Monte Carlo Simulation of Step Fluctuation Hailu Gebremariam Bantu

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Monte Carlo Simulation of Step FluctuationHailu
Gebremariam Bantu

• Vicinal surfaces - isolated step

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Basics of the fluctuationThree physical
quantities consideredWidth, correlation
function, survival probability

• Width
• Small t large t -
• Dynamic constant z a/ß

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• Correlation function
• - G(t) lt(dx(i,t) dx(i,0))2gt
• G(t) t(1/z)
• - C(t) ltdx(i,t)dx(i,0)gt
• for large t C(t) exp(-t/tc)
• where tc is correlation time
• Survival probability a point doesnt return to
  average value over time t.
• for large time S(t) exp(-t/ts)

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The simulation

• Initialization straight step
• Updating - metropolis algorithm
• . Randomly choose a point x(i)
• . Choose a direction dir 1 or 1
• . Calculate the energy difference
• dE abs(x(i1) x(i) dir)
  abs(x(i-1) x(i) dir) abs(x(i1) x(i))
  abs(x(i-1) x(i))
• If dE lt0, the move is accepted.
• if dE gt 0, generate random number
  0ltr2lt1
• if exp(-dE/T) gt r2, the move
  is accepted.
• T temperature parameter
• create array of exp(-dE/T) with 5
  elements.
• 3. Measurement - width, correlation, survival
• optimizing

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Optimizing

• Width
• sum2 0.0
• sum 0.0
• for(j0jltLj)
• sumxj
• sum2xjxj
• var sqrt(sum2/L (sum/L)(sum/L))

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Optimizing -2

• Correlation
• for(j0jltLj)
• for(t0tltNDATAt)
• autocorrjt0.0
• for(k0klt(NDATA-t)k)
• autocorrjt(dhjkt
  dhjk) (dhjkt dhjk)
• autocorrjtautocorrjt/(NDATA-t)
• for(t0tltNDATAt)
• acorrfmt0.0
• for(j0jltLj)
• acorrfmtautocorrjt
• acorrfmt acorrfmt/L

• for(j0jltLj)
• for(t0tltNDATAt)
• autocorrjt0.0
• for(k0klt(NDATA-t)k)
• autocorrjt(dhjkt-dhjk)
  (dhjkt dhjk)
• autocorrtautocorrjt/(NDATA-t)
  
• average it over L and different realizations

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Results

• The speed of the code
• is improved by 75.
• G correlation
• G(t) t(1/z)
• Z 2
• A-correlation
• C(t) exp(-t/tc)
• tc 788 MCS

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Survival probability
t

• The probability that a point comes back to an
  average position.
• ts 381 MCS
• C tc/ts 0.5 lt 1
• Agrees with a theorem

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Conclusion

• The speed is improved by 75 .
• Simulation results can agree well with theory and
  experiments for large system size and with the
  speed improved this can be done easily.
• Interacting steps can be simulated.