598 Langevin and generalized Langevin equations
- If xn∆t∈B, accept the trial move, and reject it otherwise.
- If the path is rejected at steps 3 or 5, then the old trajectoryY(t) is counted
again in the calculation of averages over the transition path ensemble. Otherwise,
invert the momenta along the backward portion of the path to yield the complete
transition path X(t) and replace the old trajectory Y(t) by the new trajectory
X(t).
The shooting algorithm for stochastic paths is illustrated in Fig. 15.4.One final dif-
ference between the present shooting algorithm and that for deterministic molecular
dynamics is that for stochastic trajectories, it is not necessary to generate both for-
ward and backward segments from every shooting point. A stochastic path of higher
statistical weight in the transition path ensemble can be obtained byintegrating only
backward in time and retaining the forward part of the old trajectory or vice versa.
Of course, when this is done, the difference between old and new paths is smaller and
sampling becomes less efficient. However, some fraction of the shooting moves can be
of this type in order to give a higher average acceptance rate.
A
B
yj∆t
xj∆tFig. 15.4The shooting algorithm for stochastic paths (see also Fig. 7.7)15.7 Mori–Zwanzig theory
Our original derivation in Section 15.2 of the generalized Langevin equation was based
on the introduction of a harmonic bath as a model for a true bath. While conceptually
simple, such a derivation naturally raises the question of whether a GLE can be derived
in a more general way for an arbitrary bath. TheMori–Zwanzig theory(Mori, 1965;
Zwanzig, 1973) achieves this and gives us deeper physical insight into the quantities
that appear in the GLE (Deutsch and Silbey, 1971; Berne, 1971; Berne and Pecora,
1976).
The Mori–Zwanzig theory begins with the full classical Hamiltonian andeffectively
“integrates out” the bath degrees of freedom by using a formalismknown as the
projection operatormethod (Kuboet al., 1985). In this approach, we divide the full
set of degrees of freedom into the system and the bath, as was done for the harmonic
bath Hamiltonian. In the phase space, we consider the two axes corresponding to the
system coordinateqand its conjugate momentump, and the remaining 6N−2 axes
orthogonal to the system. In order to make this phase space picture concrete, let us
introduce a two-component system vector
A=
(
q
p