200 Canonical ensemble
In the present discussion, we will letS(∆t/2) be a primitive factorization of the
operator exp(iLNHC∆t/2). Applying eqn. (4.11.13) to exp(iLNHC∆t/2), we obtain
eiLNHC∆t/^2 ≈∏nsyα=1S(wα∆t/2). (4.11.14)Finally, RESPA is introduced very simply by applying the operatorS ntimes with a
time stepwα∆t/ 2 n, i.e.
eiLNHC∆t/^2 ≈n∏syα=1[S(wα∆t/ 2 n)]n. (4.11.15)Using the Suzuki–Yoshida scheme allows the propagator in eqn. (4.11.8) to be written
as
eiL∆t≈n∏syα=1[S(wα∆t/ 2 n)]neiL^2 ∆t/^2 eiL^1 ∆teiL^2 ∆t/^2∏nsyi=1[S(wα∆t/ 2 n)]n. (4.11.16)Finally, we need to choose a primitive factorizationS(wα∆t/ 2 n) for the operator
exp(iLNHC∆t/2). Although this choice is not unique, we must nevertheless ensure
that our factorization scheme preserves the generalized Liouville theorem. Defining
δα=wα∆t/n, one such possibility is the following:
S(δα/2) = exp[
δα
4GM
∂
∂pηM]
×
∏^1
j=M′{
exp[
−
δα
8pηj+1
Qj+1pηj∂
∂pηj]
exp[
δα
4Gj∂
∂pηj]
exp[
−
δα
8pηj+1
Qj+1pηj∂
∂pηj]}
×
∏N
i=1exp[
δα
2pη 1
Q 1pi·∂
∂pi] ∏M
j=1exp[
−
δα
2pηj
Qj∂
∂ηj]
×
∏M′
j=1{
exp[
−
δα
8pηj+1
Qj+1pηj∂
∂pηj]
exp[
δα
4Gj∂
∂pηj]
exp[
−
δα
8pηj+1
Qj+1pηj∂
∂pηj]}
×exp[
δα
4GM
∂
∂pηM]
, (4.11.17)
whereM′=M−1. In eqn. (4.11.17), the symbol
∏ 1
j=M′indicates a backward product
in whichjstarts atM′and is decremented in each term of the product untilj= 1
is reached. Eqn. (4.11.17) may look intimidating, but each of the operators appearing
in the primitive factorization has a simple effect on the phase space. In fact, one can
easily see that most of the operators are just the translation operators introduced in
Section 3.10. The only exception are operators of the general form exp(cx∂/∂x), which
also appear in the factorization. What is the effect of this type of operator?