**UFRJ**

# Monte Carlo Methods

(Parte **1** de 8)

1 MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/21/2009)

Material for these lecture notes was compiled from the references below

9L. Zhigilei, MSE 524 (Virginia Tech)

9MIT’s 3.320 course notes (Prof. G. Ceder)

9D. Chandler, Introduction to Modern Statistical Mechanics

9D.A. McQuarrie, Statistical Thermodynamics or Statistical Mechanics

9D. Frenkel, Introduction to Monte Carlo Methods

9D. Frenkeland B. Smit, Understanding molecular simulation

9M.E.J. Newman and G.T. Barkema, Monte Carlo Methods in Statistical Physics

9K. Binder and D.W. Heerman, Monte Carlo Simulation in Statistical Physics

9K. Kremer, Entangled polymers: from universal aspects to structure property relations

9J. Baschnagelet al., Monte Carlo simulation of polymers: coarse-grained models

Monte Carlo Methods

2 MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/21/2009)

Molecular Dynamics versus Monte Carlo

¾In a MD simulation, one way you calculate certain properties is by tracking them over time. If you could calculate certain properties for a given microscopic state (e.g. energy or volume), you calculate their macroscopic properties as averages (an integral over time) over the simulation. There are two issues:

¾At first, in that average you only include states that you can reach with the

MD simulation. For example, since you simulate over a finite time, if there are excitations which can determine the average energy that occur over long-time, you would not include them in that average. In this case, statistical sampling may be more efficient.

These averages only include states that occur in the time scale of the MD simulation

¾Secondly, you maybe interested to compute the average properties but not interested in simulating the system dynamics. Then sometimes you can do simpler techniques (less computing intensive) than MD. You could compute these averages by sampling.

3 MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/21/2009)

Statistical sampling versus MD

¾Sampling maybe useful not only if you care about the averages but not the dynamics itself but also in situations where you don’t know the dynamics.

¾In MD, you basically assume that the atoms move with Newton equation of motion (so you know the dynamics). There will be certain material models for which you have no idea about the dynamics (e.g. a spin model).

¾A spin model is a model of magnetic moments being oriented in space: In many cases, you don’t know the spin dynamics. If all you want to know is e.g. the average magnetic moment, there is no way of doing dynamics on the spins. You’l actualy have to sample them.

4 MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/21/2009)

A time scale problem: inter-mixing

¾Let us consider an example of a binary mixture. You may want to know the average energy of this system.

To average the energy, the system would have to go through many configurations in the simulation

Diffusion is required

¾In any given configuration, let’s assume that we have a method to calculate the energy of the system. At high temperature, the atoms will simply hop around and give you some average energy.

What kind of time scale you need to get that average right?

5 MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/21/2009)

A time scale problem: inter-mixing

¾Lets estimate the diffusion constant Drequired to get significant number of atom exchanges.

¾To get an idea of the kind of hopping rate you need, you can assume that the atoms do a random walk. Then the root mean square displacement you take is the number of jumps you take, N, times the jump distance squared (Na2 ).

Recall the relation between the diffusion constant and the root mean square displacement.

¾Now we can relate to jump rate Γ=to the diffusion constant.

<r2(t)> = N a 2

6 MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/21/2009)

A time scale problem: inter-mixing

¾If you know how long your simulation can last, you know what jump rate you need to see atoms jump in that simulation. So you can relate that to the diffusivity you need.

¾We know that the jump rate over a barrier is some vibrational frequency (frequency with which they try to go up the hill) times the success factor

(Boltzmann factor e

(Parte **1** de 8)