596 Langevin and generalized Langevin equations
0 5 10 15 20
t /T-3
0
3
q(t)0 5 10 15 20
t /T-3
0
3
q(t)-4 0 4
q-4
0
4
p-4 0 4
q-4
0
4
p-4 -2 0 2 4
q0
0.2
0.4
P(q)-4 -2 0 2 4
q0
0.2
0.4
P(q)g = 0.5 g = 8Fig. 15.3(Top) Trajectories of a harmonic oscillator withμ= 1,ω= 1, andkT= 1 coupled
to a bath via the Langevin equation forγ= 0.5 (left) andγ= 8 (right). Here,T= 2π/ω
is the period of the oscillator. (Middle) Phase space Poincar ́e sections. (Bottom) Position
probability distribution functions.
15.6 Sampling stochastic transition paths
The numerical integration algorithm of the previous subsection canbe used in conjunc-
tion with the transition path sampling approach of Section 7.7 to sample a transition
path ensemble of stochastic paths from a regionAof phase space to another regionB.
As noted in Section 7.7, the shooting algorithm is an effective method for generating
trial moves from a path Y(t) to a new path X(t). Here, we describe a simple variant
of the shooting algorithm for paths satisfying the Langevin equation.
In fact, the shooting algorithm can be applied almost unchanged from that de-
scribed in Section 7.7. However, a few differences need to be pointedout. First, the
random force term in eqn. (15.5.17) is not deterministic, which meansthat a rule such
as that given in eqn. (7.7.2) for a numerical solver such as velocity Verlet cannot be
used for Langevin dynamics. Rather, we need to account for the fact that a distribu-
tion ofq(t+∆t) values can be generated fromq(t) due to the random force. The solver
in eqn. (15.5.17) can be expressed compactly as
x(k+1)∆t= xk∆t+δxd+δxr, (15.6.19)