1549380323-Statistical Mechanics Theory and Molecular Simulation

Time evolution operator 111

theorem is somewhat involved and is, therefore, presented in Appendix C for interested
readers. Applying the symmetric Trotter theorem to the classicalpropagator yields

eiLt= e(iL^1 +iL^2 )t= lim

P→∞

[

eiL^2 t/^2 PeiL^1 t/PeiL^2 t/^2 P

]P

. (3.10.19)

Eqn. (3.10.19) can be expressed more suggestively by defining a timestep ∆t=t/P.
Introducing ∆tinto eqn. (3.10.19) yields

eiLt= lim

P→∞,∆t→ 0

[

eiL^2 ∆t/^2 eiL^1 ∆teiL^2 ∆t/^2

]P

. (3.10.20)

Eqn. (3.10.20) states that we can propagate a classical system using the separate factor
exp(iL 2 ∆t/2) and exp(iL 1 ∆t) exactly for a finite timetin the limit that we let the
number of steps we take go to infinity and the time step go to zero! Of course, this is
not practical, but if we do not take these limits, then eqn. (3.10.20) leads to a useful
approximation for classical propagation.
Note that for finiteP, eqn. (3.10.20) implies an approximation to exp(iLt):

eiLt≈

[

eiL^2 ∆t/^2 eiL^1 ∆teiL^2 ∆t/^2

]P

+O

(

P∆t^3

)

, (3.10.21)

where the leading order error is proportional toP∆t^3. SinceP =t/∆t, the error
is actually proportional to ∆t^2. According to eqn. (3.10.21), an approximate time
propagation can be generated by performingP steps offinitelength ∆tusing the
factorized propagator

eiL∆t≈eiL^2 ∆t/^2 eiL^1 ∆teiL^2 ∆t/^2 +O

(

∆t^3

)

(3.10.22)

for each step. Eqn. (3.10.22) results from taking the 1/Ppower of both sides of eqn.
(3.10.21). An important difference between eqns. (3.10.22) and (3.10.21) should be
noted. While the error in a single step of length ∆tis proportional to ∆t^3 , the error in
a trajectory ofPsteps is proportional to ∆t^2. This distinguishes thelocalerror in one
step from theglobalerror in a full trajectory ofPsteps. The utility of eqn. (3.10.22)
is that if the contributionsiL 1 andiL 2 to the Liouville operator are chosen such the
action of the operators exp(iL 1 ∆t) and exp(iL 2 ∆t/2) can be evaluated analytically,
then eqn. (3.10.22) can be used as a numerical propagation schemefor a single time
step.
In order to see how this works, consider again the example of a singleparticle mov-
ing in one dimension with a HamiltonianH=p^2 / 2 m+U(x) and the two contributions
to the overall Liouville operator given by eqn. (3.10.13). Using theseoperators in eqn.
(3.10.22) gives the approximate single-step propagator:

exp(iL∆t)≈exp

(

∆t

2

F(x)

∂

∂p

)

exp

(

∆t

p

m

∂

∂x

)

exp

(

∆t

2

F(x)

∂

∂p

)

. (3.10.23)

The exact evolution specified by eqn. (3.10.7) is now replaced by the approximation
evolution of eqn. (3.10.23). Thus, starting from an initial condition (x(0),p(0)), the
approximation evolution can be expressed as