starts with rdf

_config.yml

@@ -30,7 +30,12 @@ sphinx:
   extra_extensions:
     - sphinx_exercise
     - sphinx_togglebutton
-
+
+parse:
+  myst_enable_extensions:
+    # don't forget to list any other extensions you want enabled,
+    # including those that are enabled by default!
+    - amsmath 
 # Information about where the book exists on the web
 repository:
   url: https://github.com/BioKT/statmecsim  # Online location of your book

_toc.yml

@@ -5,6 +5,4 @@ format: jb-book
 root: intro
 chapters:
 - file: contents/classicalmechanics
-- file: markdown
-- file: notebooks
-- file: markdown-notebooks
+- file: contents/propertiesofliquids 

contents/classicalmechanics.md

@@ -10,42 +10,70 @@ We will start this course with a reminder of Newton's equations of motion,
 formulated by English physicist and mathematitian Sir Isaac Newton in 
 the *Philosophiae Naturalis Principia Mathematica* in 1687. The laws
 establish the following:
-* In the absence of external forces, a body will either be at rest or
+* In the absence of external forces, a particle $i$ will either be at rest or
 continue its motion following a straight line with a constant velocity, 
 $\mathbf{v}_i$.
 * An eternal force $\mathbf{f}_i$ on a body produces an acceleration equal
 to the force divided by the mass of the body, or
-\begin{equation}
+```{math}
+:label:
 m_i\ddot{\mathbf{r}}_i=\mathbf{f}_i(\mathbf{r}_i)
-\end{equation} 
+```
 * If a body A exerts a force on another body, B, then body B exerts a force
 in body A such that
-\begin{equation}
+```{math}
+:label:
 \bf{f}_{AB}=-\bf{f}_{BA}
-\end{equation}
+```
 Here we are already introducing a certain amount of notation. For a particle $i$
 its Cartesian coordinates are expressed using vector $\mathbf{r}_i$. Hence
 for a system with $N$ particles we will denote the position vectors collectively
 as $\mathbf{r}^N$. Also, we are using the force vector $\mathbf{f}_i$, corresponding
 to the force acting on particle $i$. This force can be obtained as a partial 
 derivative of a potential $V$, specifying the interactions in the $N$ particle
 system
-\begin{equation}
+```{math}
+:label: eq-force
 \mathbf{f}_i(\mathbf{r}^N)=-\frac{\partial V(\mathbf{r}^N)}{\partial \mathbf{r}_i}
-\label{eq:force}
-\end{equation}
+```
 For most of this course, the potential energy that we will be considering will be
 due to molecular interactions that we will consider to be pair-wise additive
 and dependent on the distance between particles. Hence,
-\begin{equation}
+```{math}
+:label:
 V(\mathbf{r}^N)=\sum_{i=1}^N\sum_{j=i+1}^Nv(r_{ij})
-\end{equation}
+```
 where we are avoiding double-counting of interactions in the sums. We can
-substitute in Equation \ref{eq:force} and see that the force acting on particle $i$
+substitute in Equation {eq}`eq-force` and see that the force acting on particle $i$
 is in fact a sum over pairwise forces
-\begin{equation}
+```{math}
+:label:
 \mathbf{f}_i=-\frac{\partial V(\mathbf{r}^N)}{\partial \mathbf{r}_i} = 
 -\sum_{i=1}^N\sum_{j=i+1}^N\frac{\partial v(r_{ij})}{\partial \mathbf{r}_i}
 =\sum_{i\neq j}^N \mathbf{f}_{ij} 
-\end{equation}
+```
+
 ## Energy conservation
+One of the most fundamental properties of the mechanical systems that we will
+consider in this course is energy conservation. In general, we can write
+the hamiltonian of the system ($\mathcal{H}$) as the sum of the kinetic energy ($K$) and the potential
+energy ($V$), 
+```{math}
+:label:
+H = K + V
+```
+where 
+```{math}
+:label:
+K=\sum_i^N\frac{1}{2}m_i\dot{\mathbf{r}}_i^2=\sum_i^N\frac{\mathbf{p}_i^2}{2m_i}
+```
+Because $H$ is a constant of motion, we have
+```{math}
+:label:
+\frac{d\mathcal{H}}{dt}=0
+```
+which can be further expanded as
+```{math}
+:label:
+\frac{d\mathcal{H}}{dt}=\frac{d}{dt}\bigg[\sum_i^N\frac{\mathbf{p}_i^2}{2m_i} + V\bigg]
+```

contents/propertiesofliquids.md

@@ -0,0 +1,53 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+---
+# Static properties of liquids
+## The radial distribution function
+Liquids are homogeneous systems, characterized by a uniform
+particle density $\rho=N/V$, where $N$ is the number of particles and
+$V$ is the volume. With simulations, we can interrogate their internal
+structure. One of the most important quantities that can be computed
+from simulations is the *radial distribution function* (in short, RDF or
+$g(r)$). Importantly,this quantity can also be accessed experimentally. 
+
+To calculate the rdf we simply need to histogram pairwise distances and
+calculate
+```{math}
+:label:
+g(r)=\frac{V}{4\pi r^2\Delta rN^2}\sum_i^Nn_i(r,\Delta r)
+```
+where $\Delta r$ is the bin width and we are averaging over
+reference particles the number of times we find another 
+particles in a bin centered at a distance $r$, $n_i(r,\Delta r)$.
+It is important to note that the RDF is dimensionless. At short
+radii, the value of $g(r)$ tends to zero due to excluded volume
+effects. At distances approaching the repulsive core diameter 
+($\sigma$) the RDF typically peaks, reflecting the high density
+coordination shell of nearest neighbours. A second peak at longer
+distances is also often found. As the distribution becomes
+homogeneous at longer $r$ (greater than the correlation length $\xi$),
+the density reaches that of the bulk and the value of the RDF plateaus
+at 1.
+
+When using a rigorous definition of the RDF, we can relate it to the 
+``effective pair potential'' or potential of mean force $v(r)$ as 
+```{math}
+:label:
+g(r)=\exp{(-\beta v(r))}
+```
+or equivalently
+```{math}
+:label:
+-k_BT\ln g(r)=v(r)
+```
+
+## Useful properties related to the radial distribution function 
+The number of particles within a distance $r_c$ can be estimated
+easily from the RDF as 
+```{math}
+:label
+n_c = 4\pi\rho\int_0^{r_c}drr^2g(r)
+```