With GREAT THANKS To - Weinan E ,Weiqing Ren , L.R. Pratt
without whom the following findings cud not b published on web .
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The mechanism of transition (reaction coordinate) during an activated process is best described in terms of the isocommittor
surfaces. These surfaces can be used to identify effective transition tubes inside which the reactive trajectories involved in the activated
process stay confined. It is shown that the isocommittor surfaces can be identified directly, i.e., without ever sampling actual
reactive trajectories, and some procedures to turn this observation into practical algorithms such as the finite temperature string
method are discussed.
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Activated processes such as nucleation events during
phase transition, conformational changes of macromolecules,
or chemical reactions are rare events relative to
the time scale of the atomic vibrations. The reason is
that these events require the system to find its way
through dynamical bottlenecks such as energetic or
entropic barriers which separate metastable sets in configuration
space. The wide separation of time-scales
makes conventional molecular dynamics simulations
ineffective for these rare events and there is a crucial
need for alternative techniques that are capable of determining
the transition pathways of activated processes
arising in complex systems.
In this Letter, we argue that the best reaction coordinate
to describe the activated process, in fact the reactioncoordinate, consists of the isocommittor surfaces,
defined in such a way that a trajectory launched anywhere
on one of these surfaces has the same probabilityto reach first one of the metastable sets rather than the
other. The idea of using the isocommittor surfaces as
reaction coordinates is not new (see, e.g. [2,3,5,6]), and
yet more can be said about these surfaces.
First, we show that the isocommittor surfaces are the
only surfaces such that the distribution of where thereactive trajectories hit these surfaces is the same as
the distribution of where all trajectories hit. Therefore,
this distribution is given by the equilibrium distribution
restricted to the isocommittor surfaces – on any other
set of surfaces, there is a biasing factor to add to the
equilibrium distribution to describe the distribution of
hits by reactive trajectories. This property can be used
to define tubes in configuration space inside which the
reactive trajectories stay confined with high probability.
This is done by weighting the isocommittor surfaces by
the equilibrium probability distribution and identifying
regions of large probability within these surfaces. The
transition tubes generalize the notion of minimum energy
path to situations where the energy landscape is
rough and/or entropic effects dominate.
Second, we discuss how to identify the isocommittor
surfaces directly, that is without ever having to
sample actual reactive trajectories. The direct identification
is based on the observation that the isocommittor
surfaces are the level sets (isosurfaces) of the
solution of the backward Kolmogorov equation, a
well known equation in stochastic process theory.
The backward Kolmogorov equation admits a variational
formulation (least square principle), which
serves as a starting point to use appropriate test functions
to identify the best approximation to the isocommittor
surfaces within a certain class (like, e.g.,
hyperplanes). This way, practical algorithms can bedesigned. We shall discuss in particular the finite temperature
string method (FTS), which can be thought
of as an adaptive version of the blue moon samplingtechnique where one samples the equilibrium distribution
within a family of hyperplanes, and updates these
planes according to some criterion until they converge
towards the isocommittor surfaces.
Identifying the isocommittor surfaces directly is
quite different from what is usually done to determinetransition pathways. Indeed, the idea behind
most of the current techniques is to introduce some
bias on the dynamics to enhance the probability to
observe reactive trajectories. Following Pratt!s original
suggestion [1], this can be done by using Monte
Carlo sampling in trajectory space, with the constraint
that the end points of the trajectories belong
to different metastable sets. Examples of techniques
which sample reactive trajectories directly include
transition path sampling (TPS) [2,3], or the action
based methods introduced by Elber and collaborators
[4].
Figure :-. Metastable sets (regions within the green curves), the mean
transition path æ (center red curve), and the boundaries of the effective
switching tube (external red curves) shown on the isolines of the
potential in Figure 1. Also shown in blue are two dynamical paths by
which the system switches from the left metastable set to the right one
and conversely. Here the thermal energy is on the order of the smallscale
features in the potential, and the notion of minimal energy path
(MEP) is insufficient; instead, the rather complicated dynamical
switching paths lie within the much simpler switching tube, which is
identified by the finite temperature string method.
It should be noted, however, that the reactive trajectories
can be very complicated, and the information
these trajectories provide on the mechanism of the
transition is very indirect. In particular, the isocommittor
surfaces cannot be identified directly from the
ensemble of reactive trajectories since, by definition,
these trajectories always connect one metastable set
to the other and therefore any point along these trajectories
seems to have committor value either one
or zero. Another way to see the problem is to realize
that the reactive trajectories are parametrized by time,
and time is not a good indicator of the advancement
of the reaction. In general, two trajectories leaving a
set at the same time will hit an isocommittor surface
at different times. Even worse, a single reactive trajectory
may hit an isocommittor surface many times during
a transition. This makes the reconstruction of the
isocommittor surfaces from the reactive trajectories a
very challenging numerical task [6]. By identifying
the isocommittor surfaces directly we avoid these difficulties
altogether.
A few words about the organization of the Letter.We
are primarily interested in systems governed by the
Langevin equation
The Equation r currently non visible due to fonts problem in this Blog :(
where ðxðtÞ; vðtÞÞ 2 Rn & Rn (n degrees of freedom with n
large in general), V(x) is the potential, b = 1/kBT, the inverse
temperature, c, the friction coefficient, M, the diagonal
mass matrix, and g(t), a white-noise satisfying
Ægi(t)gj(t0)æ = dijd(t $ t0). Most of our results formally
apply in the limit of zero friction, c ! 0, when (1) reduces
to Hamilton!s equation of motion, but it should
be stressed that the results below are rigorous only when
the friction coefficient is positive. For simplicity of presentation,
we will consider first in Sections 2 and 3 the
high friction situation when c ' 1 which is technically
simpler. Then, in Section 4 we will show that the conclusions
we draw from the high friction dynamics in terms
of mechanism of transition also apply to (1), provided
that the mechanism of transition can be described accurately
in configuration space alone. Finally, in Section 5
we discuss algorithms like the finite temperature string
method (FTS) which allow to identify the isocommittor
surfaces.
2. Isocommittor surfaces and backward Kolmogorov
equation
Consider a system governed by
The Equation r currently non visible due to fonts problem in this Blog :(
This equation arises from (1) in the high friction limit,
c ' 1, and we will return to (1) in Section 4.
The dynamics in (2) is ergodic with respect to the
Boltzmann–Gibbs probability density function (NVT
ensemble)
The Equation r currently non visible due to fonts problem in this Blog :(We will assume that it has been established that (1) is
metastable over the two sets A ( Rn and B ( Rn in configuration
space. By this we mean that the volume of
these sets may be relatively small, and yet the probability
to find the system inside one of these sets is close to
one:
The Equation r currently non visible due to fonts problem in this Blog :(
By ergodicity, transitions between these sets must occur,
and our main purpose is to understand how they occur.
How to systematically identify A and B may for instance
be done by analyzing the spectrum of the infinitesimal
generator associated with (1) – metastability is related
to the existence of a spectral gap, and the metastable sets
can be identified from the eigenfunctions associated with
More would b posted when i find proper fonts to support my blog
Bcoz without equations this blog is useless...................... wait for updates .
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