1549380323-Statistical Mechanics Theory and Molecular Simulation

478 The Feynman path integral

coordinate transformation capable of uncoupling the harmonic term in eqn. (12.6.3),
then we can replace the single fictitious massm′ in the classical Hamiltonian with
a set of massesm′ 1 ,...,m′P such that only one harmonic frequency remains. Finally,
we can then adjust the time step for stable integration of motion having that charac-
teristic frequency and/or employ RESPA to the problem. This will ensure adequate
sampling of all modes of the cyclic chain. In fact, we have already seen an example
of such a transformation in Section 4.5. Eqns. (4.5.38) and (4.5.39) illustrate how a
simple transformation uncouples the harmonic term for a model polymer that, when
made cyclic, is identical to the discrete path integral. The one-dimensional analog of
this transformation appropriate for eqn. (12.6.2) is

u 1 =x 1

uk=xk−

(k−1)xk+1+x 1

k

, k= 2,...,P, (12.6.5)

the inverse of which is

x 1 =u 1

xk=uk+

k− 1

k

xk+1+

1

k

u 1 , k= 2,...,P. (12.6.6)

Note that, as in eqn. (4.5.39), eqn. (12.6.6) is defined recursively. Becausex 1 =u 1 ,
the recursion can be seeded by starting with thek=Pterm and working backwards
tok= 2. The inverse can also be expressed in closed form as

x 1 =u 1

xk=u 1 +

∑P

l=k

k− 1

l− 1

ul, k= 2,...,P. (12.6.7)

The transformation defined in eqns. (12.6.5), (12.6.6), and (12.6.7)is known as a
staging transformationbecause of its connection to a particular path-integral Monte
Carlo algorithm (Ceperley and Pollock, 1984), which we will discuss in the next section.
The staging transformation was first introduced for path-integral molecular dynamics
by Tuckermanet al.(1993). The variablesu 1 ,...,uPare known asstaging variables, as
distinguished from the original variables, which are referred to asprimitive variables.
We now proceed to develop a molecular dynamics scheme in terms of the staging
variables. When the harmonic coupling term is evaluated using these variables, the
result is
∑P

k=1

(xk−xk+1)^2 =

∑P

k=2

k

k− 1

u^2 k, (12.6.8)

which is completely separable. Since the Jacobian of the transformation is 1, as we
showed in eqn. (4.5.45), the discrete partition function becomes

QP(L,T) =

∫

dp 1 ···dpP

∫

du 1 ···duP