10A The Onsager-Machlup path-integral revisited
In Sect. 10.2 we considered a path-integral representation of solutions to the SDE
√
dX(t) = A(X)dt + εdW (t),
(10A.1)
for 0 ≤ t ≤ T and initial condition X(0) = x0 . Here W (t) is a Wiener process and
the noise is taken to be weak (ε 1). Using the Ito version of time-discretization
we found that
"
2 #
1 N−1 xm+1 − xm
1
− A(xm ) ∆t .
(10A.2)
exp −
P(x) =
∑
2ε m=0
∆t
(2πε∆t)N/2
Integrating over intermediate states and then formally taking the continuum limit,
leads to the so-called Onsager-Machlup (OM) path integral [232, 233]
Z x(τ)=x
Z
1 τ
2
P(x, τ|x0 ) =
(ẋ − A(x)) dt D[x].
(10A.3)
exp −
2ε 0
x(0)=x0
However, there is a problem with interpreting the meaning of the term ẋ in this pathintegral, since the majority of paths in a diffusion process are non-differentiable. It
turns out that within the context of large deviation theory in the limit ε → 0, the
optimal paths are smooth so one can use the classical action
S[x] =
1
2
Z τ
(ẋ − A(x))2 dt,
(10A.4)
0
Nevertheless, it is important to clarify the meaning of the OM path-integral. Here
we follow the particular formulation of Adib [1], see also [3, 2].
The basic idea is to expand the quadratic term in equation (10A.2) before taking
the continuum limit:
2 xm+1 − xm
xm+1 − xm 2
xm+1 − xm
− A(xm ) =
−2
A(xm ) + A(xm )2 .
∆t
∆t
∆t
The term quadratic in ∆t can still be written formally as ẋ2 in the continuum limit,
but it is really shorthand for the purely diffusive contribution to the path-integral,
which determines the Wiener measure. On the other hand, the cross-term becomes
an Ito integral
N−1
lim lim
∑ (xm+1 − xm )A(xm ) =
N→∞∆t→0
m=0
Z x
A(x)dx.
x0
Recall from Sect. 2.5 that the usual rules of calculus do not apply. In particular,
suppose that we set A(x) = −U 0 (x). Then
1
N−1
U(x) −U(x0 ) = lim lim
∑ U(xm+1 ) −U(xm )
N→∞∆t→0
m=0
N−1
= lim lim
∑ U(xm + ∆ xm ) −U(xm )
N→∞∆t→0
m=0
N−1
1 00
2
= lim lim ∑ U (xm ) ∆ xm + U (xm )∆ xm
N→∞∆t→0
2
m=0
i
h
N−1
ε
= lim lim ∑ U 0 (xm ) ∆ xm + U 00 (xm )∆t
N→∞∆t→0
2
m=0
Z x
=
U 0 (x)dx +
x0
Therefore,
Z x
0
ε
2
Z t
U 00 (x(t))dt.
0
ε
A(x)dx = −∆U −
2
x0
Z t
A0 (x(t))dt
0
so that, we can rewrite the action as
S[x] = ∆U +
1
2
Z τ
ẋ2 + A(x)2 + εA0 (x) dt.
(10A.5)
0
Except for the formal ẋ term in the above action, the remaining terms involve ordinary integrals only (assuming that A is smooth), and are thus insensitive to the
particular choice of discretization.
Supplementary references
1. Adib, A.: Stochastic actions for diffusive dynamics:? Reweighting, dampling, and minimization.
J. Phys. Chem. B 112, 59105916 (2008)
2. Faccioli, P., Sega, M., Pederiva, F., Orland, H.: Dominant pathways in protein folding. Phys.
Rev. Lett.97, 108101 (2006)
3. Zuckerman, D. M., Woolf, T. B.: Efficient dynamic importance sampling of rare events in one
dimension. Phys. Rev. E 63 016702 (2000)
2