removed some more trajectories and started tutorial 2

tutorials/tutorial2-van-hove-function/README.md

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+# MDANSE Tutorial 2: The van Hove functions
+
+This tutorial will show you:
+* how to run an analysis related to the van Hove functions,
+* how to plot the results of the analysis
+
+**Questions** will be asked in different sections
+of this tutorial. The **answers** will be provided
+at the end of the tutorial.
+
+## Background
+
+### Pair distribution function
+The pair distribution function (PDF) provides us with some information about 
+how atoms are distributed in the system. It tells us what the average 
+number of particles will be at a distance $\mathbf{r}$ in volume 
+$\mathrm{d}\mathbf{r}$ from an atom.
+
+```math
+n_0 g(\mathbf{r}) \mathrm{d}\mathbf{r} = \mathrm{d}N(\mathbf{r})$$
+```
+
+Here $n_0$ is the bulk density, $g(\mathbf{r})$ is the PDF and 
+$\mathrm{d}N(\mathbf{r})$ is the average number of particles in the 
+volume $\mathrm{d}\mathbf{r}$. The PDF is can be written so that it is a 
+function of the distance atoms. In this case the pair distribution 
+function tells us the average number of particles in the shell volume 
+$4 \pi r^2 \mathrm{d}r$ from a distance $r$ of an atom.
+
+```math
+n_0 g(r) 4 \pi r^2 \mathrm{d}r = \mathrm{d}N(r)
+```
+
+Now the PDF $g(r)$ is a function of distance and 
+$\mathrm{d}N(r)$ is the average number of particles in the shell volume 
+$\mathrm{d}\mathbf{r}$. The PDF can be written as a sum of delta 
+functions. 
+
+```math
+n_0 g(r) = \frac{1}{4 \pi r^2} \frac{1}{N} \sum_{k \neq j} \langle \delta (r - \vert \mathbf{r}_k - \mathbf{r}_j \vert) \rangle
+```
+
+**Question 1**: Try to derive this equation. Think about what $N(r)$ is 
+and therefore what $\mathrm{d}N(r) = N(r + \mathrm{d}r) - N(r)$ should be.
+
+For more details on the PDF see **MDANSE Tutorial 1: a phase transition**.
+
+### The van Hove function
+
+```math
+n_0 g(\mathbf{r}) = \frac{1}{N} \sum_{k \neq j} \langle \delta (\mathbf{r} - \mathbf{r}_k - \mathbf{r}_j) \rangle
+```
+
+Is the PDF as a function of $\mathbf{r}$, the van Hove function is closely 
+related and is also a sum of delta functions.
+
+```math
+G(\mathbf{r}, t) = \frac{1}{N} \sum_{k j} \langle \delta (\mathbf{r} - \mathbf{r}_k(t) - \mathbf{r}_j(0)) \rangle
+```
+
+The PDF gives use some insights into the structure of our system while 
+the van Hove function gives us insights into the structure and dynamics 
+of the system. The van Hove function can be split into self and distinct 
+parts.
+
+```math
+G_{\mathrm{s}}(\mathbf{r}, t) = \frac{1}{N} \sum_{j} \langle \delta (\mathbf{r} - \mathbf{r}_j(t) - \mathbf{r}_j(0)) \rangle
+G_{\mathrm{d}}(\mathbf{r}, t) = \frac{1}{N} \sum_{k \neq j} \langle \delta (\mathbf{r} - \mathbf{r}_k(t) - \mathbf{r}_j(0)) \rangle
+```
+
+At $t=0$ the distinct-part of the van Hove function is the PDF 
+$G_{\mathrm{d}}(\mathbf{r}, 0) = n_0 g(\mathbf{r})$. At other times
+the distinct-part of the van Hove function describes distance between 
+atom at different times.
+
+![temperature_analysis](pictures/vhd_diagram.png)
+
+The above figure shows the distances (red arrows) that the distinct-part 
+of the van Hove function depends on. At $t=0$ the van Hove function is simply 
+the PDF. After a some time the blue and green atoms move 
+some distance, the distinct-part of the van Hove function depends on the 
+distances between atoms at $t=0$ and $t=1$.
+
+![temperature_analysis](pictures/vhs_diagram.png)
+
+The self-part of the van Hove function is closely related to the 
+diffusion of a particle. The above figure shows the distances (red arrows) 
+that the self-part of the van Hove function depends on. At $t=0$ there 
+are no arrows since the distance of at atom with itself is zero so that 
+the van Hove function is a delta function 
+$G_{\mathrm{s}}(\mathbf{r}, 0) = \delta(\mathbf{r})$. At $t \neq 0$ the 
+self-part of the van Hove function depends on distances between atoms 
+with itself at different times.
+
+## Scenario of this tutorial
+
+We will analyse a trajectory of liquid argon using the self and 
+distinct-parts of the van Hove functions and see how they change as a 
+function of time.
+
+# Files
+
+This tutorial contains the following files:
+
+## md_inputs
+These are the input files needed to re-run the MD simulation:
+* argon_start_structure.txt - a LAMMPS structure file
+* argon.lmp - a LAMMPS script
+
+## md_outputs
+These are the LAMMPS files:
+* argon_traj_120fs_85k.txt - a LAMMPS custom format trajectory
+
+## mdanse_inputs
+
+## mdanse_outputs
+All the files created by MDANSE will be written here.
+
+# The actual tutorial, step by step.
+In the text of the tutorial, we will concentrate on the
+MDANSE GUI. However, the conversion and analysis jobs can
+be run also without the GUI. The scripts for running all
+the parts of the tutorial are provided in `md_inputs/script*`.
+
+## Convert and load the trajectory
+The trajectory in this example has been created using LAMMPS.
+The LAMMPS script which produced this trajectory is available
+in `md_inputs/argon.lmp` and you will need to open it
+to find some information needed to convert the trajectory
+correctly.
+
+Go to the 'Converters' tab in the GUI, and pick the LAMMPS
+converter. Now you have to pass the correct inputs to the
+converter. The LAMMPS configuration file is
+`md_inputs/argon_start_structure.txt`, and the LAMMPS trajectory file
+is `md_outputs/argon_traj_120fs_85k.txt`. Change the LAMMPS time step to '2',
+since this is the value found in the LAMMPS script for this simulation. Use 
+the generic output filename `mdanse_outputs/converted_trajectory.mdt`.
+
+![temperature_analysis](pictures/conversion_gui.png)
+
+## Calculate the distinct part of the van Hove function
+Select the VanHoveFunctionDistinct job leaving the setting to the 
+defaults. To speed the calculation up you may want to swtich the 
+running mode to multicore and the number of processes to a number 
+greater than 1. Set the outputs file to saving the output
+to `mdanse_outputs/vanhovefunctiondistinct.mdt`, hit run and wait 
+for the job to complete, you check the running jobs tab to check the 
+progress of the job.
+
+![temperature_analysis](pictures/vhd_gui.png)
+
+Once complete the results would load up automatically. Go to the 
+plot creator and plot the `g(r,t)_total` result. Go to the plot holder 
+and in the dataset table set the `g(r,t)_total` to have a main axis for 
+`r` and the Use it? setting to `0,10,20`.
+
+![temperature_analysis](pictures/vhd_plotting_gui.png)
+
+We can see that time advances, the van Hove function begins to flatten 
+and the correlation hole around each atom begins to fill up. The 
+correlation hole is 
+
+**Question 2**: We calculated the distinct-part of the van Hove 
+function for a liquid. What do you think the distinct-part of 
+the van Hove function would look like for a solid?
+
+**Question 3**: We know that at $t=0$ that the distinct-part of the van 
+Hove function is the PDF. What does the van Hove function become as 
+$t \rightarrow infty$ for solid, liquid and gaseous systems?
+
+## Question 1:
+
+```math
+n_0 g(\mathbf{r}) 4 \pi r^2 \mathrm{d}r = \mathrm{d}N(r)
+```
+
+The function $N(r)$ is a sum of step functions so that $N(r)$
+increases by one as $r$ increases each time it comes across an atom on average. 
+Therefore, $\mathrm{d}N(r) = N(r + \mathrm{d}r) - N(r)$ and $\mathrm{d}N(r)$ will be the 
+average number of particles in the shell volume. The derivative of $N(r)$ will be 
+the derivative of the step functions which are delta functions each 
+centered on the distances away to another atom.
+
+```math
+\frac{\mathrm{d}N(r)}{\mathrm{d}r} = \frac{1}{N} \sum_{k \neq j} \langle \delta (r - \vert \mathbf{r}_k - \mathbf{r}_j \vert) \rangle
+```
+
+Where we have written the sum of delta functions inside $\langle \cdot \rangle$ 
+so that the positions are thermally averaged and average the thermal average
+over all possible atom origins.

tutorials/tutorial2-van-hove-function/md_inputs/argon.lmp

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+
+# LAMMPS simulation of 36Ar isotope
+
+# Intialization
+units          real
+dimension      3
+boundary       p p p
+atom_style     full
+
+package omp 0
+suffix omp
+
+atom_modify map array sort 100 2.0
+
+# Atom Definition
+read_data      start_structure.txt
+replicate      4 4 4
+
+pair_style lj/cut 12.0
+pair_coeff 1 1 0.237907 3.405
+
+neighbor        0.5 bin
+neigh_modify    every 10 delay 0 check no
+
+timestep       2
+compute 1 all pressure thermo_temp
+
+thermo_style   custom step temp etotal c_1[*]
+thermo         100
+
+minimize 1.0e-4 1.0e-6 3 5
+
+velocity       all create 130.0 4928459 rot yes mom yes dist gaussian
+fix            1 all nvt temp 130.0 130.0 100.0 tchain 1 
+
+# Settings
+
+# Run the simulation
+run 100
+
+minimize 1.0e-4 1.0e-6 3 5
+
+run 100
+
+minimize 1.0e-4 1.0e-6 3 5
+
+run 1500
+
+unfix 1
+fix            1 all nvt temp 130.0 85.0 100.0 tchain 1
+
+
+run 6000
+unfix 1
+fix            1 all nvt temp 85.0 85.0 100.0 tchain 1
+
+run 120000
+
+unfix 1
+fix 1 all nve
+
+reset_timestep 0
+
+dump           dumpTraj all custom 60 argon_traj_120fs_85k.txt id type x y z
+
+run 60000
+

tutorials/tutorial2-van-hove-function/md_inputs/argon_start_structure.txt

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+# STO frame
+
+        4 atoms
+
+        1  atom types
+
+0 5.73 xlo xhi
+0 5.73 ylo yhi
+0 5.73 zlo zhi
+
+Masses
+
+1 36.0
+
+
+Atoms 
+
+1 1 1 0.0 0.0 0.0 0.0
+2 1 1 0.0 2.86 2.86 0.0
+3 1 1 0.0 2.86 0.0 2.86
+4 1 1 0.0 0.0 2.86 2.86