Lecture 5: MD1 Flashcards
What does statistical mechanics bridge?
Macroscopic and microscopic
What are the aims of Statistical mechanics?
-predict behaviour of bulk matter from knowledge of molecular properties
-extract microscopic properties from a knowledge of bulk behaviour
What is a macro state?
A particular set of values of energy, the number of particles and then volume of an isolated thermodynamic system
What is a Microstate?
Specific microscopic configuration of the system
are thermodynamic properties of individual micro states measureable?
No- we observe averages instead
What does movement that changed the arrangement or particles cause?
A new micro state
Are all micro states equally probable?
Yes
Are all macro states equally probable?
No- prob is directly proportional to number of associated micro states
The more ways a microstate can occur the more probable it is
How can classical systems of N particles be described?
In 6N-dimensional phase space
What’s the equation for 6N phase space?
X(r,p)
r=(x1,y1,z1…) position
p=(px1, py1, pz1…) momentum
What’s a good way to visualise phase space?
-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
What is the ergodic hypothesis?
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
What is the ergodic hypothesis simplified?
The time average is the same as the ensemble average
Is there proof of the ergodic hypothesis?
No but there is overwhelming practical evidence
What is an ensemble?
A system at equilibrium corresponding to different physical requirements
Describe the microscopic va macroscopic view of ensembles
Macroscopic view: looks unchanging
Microscopic view: different microstates
What statistical ensemble would the coefficient of thermal expansion use?
Isobaric-isothermal NPT ensemble that allows for volume changes
Describe the process of MD program
- Read in parameters (specify system conditions)
- Initialise system (initial positions and velocities)
- Compute forces on all properties
- Integrate through time (use Newtons equations of motions)
- Compute and print averages
What is important to remember when picking the right approach
Is the simulation sensible?
What will it tell me?
Why is it important to initialise the system?
-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
How are positions assigned for a crystal?
Place molecules in desired position along a lattice
How are positions assigned for a liquid?
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
How are positions assigned for a gas?
Randomly
What are some options for assigning momentum?
-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
How is force calculated in MD simulations?
F=-dU/dr
What’s mathematically good about F and U?
(sigma/r)^6 can be calculated and then used for both calculations rather than calculating both from scratch
What is the equation for energy in the Harmonic potential?
U(r) =1/2 k(r-r0)^2
Why do we need to integrate through time?
To get the ensemble averages of our property of interest
What is Newton’s second law?
F=ma
Describe the method used for solving time integration
-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
What are two time integration errors?
-truncation errors (due to accuracy of finite difference method)
-round off errors (due to implementation of algorithm)
How can time integration errors be reduced?
By reducing dt
What is the Verdot algorithm?
-calculated both forward and backwards step
-more accurate than forward simple step time integration
What does the verlet algorithm not generate?
Velocities needed for kinetic energy and temp
Why is verlet algorithm computationally cheaper?
Lower data storage required
What is the leapfrog algorithm
-modification of verlet algorithm
-allows velocities and positions to leap over one another ensuring velocities are accurately calculated
What does leapfrog Algorithm not generate?
Velocities at time t
Instead need to take means from t+/-1/2 dt
What is the velocity verlet algorithm
-more accurate trajectories and less drift in total energy
-increased computational cost
Describe the velocity verlet algorithm
-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
