Lecture 5: MD1

Question 1

What does statistical mechanics bridge?

Answer 1

Macroscopic and microscopic

Question 2

What are the aims of Statistical mechanics?

Answer 2

-predict behaviour of bulk matter from knowledge of molecular properties
-extract microscopic properties from a knowledge of bulk behaviour

Question 3

What is a macro state?

Answer 3

A particular set of values of energy, the number of particles and then volume of an isolated thermodynamic system

Question 4

What is a Microstate?

Answer 4

Specific microscopic configuration of the system

Question 5

are thermodynamic properties of individual micro states measureable?

Answer 5

No- we observe averages instead

Question 6

What does movement that changed the arrangement or particles cause?

Answer 6

A new micro state

Question 7

Are all micro states equally probable?

Answer 7

Yes

Question 8

Are all macro states equally probable?

Answer 8

No- prob is directly proportional to number of associated micro states
The more ways a microstate can occur the more probable it is

Question 9

How can classical systems of N particles be described?

Answer 9

In 6N-dimensional phase space

Question 10

What’s the equation for 6N phase space?

Answer 10

X(r,p)
r=(x1,y1,z1…) position
p=(px1, py1, pz1…) momentum

Question 11

What’s a good way to visualise phase space?

Answer 11

-Consider a ball bouncing in a box
-each frame is a microstate
-phase space is X=(x,y,z,px,py,pz)
-over time the ball bounces all over the box exploring the inside of the box

Question 12

What is the ergodic hypothesis?

Answer 12

A system/microstate is equally likely to be found in any microstate, or, that over long periods of time all accessible micro states are equiprobable

Question 13

What is the ergodic hypothesis simplified?

Answer 13

The time average is the same as the ensemble average

Question 14

Is there proof of the ergodic hypothesis?

Answer 14

No but there is overwhelming practical evidence

Question 15

What is an ensemble?

Answer 15

A system at equilibrium corresponding to different physical requirements

Question 16

Describe the microscopic va macroscopic view of ensembles

Answer 16

Macroscopic view: looks unchanging
Microscopic view: different microstates

Question 17

What statistical ensemble would the coefficient of thermal expansion use?

Answer 17

Isobaric-isothermal NPT ensemble that allows for volume changes

Question 18

Describe the process of MD program

Answer 18

1. Read in parameters (specify system conditions)
2. Initialise system (initial positions and velocities)
3. Compute forces on all properties
4. Integrate through time (use Newtons equations of motions)
5. Compute and print averages

Question 19

What is important to remember when picking the right approach

Answer 19

Is the simulation sensible?
What will it tell me?

Question 20

Why is it important to initialise the system?

Answer 20

-There is a large unfavourable potential energy of the system that leads to the system exploding/breaking
-there is particle-particle overlap
-there is unfavourable intramolecular interactions that can lead to poor initial choices eg long bonds

Question 21

How are positions assigned for a crystal?

Answer 21

Place molecules in desired position along a lattice

Question 22

How are positions assigned for a liquid?

Answer 22

Start from a crystal, run at very high temp, cool to desired temp
Place at random at a slightly lower density, run at constant pressure to get desired density

Question 23

How are positions assigned for a gas?

Answer 23

Randomly

Question 24

What are some options for assigning momentum?

Answer 24

-temp ramp from 0K
-assign velocities from a uniform distribution between two values and scales to produce target temp
-assign velocities from a Gaussian distribution with mean=0 and variance scaled to produce target temp

Question 25

How is force calculated in MD simulations?

Answer 25

F=-dU/dr

Question 26

What’s mathematically good about F and U?

Answer 26

(sigma/r)^6 can be calculated and then used for both calculations rather than calculating both from scratch

Question 27

What is the equation for energy in the Harmonic potential?

Answer 27

U(r) =1/2 k(r-r0)^2

Question 28

Why do we need to integrate through time?

Answer 28

To get the ensemble averages of our property of interest

Question 29

What is Newton’s second law?

Answer 29

F=ma

Question 30

Describe the method used for solving time integration

Answer 30

-the equations must be solved numerically rather than analytically
-time is quantised
-simulated time goes in time steps dt
-need to know positions and time derivatives

Question 31

What are two time integration errors?

Answer 31

-truncation errors (due to accuracy of finite difference method)
-round off errors (due to implementation of algorithm)

Question 32

How can time integration errors be reduced?

Answer 32

By reducing dt

Question 33

What is the Verdot algorithm?

Answer 33

-calculated both forward and backwards step
-more accurate than forward simple step time integration

Question 34

What does the verlet algorithm not generate?

Answer 34

Velocities needed for kinetic energy and temp

Question 35

Why is verlet algorithm computationally cheaper?

Answer 35

Lower data storage required

Question 36

What is the leapfrog algorithm

Answer 36

-modification of verlet algorithm
-allows velocities and positions to leap over one another ensuring velocities are accurately calculated

Question 37

What does leapfrog Algorithm not generate?

Answer 37

Velocities at time t
Instead need to take means from t+/-1/2 dt

Question 38

What is the velocity verlet algorithm

Answer 38

-more accurate trajectories and less drift in total energy
-increased computational cost

Question 39

Describe the velocity verlet algorithm

Answer 39

-calculate position at t+dt
-calculate half step velocities t+1/2 dt
-calculate acceleration at t+dt from forces using r(t+dt)
-calculate velocities at t+dt