The Hard Sphere Potential

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The Hard Sphere Potential
?
U(r)
?
0
r
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Phase Transition in Hard Sphere Simulation
Compressibility
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Effect of Packing Density On Order Parameter
Order Parameter
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Radial Distribution Function
Probability that an atomic center lies in a
spherical shell of radius r and thickness dr with
the shell centered on another atom
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Radial Distribution Function
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Reduced Units
Set mass to be a fundamental unit
Then momenta
Force

• Use of reduced units avoids the need to conduct
  essentially duplicate simulations
• Time saved in the calculation of potential
  energy, forces etc

In reduced units
Coulombs Law
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Why Finite Difference? Interaction Potentials
Hard Sphere Potential
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(12, 6) Function
Overlap forces Repulsive
Dispersive forces Attractive
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Taylor Series
Truncation error
Finite-difference methods

• Replace differentials with differences
• Replace differential equations with
  finite-difference equations

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Total global Error vs Step Size
Truncation error

Round-off error
Determine
Algorithmic stability
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Eulers method

• First order term of Taylor expansion

12
Phase-Space of 1-DHO
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1DHO Algorithm Stability And Step Size
dt0.001
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1DHO Algorithm Stability And Step Size
dt0.005
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1DHO Algorithm Stability And Step Size
dt0.05
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Verlet Algorithm

• Third order truncation error
• Special setup of initial conditions

Leapfrog Algorithm
Velocity Verlet Algorithm
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(a) Verlet (b) Leapfrog (c) Velocity Verlet
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Predictor-Corrector Algorithms

• From the current position x(t) and velocity v(t)
• Predict the position x(t?t) and velocity(t?t)
  at the end of next step
• Evaluate the forces at t ?t using the predicted
  position
• Correct the predictions using some combination of
  predicted and previous values of positions and
  velocity

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1DHO Algorithm Stability And Step Size
Velocity Verlet Gear Predictor-Corrector
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1DHO Algorithm And Time Step
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Choosing a time step
Small Steps phase space is covered too slowly
Large steps causes instabilities and errors
from the approximations
Appropriate step size Efficient simulation
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Comparison of MD Algorithms

• Velocity Verlet
• Time reversible
• Symplectic Error in total energy is "bounded"
  (valid only when PE is indpt of momenta and KE is
  indpt of coordinates)
• Does not work well in other coordinate systems.

• Gear Predictor Corrector
• Not time reversible
• Not symplectic Errors are much smaller, but
  continue to grow with the no of steps
• More accurate for short time steps
• Implicit Method ie the equation at n1th point is
  defined in terms of both terms of n and of the
  n1th point.

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Comparison of V-Verlet Gears PC
System Box of 256 Argon atoms
Source http//www.teoroo.mkem.uu.se/daniels/ngssc
_numana/
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Role of Arithmetic Precision
Float
Double
Simulation Based on Gears predictor-corrector
algorithm
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Simulation of a Box of Argon Particles