Subject: RE: G:Histogram method, redux

From: Jan Reimers <[email-masked]>
Date: Thu, 4 Feb 1999 14:49:36 -0800

I will try once more to explain with the magnetism analogy (as used by
Ferrenberg and Swendsen).
		
			Magnetism			Protein folding
You tune		Temp & Field (T,Hm)		Temp &
"FoldingField" (T,Hfold)
MC measures		Energy & Magnetization (E,m)	Energy & Degree
of folding (E,f)

In the magnetic system below Tc the free energy F(m) becomes bimodal
with minima at +m and -m.  The MC time required to jump from the +m to
-m grows exponentially with (Tc-T), so it is hopeless to measure the
relative probability of the two states at low T.  You need a way to get
around this.  
	For the multi histogram method we have to options for data
collection, 1) sweep various T values and store energy histograms H(E),
m(e), m^2(e) etc. (as do Ferrenberg and Swendsen) or 2) sweep H values
and store H(m), E(m), E^2(m) etc.  Joining histograms H(E) provides
information on the relative probabilities of the various energy states
P(E), not the probabilities of magnetization states.  For the latter you
need to build up H(m), which is most efficiently done sweeping the
external field that couples to m in the Hamiltonian, Hm.
	Enough magnetism.  For protein folding the story is similar,
joining H(E) histograms does not help you determine H(f). f=degree of
folding).  You need to define a folding parameter (doesn't sound too
hard) and add a term Hfold*f to your Hamiltonian.  Hfold now  "pushes"
your system towards folded (Hfold>0) and unfolded (Hfold<0) states.  In
the end you calculate P(f) (or the log) using the multi histogram method
for a value of Hfold=0, so the fact that Hfold is unphysical is
irrelvant to the final number, it just forces the metropolis algorithm
to sample states that are extremly low probability when Hfold=0.
________________________
                                                  
Jan N. Reimers (Ph.D.)
Manager, Materials Research 
NEC Moli Energy (Canada) Ltd.
________________________