Project 4/5 - Molecular dynamics part II: advanced study of Lennard-Jones fluids, deadline December 1st (noon)

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Jennifer Summers
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1 Format for delivery of report and programs The format of the project is that of a printed file or hand-written report. The programs should also be included with the report. Write only your candidate number on the first page of the report and state clearly that this is your report for the last project of FYS3150/FYS4150, fall The project counts 50% of the final mark. If you have collaborated with one or more fellow students, please state the respective candidate number(s). There will be a box marked FYS3150/FYS4150 at the reception of the Department of Physics (room FV128). We encourage you to work two and two together. Optimal working groups consist of 2-3 students. You can then hand in a common report. Project 4/5 - Molecular dynamics part II: advanced study of Lennard-Jones fluids, deadline December 1st (noon) So far you should have a working MD code that simulates argon atoms starting in an FCC lattice. Your code should use cell lists to improve the speed of calculating forces by only counting atom pairs closer than a cutoff length r cut. Here it is also important that you shift the energy as mentioned in the first project. Before you start working on this project, be sure that the energy is conserved fairly well (fluctuations in the 5th leading digit in total energy) and that you can simulate a few thousand atoms with a decent speed. Note: the vec3 class has been updated to support more of the expected features/operators on vectors. In this project we will improve the previous code by adding essential features such as save/load functions and a Python framework for running more advanced simulations. We will also measure the pressure and the heat capacity. a) Save/load function: An MD simulation typically consists of several steps. First we create the initial state (i.e. an FCC lattice), then we heat the system to some temperature before we let the system equilibrate before the measuring of statistical properties begins. We might want to reuse some of these states for other studies, so we should create a save/load function. Since ASCII text files are slow, we will write/read binary files (see appendix ). Create two functions on the system class as described in the appendix. b) Save statistical properties to files: In the StatisticsSampler class, you should now have functions measuring kinetic energy, potential energy and temperature. Add another function that measures the number density. Also, you should write the measured numbers to file so you easily can plot them with your favorite plotting tool. This is important to be able to plot 1

2 the properties and calculate more advanced properties based on the fluctuations/variance of the statistical properties (the heat capacity can be calculated as a function of fluctuations in the kinetic energy). c) Measure the pressure P : The pressure of an MD system can be calculated as (it is actually for the NV T ensemble, but it works fairly well in NV E as well) P = ρ n k B T + 1 N F i r i, (1) 3V i=1 where ρ n is the number density, k B is the Boltzmann constant, T is the temperature, V is the volume, r i is the position of atom i and F i is the total force on atom i. The brackets mean that the pressure should be computed by taking the average value of the sum over many timesteps. Assuming we are using two particle forces only, show that the above equation can be rewritten as P = ρ n k B T + 1 3V r ij F ij, (2) where r ij is the relative vector between two atoms i and j and F ij is the force between them. Implement this (you will have to compute this inside the force loop, just as the potential energy). d) Initial behaviour on new systems : Create a new system with 4000 atoms ( unit cells) and an initial temperature T = 300 K. Plot the energy (kinetic, potential and total), temperature and the pressure as functions of time and discuss what happens. Why is the temperature decreasing? What happens to the pressure? How many picoseconds does the system need to stabilize? e) Create a (small) Python framework: With your program, you can now do one simulation, save the state and continue the state with a new set med parameters (such as with or without the Berendsen thermostat). Instead of writing a complete simulation inside your main function, you should write a Python framework (a sample framework can be found on github.com/andeplane/simulation-framework-fys3150) so that you more easily can run more complicated simulations. The Python code should run your executable with the set of parameters sent as arguments to the program. This will allow you to quickly set up a set of simulations with a Python script and let it run automatically for days. You can also, by using for example matplotlib, create all the plots within the same script. f) Measure the heat capacity C V : i>j 2

3 The heat capacity can be calculated through the relation Ek 2 E k 2 = 3k ( BT 2 1 3k ) B, (3) 2N 2C V where we recognize the left hand side as the variance of the kinetic energy and C V as the heat capacity. Does this correlate with experimental values? g) P ρ n diagram (optional, but gives an extra 30% on your final score): It is now time to make full use of the Python framework. We will now do a series of numerical experiments. In each of these experiments, you should create a new state with a given density (you can control this with the FCC lattice parameter b), use the Berendsen thermostat to reach a desired temperature, let the system equilibrate before you sample statistics. Before we start, we should figure out a few things first: 1) Find the number density ρ n as a function of the lattice constant b. 2) How long should the thermostat be enabled before we have reached the temperature we want? 3) How long do we need to equilibrate before we can start sample statistics, and why not immediately after the thermostat has been turned off? 4) Since each state is very correlated to the state of the previous timestep we shouldn t measure statistical properties every timestep, but rather every n th timestep. How long should we wait between each measurement? Now create a run script using your Python framework and prepare a series of experiments. We want to plot a P ρ n diagram. This means that you should measure the pressure for different densities for a given temperature T giving you a isotherm. Repeat this for different temperatures so that you you get many such isotherms. Remember that for each simulation, the measured value for the pressure should be averaged over many measurements. At what temperature do we see a change? Does this indicate a phase transition? Does the diagram look like what you might expect from a van der Waals gas? Appendices Reading/writing binary files We want to store the atom data (positions and velocities) in a binary format. Compared to ASCII text format, the file size will be much smaller. Here follows an example code on 3

4 how to read and write binary files. void save ( s t r i n g filename, System system ) { ofstream f i l e ( filename, i o s : : out i o s : : binary ) ; i f (! f i l e. is_open ( ) ) { c e r r << "Could not open f i l e " << f i l e n a m e << ". Aborting! " << endl ; e x i t ( 1 ) ; int numberofphasespacecoordinates = 6 system >atoms ( ). s i z e ( ) ; double phasespace = new double [ numberofphasespacecoordinates ] ; vec3 systemsize = system >systemsize ( ) ; int phasespacecounter = 0 ; for ( int i =0; i <system >num_atoms_local ; i ++) { phasespace [ phasespacecounter++] = system >atoms ( ) [ i ]. p o s i t i o n [ 0 ] ; phasespace [ phasespacecounter++] = system >atoms ( ) [ i ]. p o s i t i o n [ 1 ] ; phasespace [ phasespacecounter++] = system >atoms ( ) [ i ]. p o s i t i o n [ 2 ] ; phasespace [ phasespacecounter++] = system >atoms ( ) [ i ]. v e l o c i t y [ 0 ] ; phasespace [ phasespacecounter++] = system >atoms ( ) [ i ]. v e l o c i t y [ 1 ] ; phasespace [ phasespacecounter++] = system >atoms ( ) [ i ]. v e l o c i t y [ 2 ] ; int numberofatoms = system >atoms ( ). s i z e ( ) ; f i l e. w r i t e ( reinterpret_cast<char >&numberofatoms, sizeof ( int ) ) ; f i l e. w r i t e ( reinterpret_cast<char >(phasespace ), numberofphasespacecoordinates sizeof ( double ) ) ; f i l e. w r i t e ( reinterpret_cast<char >(&systemsize [ 0 ] ), 3 sizeof ( double ) ) ; f i l e. c l o s e ( ) ; delete phasespace ; void load ( s t r i n g filename, System system ) { i f s t r e a m f i l e ( filename, i o s : : in i o s : : binary ) ; i f (! f i l e. is_open ( ) ) { c e r r << "Could not open f i l e " << f i l e n a m e << ". Aborting! " << endl ; e x i t ( 1 ) ; int numberofatoms ; f i l e. read ( reinterpret_cast<char >(&numberofatoms ), sizeof ( int ) ) ; double phasespace = new double [ 6 numberofatoms ] ; 4

5 f i l e. read ( reinterpret_cast<char >(phasespace ), 6 numberofatoms sizeof ( dou vec3 systemsize ; f i l e. read ( reinterpret_cast<char >(&systemsize [ 0 ] ), 3 sizeof ( double ) ) ; system >setsystemsize ( systemsize ) ; int phasespacecounter = 0 ; double mass = ; system >atoms ( ). c l e a r ( ) ; for ( int i =0; i<numberofatoms ; i++) { Atom atom = new Atom( mass ) ; atom >p o s i t i o n = vec3 ( phasespace [ phasespacecounter ++], phasespace [ phasespacecounter ++], phasespace [ phasespacecounter ++]); atom >v e l o c i t y = vec3 ( phasespace [ phasespacecounter ++], phasespace [ phasespacecounter ++], phasespace [ phasespacecounter ++]); system >atoms ( ). push_back ( atom ) ; f i l e. c l o s e ( ) ; delete phasespace ; 5

What is Classical Molecular Dynamics?

What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential functions Newton s equations of motion are integrated