So hello, welcome also from my side for the master course of monitor modeling.

Most of you will remember me from the bachelor course of molecular modeling.

And you will hopefully also remember the key methods of molecular simulations that I was discussing with you.

That is molecular dynamic simulations and Monte Carlo simulation.

And here I would like to wind up what we did in the bachelor classes on this method.

And introduce a few more details on how to set the constant temperature as a thermal star and how to set, how to impose constant pressure as a thermal star.

So let's quickly wind up what is molecular dynamic simulations about.

The key idea is to solve the new distributions of motions as mass times acceleration is the velocities acting on the individual atoms.

And this has to be used in combination with a thermostat for manipulating velocities and a barostat for manipulating atomic positions.

Depending on what kind of ensemble we discuss on the boundary conditions of our simulation cells that should be constant energy, constant temperature, constant pressure and so forth.

Likewise with Monte Carlo simulations.

Here the key idea was to randomly generate new configurations.

So we have a given atomic configurations here.

We know that as the ensemble as manifold of all atomic positions are high.

We know that as all and new ones are being generated randomly by random modifications.

And these modifications will be accepted or rejected according to criteria.

And these criteria are defined by the ensemble heat, constant energy, constant temperature, constant volume or concentration, constant temperature.

Now I would like to go through all the different ensembles.

Starting with the most easy one.

With the isolated system the ZSC also has constant number of particles, constant volume and constant energy.

Now obviously constant energy and constant volume is typical.

So it's just don't remove atoms or don't introduce atoms.

The number of particles is constant.

Same with the volume. If you don't change it, it's going to be constant.

And now we need to consider how to implement constant energy.

And I will do this for both of the methods that we discussed.

Always A will be Monte Carlo simulations and B Monte Carlo simulations.

With Monte Carlo simulations we have to solve the new equations of motion.

And maybe you remember from the bachelor classes, actually once you solve the new equations of motion, no further action is required to keep energy constant.

Actually you can derive the new equations of motions from demanding that total energy is constant.

I was recapitulating this here.

So if total energy, sum of kinetic and potential energy is supposed to be constant, then it should not change as a function of time.

So like this, the first derivative with respect to time of the sum of kinetic energy and potential energy is zero.

Now we can use this derivative considering that both the velocities and the position vectors are time dependent.

And like this, we can do this derivative. Looks like this.

So in order to keep this zero, either all the velocities have to be zero, which is not the general case, or this bracket here has to be zero.

And this directly reads mass times velocity is equal to minus the derivative of the potential energy, which is the acting force of the different particles.

So we actually do not need to do anything to keep the total energy constant in monodynamic simulations, but just solve these equations of motions.

Let's go to B, Monte Carlo simulations.

Here's an example. We randomly generate new configurations, and then we have to decide whether to accept or reject the modification.

And if you want to have constant energy, all you need to do is to compare the energy before and after, and if they are identical, then the new configuration can be accepted.

Let's go to more complicated cases. Now to the ensemble with constant number of particles, constant volume, and constant temperature.

That's the so-called canonical ensemble. So right now our system is considered as in contact with a heat bath, some environment which is used to heat up or to cool down the system.

And for that reason, the energy is no longer constant, but instead the temperature.

With monodynamic simulations, we were achieving a constant temperature by rescaling the velocities.

This is denoted here. So the velocities are changed by a factor s, scaled down by a factor s, so meaning that all the individual velocities in x, y, and z direction are reduced or increased depending if this factor s is larger or smaller than one.

The idea is that temperature is of course related to kinetic energy, so I denoted this here.

This is the sum of kinetic energy taken from all the particles n, and this should be equal to one-half kVT per degree of freedom.

And we have n particles, and each of these particles can move along x, y, and z, so we have 3 times n one-half kVT, which should be the average of the total kinetic energy.

And like this we can calculate the actual temperature. And now we can also see if the temperature is, let's say, too high, what we need to do is to reduce kinetic energy.