Finite Difference Methods and ODE

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• Finite Difference Methods and ODE
• Molecular Dynamics
• (e.g. D W Heermann, Computer Simulation Methods
  in Theoretical Physics, 2nd ed. (Springer-Verlag,
  1990) 3.1)

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Finite difference replacement

• x-h x xz

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Applications

• Ordinary Differential Equations

• Partial Differential Equations

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Typical coupled ordinary differential equations
We could start to integrate the first equation
for y(x) if we knew the function z(x). We could
find z(x) if we could solve the second equation,
unfortunately this requires that we already know
y(x)! just what we set out to find.
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If number of equations not too large and
functions F, G etc. not too complicated can use
predictor-corrector methods, etc.
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Many-body problemsthese conditions are not
satisfied.
Prototype 2nd order ODE is Newtons Law,
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In truncated form these give us Verlets algorithm
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Verlets algorithm
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Can calculate phase space trajectory of a system
of particles, x(t), v(t)

• Start with X(0), V(0)
• Calculate x(t)
• Step the calculation forward, in steps of t at a
  time to find x(t)

Truncation error x(t)-X(t) depending on t
But unstable due to round-off error, x(t) not
near X(t) !
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Verlets Velocity Algorithm
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Truncating (2a) and (2b) gives us the stable
Verlet velocity algorithm
Application to classical Molecular Dynamics
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Molecular Dynamics
In a system of N particles the number of terms in
calculating the forces (-grad(U)) at any one time
is 3N(N-1)/2
A sample problem to calculate the thermal
properties of an isolated cluster of atoms
interacting through a Lennard-Jones
potential. The energy is fixed, this is
microcanonical molecular dynamics
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For simplicity we restrict the motions to two
dimensions e.g. on a surface.
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Lennard-Jones Potential
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Converted by Mathematica      September 26, 2001
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For a system of N particles, the interaction
energy
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with a similar N equations for the y-components.
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These equations are to be integrated, from an
initially specified configuration, using Verlets
velocity algorithm.
What value should be used for t?
As small as possible to minimise truncation error
which will eventually cause the energy not to
be conserved as time goes on
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Argon r03.8 10-10 m, ?/kB119K, M39.948u
(1u931.44MeV/c2)
Natural unit of time
t0.01 corresponds to a physical time of 2.45
10-14 secs
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Procedure

• Setup
• Assign 2N initial positions and velocities at
  t0
• Calculate the total energy
• Calculate the 2N components of force, equation
  (3) above
• Find positions at time t

• Iteration
• Calculate the 2N force components
• Calculate positions at next time step
• Calculate the 2N velocity components

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kinetic energy K is
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Iterate for a number of steps, monitoring the
kinetic energy and when it seems to have
stabilised estimate the equivalent temperature
from, (valid at equilibrium)
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Studying thermal properties
The thermal average of any physical quantity,
A(x), is defined by
where p(x,T)dx is the probability distribution of
its configurations, e.g. the Boltzmann
distribution for the canonical ensemble.
If the system is ergodic this can also be
evaluated by,
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This is so provided the system has relaxed to the
desired temperature, thermal properties can be
estimated, e.g. the specific heat by finding the
average of the kinetic energy, and its
reciprocal, and using the formula,
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Minimisation by Molecular Dynamics
Simulated Annealing

• Start with the system at a high temperature

• Explore the configuration space until in
  equilibrium region
• Reduce temperature, i.e. rescale velocities

• Repeat above steps