View source for Third year simulation experiment/Dynamical properties and the diffusion coefficient

'''<big><span style="color:blue; ">This is the sixth (and final) section of the third year simulation experiment. You can return to the previous page, [[Third year simulation experiment/Structural properties and the radial distribution function|Structural properties and the radial distribution function]], or go back to the [[Third_year_simulation_experiment|Introduction]].</span></big>'''

In this final section, we are going to make measurements to get some idea of how much the atoms in our simulation move around. We can characterise this by the diffusion coefficient, which we will calculate by two different approaches.

==Simulations in this Section==

'''<big>TASK 10: In the D subfolder, there is a file ''liq.in'' that will run a simulation at specified density and temperature to calculate the mean squared displacement and velocity autocorrelation function of your system. Run one of these simulations for a vapour, liquid, and solid. You have also been given some simulated data from much larger systems (approximately one million atoms). You will need these files later. make a plot for each of your simulations (solid, liquid, and gas), showing the mean squared displacement (the "total" MSD) as a function of timestep. Are these as you would expect? Estimate <math>D</math> in each case. Be careful with the units! Repeat this procedure for the MSD data that you were given from the one million atom simulations. Compare your data to the million atoms. <span style="color:red">[0 marks - voluntary]</span> </big>'''

==Mean Squared Displacement==
The easiest way to measure <math>D</math> is by exploiting its connection to the [http://en.wikipedia.org/wiki/Mean_squared_displacement mean squared displacement].

<math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math>

Note that in general, we expect the simulation to take a little time to establish this linear behaviour!

The simulations that you have already performed have recorded the MSD for you. Download the "optional output-2" file for each of the simulations, and give it a suitable name. These are text files which contain 5 columns: the first is the number of timesteps since the start of the simulation (*not* the elapsed time in reduced units), the next three contain the mean squared displacement for each of the Cartesian directions (x,y,z), and the final column contains the "total" mean squared displacement.

'''<big><span style="color:blue; ">This is the seventh (and final) section of the third year simulation experiment. You can return to the previous page, [[Third year simulation experiment/Structural properties and the radial distribution function|Structural properties and the radial distribution function]], or go back to the [[Third_year_simulation_experiment|Introduction]].</span></big>'''