Hydrodynamic average (with delta function) - lack of understanding

Question

Consider $\Omega_N=\left\{0\leq x_1\leq x_2\leq\ldots\leq x_N\leq L\right\}$, where $x_i, 1\leq i\leq N$ are, for example, some objects on $[0,L]$.

For phenomena occuring on a length scale large compared to the average distance between the objects, it is convenient to go over to a hydrodynamic description, averaging the microscopic quantities over an interval $\Delta R$ which is small on the macroscopic length, but still contains many objects. Therefore, we introduce the local density of the objects $$ m(x,t)=\frac{1}{\Delta R}\int_{\Delta R}dx\sum_{_i}\delta[x-x_i(t)]~~(1) $$

I do not understand (1). What is done there? in which way is this an average?

Answer 1

You are taking an interval of length $\Delta R$ and counting the objects within the interval. You started with all the $x_i$ as coordinates of the objects. The integral results in the number of objects because for objects outside the interval the delta function is $0$, for objects within the interval the delta function integrates to $1$. Then dividing by $\Delta R$ gives the density of objects at $x$

Answer 2

$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{\mathrm{i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\Li}[1]{\,\mathrm{Li}_{#1}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ $$ \begin{array}{c} {x\color{#f00}{\mid}\phantom{\Huge AAAAAAAAAA}\color{#f00}{\mid} x + \Delta R} \\ {\LARGE\bullet\quad\bullet\qquad\color{#f00}{\bullet}\quad\quad\color{#f00}{\bullet}\ \color{#f00}{\bullet}\ \color{#f00}{\bullet}\quad \color{#f00}{\bullet}\quad\qquad\qquad\color{#f00}{\bullet}\quad\bullet\quad\bullet\quad\bullet\quad\bullet} \end{array} $$ In the interval $\ds{\Delta R}$ you are replacing the detailed description by an 'average' over the interval $\ds{\pars{x,x + \Delta R}}$ which is called a 'coarse-graining'. It just 'counts' positions in that interval. Namely, the $\color{#f00}{red}$ points $\ds{\color{#f00}{\LARGE\bullet}}$: \begin{align} \mrm{m}\pars{x,t} & \equiv \left.{1 \over \Delta R}\sum_{i}1 \,\right\vert_{\ \mrm{x}_{\,i}\,\pars{t}\ \in\ \pars{x\,,\,x + \Delta R}} \label{1}\tag{1} \end{align} Hydrodynamic is a macroscopic discipline which is valid at length scales larger that the microscopical typical lengths.

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However, \begin{align} \int_{x}^{x + \Delta R}\delta\pars{x - \mrm{x}_{i}\pars{t}}\,\dd x & = \left\{\begin{array}{rcl} \ds{1} & \mbox{if} & \ds{\mrm{x}_{i}\pars{t} \in \pars{x,x + \Delta R}} \\[1mm] \ds{0}&& \mbox{otherwise} \end{array}\right. \\[1cm] \mbox{and}\quad \sum_{i}\int_{x}^{x + \Delta R}\delta\pars{x - \mrm{x}_{i}\pars{t}}\,\dd x & \,\,\,\, =\,\,\,\, \pars{\begin{array}{l} \mbox{Number of particles} \\ \mbox{'inside'}\ds{\ \pars{x,x + \Delta R}} \end{array}} \\ &\,\,\,\, =\,\,\,\, \left.\sum_{i}1 \,\right\vert_{\ \mrm{x}_{\,i}\,\pars{t}\ \in\ \pars{x\,,\,x + \Delta R}} \end{align}

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Replacing this expression in \eqref{1}: \begin{align} \mrm{m}\pars{x,t} & \equiv {1 \over \Delta R}\sum_{i}\int_{x}^{x + \Delta R}\delta\pars{x - \mrm{x}_{i}\pars{t}}\,\dd x = {1 \over \Delta R}\int_{x}^{x + \Delta R} \sum_{i}\delta\pars{x - \mrm{x}_{i}\pars{t}}\,\dd x \end{align} From this expression, we can see that $$ \begin{array}{|l|}\hline\mbox{}\\ \ds{\quad\mrm{m}\pars{x,t}}\ \mbox{is the}\ Number\ of\ Particles\ per\ Unit\ Length\,,\ \mbox{in the interval}\ \ds{\pars{x,x + \Delta R}}\,,\quad \\ \ds{\quad}\mbox{at time}\ \ds{t}. \\ \mbox{}\\ \hline \end{array} $$

Indeed, $\ds{\rho\pars{x,t} \equiv \sum_{i}\delta\pars{x - \mrm{x}_{i}\pars{t}}}$ is the Linear Density of Particles or/and the Number of Particles per unit length because: $$ \left\{\begin{array}{l} \ds{\rho\pars{x,t} = 0 \quad\mbox{if}\quad\mrm{x}_{i}\pars{t} \not= x\,,\ \forall\ i} \\[3mm] \mbox{and}\ds{\quad\int_{-\infty}^{\infty}\rho\pars{x,t}\,\dd x =}\quad \mbox{Total Number of Particles.} \end{array}\right. $$