Thermal properties of a system by Simulation

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Lecture 10

• Thermal properties of a system by Simulation
• Optimisation by Simulated Annealing (SSP 6.1,
  6.2)

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An aside on Monte Carlo integration (SSP 7.1-7.4)
An estimate of the mean value of any function
f(x) over an interval a,b in which x occurs
with the probability ?(x)dx is given by
where the K values xi are selected from the
distribution ?.
This can also be thought of as an estimate of the
integral
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An unbiased estimate of any integral, ?f(x)dx
over the region (0,1) is
where the variates ?i are selected from the
uniform distribution on (0,1)
For a given size of sample N, this estimate will
have an uncertainty (standard error) associated
with it.
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Importance sampling
It will often be possible to reduce the
uncertainty (variance reduction) by rewiting the
integral as
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The estimate is then
For a suitably chosen ?(x) the uncertainty in
this estimate may, for the same sample size N, be
much smaller than in the simpler case.
Integrals of this type are important in
Statistical mechanics and other fields.
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Monte Carlo Simulation of thermal systems
Aim to predict thermal properties of a system of
particles with known interactions using classical
statistical mechanics by Monte Carlo simulation
? Isolated system, constant energy,
microcanonical ensemble Thermal demon (Molecular
Dynamics)
? System in contact with thermal reservoir at
temperature T, canonical ensemble Heat Bath
method Metropolis method
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System is specified by some set of
configurational coordinates q, Positions, Angles
, Spin orientations . . .
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Canonical ensemble using Metropolis method
At equilibrium, configurations q occur
distributed according to the Boltzmann
distribution,
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For comparison with experiment, calculate
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Thermal properties of a cluster
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Example A cluster of N atoms interacting through
the Lennard-Jones potential.
At T0 the atoms are in a fixed configuration of
lowest energy (classically)
At T?0 not a unique configuration, but some 3N
component configuration r(i)(r1, r2, rN)
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The probability of a given configuration r(i)
occuring at that temperature is given by the
Boltzmann distribution,
Where the normalising constant is 1/Q, with Q the
configurational partition function,
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In the case of an N-particle L-J cluster,
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Typical configuration of an N13 particle cluster
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If we want to calculate, for example, the mean
potential energy,
we have to be able to sample from the
distribution pB, then our estimate for ltUgt is
In simulating annealing, we have to be able to
sample from the distribution at the different
temperatures.
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Selection from the Boltzmann distribution using
the Metropolis algorithm
Starting from an arbitrary configuration r(0) a
chain of configurations is evolved,
Select a tentative config. r(t) r(i)? Call a
uniform random number ?
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• After the influence of the initial arbitrary
  configuration has died away, sequences of
  configurations in the chain will be
  representative of the equilibrium Boltzmann
  distribution at that temperature.

• Note that at any temperature, configurations
  with a range of energy are being sampled, with
  heat being taken or returned to the heat
  reservoir.

• Note that these configurations, in this case,
  could also be generated by Molecular Dynamics
  using the Newtonian dynamics dictated by the
  forces involved plus a random Brownian motion
  appropriate to the temperature T.

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Optimisation by Simulated Annealing
To find the configuration at T0, i.e. the
minimum energy arrangement of the atoms, start at
a high temperature and anneal the cluster
Why not quench?
Same as the method of steepest descent!
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Problems
Annealing schedule How fast to reduce
temperature? How much sampling to do at each
temperature?
Absolute minimum not guaranteed to be encountered!
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Annealing algorithm
(1) Input coordinates of initial configuration
r(0), calculate U(r(0)) (2) Input initial
temperature (3) Use Metropolis algorithm to
generate configurations (archive minimumum to
date) until in equilibrium, then find average ltUgt
and its variance ?2 (4) Reduce temperature by
?T (5) Repeat steps (3) and (4) until difference
between ltUgt and archived minimum is lt?
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Examples of optimisation problems

• Folding configuration of polymer molecules

• Layout of circuit elements on computer chip

• Traveling salesman problem

• Ground state of spin-glass systems

• Configuration of charges on a disk (SSP6.2.2)

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• In each case there is a cost function U which
  depends on a set of parameters, positions,
  angles, spin directions, etc.

• A temperature-like parameter is introduced and
  the thermal behaviour of the system simulated
  as though it was a system of particles in the
  canonical ensemble.

• As in simulating real annealing, the parameter
  space is sampled as the temperature is
  gradually reduced and the lowest energy of the
  cost function monitored.

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Traveling Salesman Problem