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Wang-Landau sampling method
Wang­Landau method is the random walk in the energy space with a flat histogram. F. Wang and D.P. Landau, PRL 86 (2001) 2050 and PRE 64 (2001) 056101 D.P. Landau and K. Bunder, A guide to Monte Carlo simulations in statistical physics, Cambridge University Press, 2009


Wang-Landau algorithm
· · · · set

g(E)=1

for all energy values

E; f
= 2.718281828;

fix value of the modification factor generate random state

s

i

;

calculate energy of the state

E

k

;

· set auxiliary histogram H(E)=0 for all E; _______________________________________________________________ · choose randomly spin si and calculate energy · If

E

k+1

of the state with the flipped spin

si -s

i

;

g(Ek+1) < g(Ek),

then accept the new state.

If not, accept the new state with probability g(Ek) / g(Ek+1), Otherwise keep the current state unchanged (not-accept). -----------------------------------------------------------------------------------------------------------


Wang-Landau algorithm /2
· Acceptance of the new state consists with the following steps: flip spin

s

i

:= -

s

i

update DoS entry update histogram

g(Ek+1) := f g(Ek) H(Ek+1) := H(Ek) +1, g(Ek) := f g(Ek) H(Ek) := H(Ek) +1, MN say 104
times, where

· Non-acceptance of the new state consists with the following steps: update DoS entry update histogram

· Repeats steps between lines on the previous slide some

N

is the number of spins and M is some large value ( ) and check the "flatness" of the histogram (it is an empirical way to treat flatness and originally some 5% of flatness recommended).


Wang-Landau algorithm /3
If histogram is not "flat" enough, repeat additional If histogram is flat, decrease factor normalize function g(E) such that and with proceed the process ...

MN

times, ...

f = f1/2, g(E0) = 1

, reset histogram

H(E)=0

The algorithm may be stopped with some small value of f.e.

f

close enough to unity,

log f = 10-9


DoS of Ising model
Exact - Beale, PRL 76 (1996) 78

DoS g(E) for Ising model 16x16. Circles - exact, line - Wang-Landau sampling.


DoS of Ising model


DoS of Ising model


DoS of Ising model


Transition matrix

Transition from level energy Ek to level energy Ek

'

Count the number of transitions (k,k'). Build Matrix of that number M(k,k'). The sum over the rows is the same in the large limit of events. I.e. it is the Markov process in the energy space. Result: instead of checking the flatness of the histogram, one has to check variation of the sum on the rows.


Stability of DoS

Exact DoS is not stable with respect of f-process

f-process drives DoS out of the exact solutions f-process drives DoS to a little bit different limiting distribution


Parallelization







Divide energy space on a number of intervals Perform F-process within each interval (accept only jumps within the interval) up to the flattness creterium Combine spectrum, decrease f - large errors at the edges solution ­ make intervals overlaps up to the middled and "combine" histogram, decrease f "Good" scalability


Wang-Landau sampling method Discussion and Conclusion:
· Wang-Landau method is less effective for large systems in comparison with MUCA · Wang-Landau sampling method is a random walk in the configuration space and the Markov process in the energy space. · What is the right algorithm with respect of the f-function? ­ ? the one which drives the system to the true DoS? Parallelization: + scalability is better for large systems - does not work properly for "complicated" systems + good for 1-st order phase transition