@ Spectral radius $r$ of the bounded operator \end{frame} -\begin{frame}\frametitle{Generalization to other dynamics} +\begin{frame} + \begin{center} + \Large + \color{blue} + Part III: Computation of other transport coefficients + \end{center} -\bu {\bf Possible assumptions to justify the linear response} -\begin{itemize} -\item existence of invariant measure with smooth density $\psi_\eta$ -\item ergodicity $\dps \frac1t \int_0^t \varphi(x_s) \,d s \xrightarrow[t\to+\infty]{} \int_\cX \varphi \, \psi_\eta$ -\item $\mathrm{Ker}(\mathcal L_0^*) = \mathbf{1}$ and $\mathcal L_0^*$ is invertible on~$L_0^2(\psi_0)$ -\item the perturbation $\wcL$ is $\mathcal L_0$-bounded: there exist $a,b \geq 0$ such that -\[ -\| \wcL \varphi\|_{L^2(\psi_0)} \leq a \| \mathcal L_0 \varphi\|_{L^2(\psi_0)} + b \| \varphi\|_{L^2(\psi_0)} -\] -\end{itemize} + \centering + \begin{minipage}{.6\textwidth} + \begin{itemize} + \item Thermal conductivity + \item Shear viscosity + \end{itemize} + \end{minipage} +\end{frame} -\bigskip +\begin{frame} + {Thermal transport in one-dimensional chain (1)} + Consider a chain of $N$ atoms with nearest-neighbor interactions + \begin{tikzpicture} + \coordinate (origin) at (0,0); + \coordinate (shift) at (1.8,0); + \node [draw, color=red!60, fill=red!5, very thick, rectangle, minimum height=1cm] (nc) at (0,0) {$T_L$}; + \node [draw, color=blue!60, fill=blue!5, very thick, rectangle, minimum height=1cm] (nh) at ($ (origin) + 6*(shift) $) {$T_R$}; + \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n1) at ($ (origin) + 1*(shift) $) {$p_1$}; + \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n2) at ($ (origin) + 2*(shift) $) {$p_2$}; + \node [draw=none, circle, minimum size=1cm] (n3) at ($ (origin) + 3*(shift) $) {$\dotsb$}; + \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n-2) at ($ (origin) + 4*(shift) $) {$p_{N-1}$}; + \node [draw, color=black!60, fill=gray!5, very thick, circle, minimum size=1cm] (n-1) at ($ (origin) + 5*(shift) $) {$p_{N}$}; + \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n1) -- node[below=.25cm]{$r_1$} (n2); + \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n2) -- node[below=.25cm]{$r_2$} (n3); + \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n3) -- node[below=.25cm]{$r_{N-2}$} (n-2); + \draw[decoration={aspect=0.3, segment length=1mm, amplitude=1.5mm,coil},decorate] (n-2) -- node[below=.25cm]{$r_{N-1}$} (n-1); + \draw[red, ->] (nc) to [out=45,in=135] node[above]{$j_0$} (n1); + \draw[red, ->] (n1) to [out=45,in=135] node[above]{$j_1$} (n2); + \draw[red, ->] (n2) to [out=45,in=135] node[above]{$j_2$} (n3); + \draw[red, ->] (n3) to [out=45,in=135] node[above]{$j_{N-2}$} (n-2); + \draw[red, ->] (n-2) to [out=45,in=135] node[above]{$j_{N-1}$} (n-1); + \draw[red, ->] (n-2) to [out=45,in=135] node[above]{$j_{N-1}$} (n-1); + \draw[red, ->] (n-1) to [out=45,in=135] node[above]{$j_{N}$} (nh); + \end{tikzpicture} + + Mathematical model: + \begin{equation*} + \left\{ \begin{aligned} + \d r_n &= (p_{n+1} - p_n) \, \d t, \\ + \d p_1 &= v'(r_1) \, \d t - \gamma p_1 \dt + \sqrt{2 \gamma {\color{red} (T+\Delta T)}} \, \d W_t^L, \\ + \d p_n &= \bigl(v'(r_n) - v'(r_{n-1})\bigr) \, \d t, \\ + \d p_N &= -v'(r_{N-1}) \, \d t - \gamma p_N \dt + \sqrt{2 \gamma {\color{blue} (T-\Delta T)}} \, \d W_t^R, + \end{aligned} \right. + \end{equation*} + +The Hamiltonian of the system is the sum of the potential and kinetic energies: +\begin{equation*} + H(r,p) = V(r) + \sum_{n=1}^N \frac {p_n^2}{2}, + \quad V(r) = \sum_{n=1}^{N-1} v(r_n). +\end{equation*} +\end{frame} -\bu {\bf When the perturbation is not sufficiently weak?} (thermal transport) -\begin{itemize} -\item compute $\dps \int_\cX [(\mathcal L_0+\eta\wcL)\varphi ] (1+\eta\mathfrak{f}_1)\psi_0 = \mathrm{O}(\eta^2)$ -\item use a pseudo-inverse $Q_\eta = \Pi_0\mathcal L_0^{-1}\Pi_0 - \eta \Pi_0\mathcal L_0^{-1}\Pi_0\wcL\Pi_0\mathcal L_0^{-1}\Pi_0$ -\item allows to prove that $\dps \int_\cX \varphi \, \psi_\eta = \int_\cX \varphi \, \psi_0 + \eta \int_\cX \varphi \, \mathfrak{f}_1 \, \psi_0 + \eta^2 r_{\varphi,\eta}$ -\end{itemize} +\begin{frame} + {Thermal transport in one-dimensional chains (2)} -\end{frame} + \begin{itemize} + \item + When ${\red \Delta T} = 0$, + invariant distribution given by + \[ + \pi(\d r \, \d p) = Z_\beta^{-1} \exp\left(- \beta \left( \frac {|p|^2} {2} + V(r) \right)\right) \, \d r \, \d p, + \qquad \beta = T^{-1}. + \] + + \item + Generator of the dynamics: + \begin{equation*} + \begin{aligned} + \mathcal L + &= \sum_{n=1}^{N-1} (p_{n+1} - p_n) \partial_{r_n} + + \sum_{n=1}^N \Bigl(v'(r_n) - v'(r_{n-1})\Bigr) \partial_{p_n} \\ + &\qquad - \gamma p_1 \partial_{p_1} + \gamma T \partial_{p_1}^2 - \gamma p_N \partial_{p_N} + \gamma T \partial_{p_N}^2 + + {\red \gamma \Delta T (\partial_{p_1}^2 - \partial_{p_N}^2)}. + \end{aligned} + \end{equation*} + + The {\red perturbation} $\widetilde {\mathcal L} = \gamma( \partial_{p_1}^2 - \partial_{p_N}^2)$ + is not bounded relatively to $\mathcal L_0$... + \vspace{.5cm} + $\rightarrow$ Existence/uniqueness of the invariant measure more difficult to prove\footnote{P. Carmona, \emph {Stoch. Proc. Appl.} (2007)} + \end{itemize} +\end{frame} \begin{frame} - \begin{center} -\Huge{Other examples} -\end{center} + {Thermal transport in one-dimensional chains (3)} + + \bu Response function: {\blue total energy current} + \begin{block} + {Definition of the heat flux} + \[ + J = \frac{1}{N-1}\sum_{n=1}^{N-1} j_{n}, + \qquad + j_{n} = -v'(r_n)\frac{p_n+p_{n+1}}{2} + \] + \end{block} + \smallskip + + \bu Motivation: Local conservation of the energy (in the bulk $2 \leq n \leq N-1$) + \[ + \frac{\d\varepsilon_n}{\d t} = + \mathcal L \varepsilon_n = j_{i-1} - j_{i}, + \qquad + \varepsilon_n = \frac{p_n^2}{2} + \frac12 \Big( v(r_{i-1}) + v(r_n) \Big) + \] + + \bu Definition of the {\blue thermal conductivity}: linear response + \[ + \kappa_N = \lim_{\Delta T \to 0} \frac{(N-1)}{2\Delta T} \expect_{\Delta t} [J]. + \] + \end{frame} +\begin{frame} + {Shear viscosity in fluids (1)} -\begin{frame}\frametitle{Shear viscosity in fluids (1)} + Consider a fluid $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$ subjected to a sinusoidal forcing + \begin{figure} + \centering + \includegraphics[height=.5\textwidth]{figures/osc_shear.eps} + \end{figure} -\bigskip -2D system to simplify notation: $\mathcal{D} = \left( L_x\mathbb{T} \times L_y\mathbb{T} \right)^N$ -\begin{figure} -\psfrag{x}{} -\psfrag{z}{} -\psfrag{F}{force} -\center -\includegraphics[height=7cm]{figures/osc_shear.eps} -\end{figure} + Suppose that the box contains $N$ particles of mass $m$, + each subjected to a force $F$. +\end{frame} +\begin{frame} + {Shear viscosity in fluids (2)} + Macroscopic description by Navier--Stokes equation + \[ + \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) - \eta \, \laplacian \mathbf{u} = \frac{\rho}{m} F(y) \, \mathbf{e_x} + \] + Substitution of steady state ansatz $\mathbf{u} = U_x(y) \, \mathbf e_x$ gives + \[ + - \eta U_x''(y) = \overline{\rho} F(y) + \] \end{frame} -\begin{frame}\frametitle{Shear viscosity in fluids (2)} +\begin{frame} + {Shear viscosity in fluids (2)} + pairwise interactions + \[ + V(q) = \sum_{1 \leq i < j \leq N} \mathcal V(\abs{q_i - q_j}). + \] \bu Add a smooth {\blue nongradient force} in the $x$ direction, depending on~$y$ \begin{block}{Langevin dynamics under flow} \centerequation{\left \{ \begin{aligned} d q_{i,t} &= \frac{p_{i,t}}{m} \, dt,\\ d p_{xi,t} &= -\nabla_{q_{xi}} V(q_t) \, dt + {\red \eta F(q_{yi,t}) \, dt} - - \gamma_x \frac{p_{xi,t}}{m} \, dt + \sqrt{\frac{2\gamma_x}{\beta}} \, dW^{xi}_t, \\ - d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, dt - \gamma_y \frac{p_{yi,t}}{m} \, dt - + \sqrt{\frac{2\gamma_y}{\beta}} \, dW^{yi}_t, + - \gamma \frac{p_{xi,t}}{m} \, dt + \sqrt{\frac{2\gamma}{\beta}} \, dW^{xi}_t, \\ + d p_{yi,t} &= -\nabla_{q_{yi}} V(q_t) \, dt - \gamma \frac{p_{yi,t}}{m} \, dt + + \sqrt{\frac{2\gamma}{\beta}} \, dW^{yi}_t, \end{aligned} \right. } \end{block} \smallskip -\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma_x,\gamma_y>0$ +\bu {\red Existence/uniqueness of a smooth invariant} measure provided $\gamma,\gamma>0$ \smallskip @@ -631,7 +959,7 @@ Spectral radius $r$ of the bounded operator \bu Average {\red longitudinal velocity} $u_x(Y) = \dps \lim_{\varepsilon \to 0} - \lim_{\eta \to 0} \frac{\left\langle U_x^\varepsilon(Y,\cdot)\right\rangle_\eta}{\eta}$ + \lim_{\eta \to 0} \frac{\expect_{\eta} \left[ U_x^\varepsilon(Y,\cdot) \right]}{\eta}$ where \vspace{-0.3cm} \[ @@ -650,7 +978,7 @@ Spectral radius $r$ of the bounded operator \frac{1}{L_x} \left( \sum_{i=1}^N \frac{p_{xi} p_{yi}}{m}\chi_{\varepsilon}\left(q_{yi}-Y\right) - \! \! \! \! \! \! \! \! \sum_{1 \leq i < j \leq N} \! \! \! \! - \mathcal{V}'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|} + v'(|q_i-q_j|)\frac{ q_{xi}-q_{xj}}{|q_i-q_j|} \!\int_{q_{yj}}^{q_{yi}} \!\chi_{\varepsilon}(s-Y) \, ds \right) \] @@ -669,7 +997,7 @@ Spectral radius $r$ of the bounded operator \bu {\blue Definition} $\sigma_{xy}(Y) := -\eta(Y)\dfrac{du_x(Y)}{dY}$, {\red closure} assumption $\eta(Y) = \eta > 0$ \begin{block}{Velocity profile in Langevin dynamics under flow} -\centerequation{-\eta u_x''(Y) + \gamma_x \overline{\rho} u_x(Y) = \overline{\rho} F(Y)} +\centerequation{-\eta u_x''(Y) + \gamma \overline{\rho} u_x(Y) = \overline{\rho} F(Y)} \end{block} \bigskip @@ -694,71 +1022,6 @@ Spectral radius $r$ of the bounded operator \end{frame} -\begin{frame}\frametitle{Thermal transport in one-dimensional chains (1)} - -\bu Atoms at positions $q_0,\dots,q_N$ with $q_0 = 0$ fixed - -\medskip - -\bu Hamiltonian $\dps H(q,p) = \sum_{i=1}^N \frac{p_i^2}{2} + \sum_{i=1}^{N-1} v(q_{i+1} - q_i) + v(q_1)$ - -\begin{block}{Hamiltonian dynamics with Langevin thermostats at the boundaries} -\centerequation{ \left\{ \begin{aligned} -dq_i & = p_i \, dt \\ -dp_i & = \Big( v'(q_{i+1}-q_i) - v'(q_i-q_{i-1}) \Big) dt,\qquad i\neq -1, N \\[-3pt] -dp_1 & = \Big( v'(q_2-q_1) - v'(q_1) \Big) dt -- \gamma p_1 \, dt + \sqrt{2\gamma (T{\red +\Delta T})} \, dW^1_t\\[-3pt] -dp_N & = - v'(q_N-q_{N-1}) \, dt -- \gamma p_N \, dt + \sqrt{2\gamma (T{\red -\Delta T})} \, dW^N_t\\[-5pt] -\end{aligned} \right. } -\end{block} - -\medskip - -\bu {\red Perturbation} $\wcL = \gamma( \partial_{p_1}^2 - \partial_{p_N}^2)$ (not $\mathcal L_0$-bounded...) - -\medskip - -\bu Proving the existence/uniqueness of the invariant measure already requires quite some work\footnote{P. Carmona, {\emph Stoch. Proc. Appl.} (2007)} - -\bigskip - -\end{frame} - - -\begin{frame}\frametitle{Thermal transport in one-dimensional chains (2)} - -\bu Response function: {\blue Total energy current} -\begin{block}{} -\centerequation{J = \frac{1}{N-1}\sum_{i=1}^{N-1} j_{i+1,i}, -\qquad -j_{i+1,i} = -v'(q_{i+1}-q_i)\frac{p_i+p_{i+1}}{2}} -\end{block} -\smallskip - -\bu Motivation: Local conservation of the energy (in the bulk) -\[ -\frac{d\varepsilon_i}{dt} = j_{i-1,i} - j_{i,i+1}, -\qquad -\varepsilon_i = \frac{p_i^2}{2} + \frac12 \Big( v(q_{i+1}-q_{i}) + v(q_i-q_{i-1}) \Big) -\] - -\bu Definition of the {\blue thermal conductivity}: linear response -\[ -\kappa_N = \lim_{\Delta T \to 0} \frac{\langle J \rangle_{\Delta T}}{\Delta T/N} -= 2\beta^2 \frac{N}{N-1}\int_0^{+\infty} \sum_{i=1}^{N-1} - \expect\Big(j_{2,1}(q_t,p_t)j_{i+1,i}(q_0,p_0)\Big)\, dt -\] - -\medskip - -\bu {\blue Synthetic dynamics}: fixed temperatures of the thermostats but external forcings -$\to$ {\red bulk driven dynamics} with $\wcL^* = -\wcL + c J$ - -\end{frame} - - \begin{frame} \begin{center} @@ -1273,40 +1536,6 @@ with $\dps \psi_{\dt,\alpha} = \Big(\I + \dt \,\mathcal A_2 \mathcal L^{-1} + \d \end{frame} -\begin{frame} - {Variance reduction techniques?} - -\bu {\blue Importance sampling?} Invariant probability measures $\psi_\infty$, $\psi_\infty^A$ for -\[ -dq_t = b(q_t) \, dt + \sigma dW_t, -\qquad -dq_t = \Big( b(q_t) + \nabla A(q_t) \Big) dt + \sigma dW_t -\] -In general {\red $\psi_\infty^A \neq Z^{-1} \psi_\infty \mathrm{e}^{A}$} -(consider $b(q) = F$ and $A = \widetilde{V}$) - -\bigskip - -\bu {\blue Stratification?} (as in TI...) Consider $q \in \mathbb{T}^2$, $\psi_\infty = \mathbf{1}_{\mathbb{T}^2}$ -\[ -\left \{ \begin{aligned} - dq^1_t & = \partial_{q_2}U(q^1_t,q^2_t) + \sqrt{2} \, dW_t^1 \\ - dq^2_t & = - \partial_{q_1}U(q^1_t,q^2_t) + \sqrt{2} \, dW_t^2 -\end{aligned} \right. -\] -Constraint $\xi(q) = q_2$, {\red constrained dynamics} -\[ -dq^1_t = f(q^1_t) \, dt + \sqrt{2} \, dW_t^1, -\qquad -f(q^1) = \partial_{q_2}U(q^1,0). -\] -Then $\dps -\psi_\infty(q^1) = Z^{-1} \int_0^{1} \e^{V(q^1+y)-V(q^1)-Fy} \, dy \neq \mathbf{1}_{\mathbb{T}}(q^1)$ \\ -where $\dps F = \int_0^1 f$ and $\dps V(q^1) = -\int_0^{q^1} (f(s) - F) \, ds$ - -\end{frame} -\fi - \end{document} % vim: ts=4 sw=4 |