Transition path sampling 305is determined entirely by the potential energyU. For any trial move, the acceptance
probability is taken to be
A(E 2 |E 1 ) = min[
1 ,
Ω(E 1 )
Ω(E 2 )
]
. (7.6.3)
Of course, the move is initially accepted with probability 1. After sucha move, the
system will have an energyE that is eitherE 1 orE 2 , depending on whether the
move is accepted or rejected. The key step in the Wang–Landau algorithm is that
the density of states Ω(E) is modified after each move according to Ω(E)→Ω(E)f,
wherefis a scaling factor withf >1. Note that the scaling is appliedonlyto the
energy bin in whichEhappens to fall. All other bins remain unchanged. In addition,
we accumulate a histogramh(E) of each energy visited as a result of such moves. After
many iterations of this procedureh(E) starts to “flatten out,” meaning that it has
roughly the same value in each energy bin. Once we decide thath(E) is “flat enough”
for the given value off, we start refining the procedure. We choose a new value of
f, for example,fnew=
√
fold(an arbitrary formula suggested by Wang and Landau)
and begin a new cycle. As before, we wait untilh(E) is flat enough and then switch
to a new value off(using the square-root formula again, for example). After many
refinement cycles, we will find thatf →1 andh(E) becomes smoothly flat. When
this happens, Ω(E) is a converged density of states. The Wang–Landau approach can
be used for both discrete lattice-based models as well as continuous systems such as
simple fluids (Yanet al., 2002) and proteins (Rathoreet al., 2003).
An interesting point concerning the Wang–Landau algorithm is that itdoes not
satisfy detailed balance due to the application of the scaling factor to the density
of states Ω(E), causing the latter to change continually throughout the calculation.
Asfapproaches unity, the algorithm just starts to satisfy detailed balance. Thus,
the Wang–Landau approach represents a Monte Carlo method that can work without
strict adherence to detailed balance throughout the sampling procedure. Note that it
is also possible to use molecular dynamics to generate trial moves before application
of eqn. (7.6.3). If a large time step is used, for example, then aftermsteps, energy
will not be conserved (which also occurs in the hybrid Monte Carlo scheme), meaning
that there will be an energy change fromE 1 toE 2. Thus, the potential energy will
change fromU 1 toU 2 , and the kinetic energy will also change fromK 1 toK 2 , with
E 1 =K 1 +U 1 andE 2 =K 2 +U 2. In this case, the acceptance criterion should be
modified, as suggested by Rathmoreet al.(2003), according to
A(E 2 |E 1 ) = min[
1 ,e−β∆KΩ(U 1 )
Ω(U 2 )
]
, (7.6.4)
where ∆K=K 2 −K 1.
7.7 Transition path sampling and the transition path ensemble
The last technique we will describe in this chapter is something of a departure from
the methods we have discussed thus far. Up to now, we have discussed approaches
for sampling configurations from a specified probability distribution,and because our