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attraction of dierent minima and are separated from each other by saddles whose
energy is still signicantly higher than the temperature.
In [6] a coarse graining procedure has been put forward, aiming at dening a
reasonable approximation of the connectivity graph at a given temperature. The
procedure goes as follows. We rst of all preliminarily sort all saddles in ascending
order according to their minimal energy barrier, that is, the barrier experienced
by crossing each saddle coming from the minimum of higher energy between the
two minima it connects. Then, for each saddle of minimal energy barrier lower
than a temperature dependent renormalization threshold, we perform the following
operations:
after identifying the lower in energy between the two minima it connects,
we substitute its index for that of the higher one in the annotations of all
the saddles (that is: if the saddle connects the i-th and the j-th minimum,
and i is lower in than j, we rename i for j everywhere).
we erase all self connecting saddles (that is, the saddles whose header states
that they connect minimum k with minimum k).
After this procedure has been iteratively repeated for the saddles with the appropri-
ate minimal energy barrier, all saddles are screened for multiple connections and, in
case that more than one saddle is found between two minima, the lower energy one
is chosen. The nodes that survive this procedure actually correspond to clusters of
minima that approximate the metastable states of the system.
The rate of transition between such metastable states can be assigned by speci-
fying a procedure for the updating of the weights of the graph connections during
the renormalization procedure. We follow the prescription given in [6]: we rst of
all assign to each node of the zero{temperature graph a reference probability
i
that corresponds to the stationary probability of the corresponding minimum of the
potential energy, then, each time two dierent metastable states i and j coalesce to
a new one i
0
, we perform the following operations:
the rate
i
0
;k
of the connection from i
0
to k is assigned as the weighted mean
of the rates of the connections from i to k and from j to k, the weights being
the reference probabilities of i and j respectively:
i
0
;k
=
i
i;k
+
j
j;k
i
+
j
(6.5)
the rate of the connection from k to i
0
is assigned as the sum of the rates
of the connections from k to i and from k to j
k;i
0
=
i;k
+
j;k
(6.6)
the new reference probability of i
0
is assigned as the sum of the reference
probabilities of i and j:
i
0
=
i
+
j
(6.7)
