Computing Freidlin’s Cycles for the Overdamped Langevin Dynamics. Application to the Lennard-Jones-38 Cluster

Abstract

The large time behavior of a stochastic system with infinitesimally small noise can be described in terms of Freidlin’s cycles. We show that if the system is gradient and the potential satisfies certain non-restrictive conditions, the hierarchy of cycles has a structure of a full binary tree, and each cycle is exited via the lowest saddle adjacent to it. Exploiting this property, we propose an algorithm for computing the asymptotic zero-temperature path and building a hierarchy of Freidlin’s cycles associated with the transition process between two given local equilibria. This algorithm is suitable for systems with a complex potential energy landscape with numerous minima. We apply it to find the asymptotic zero-temperature path and Freidlin’s cycles involved into the transition process between the two lowest minima of the Lennard-Jones cluster of 38 atoms. D. Wales’s stochastic network of minima and transition states of this cluster is used as an input.

Appendix

Proof

(Lemma 3) For zero order cycles consisting of a single state, the main state is the state itself. Hence Lemma 3 holds straightforwardly. For first order cycles, the main state is defined as follows [7]. Let C be a first order cycle. Then its main state is

$$ M(C)=\arg\max_{i\in C}U_{iJ(i)},\quad \mathrm{where}\ J(i)=\arg\min_{j\neq i}U_{ij}. $$

(45)

Using Eq. (3) we obtain

$$ M(C)=\arg\max_{i\in C} (V_{iJ(i)}-V_{i}). $$

(46)

First order cycles are constructed identically in this work and in [6–8]. Hence Lemma 1 claiming that first order cycles consist of exactly two states holds. Let C={i,j} be a first order cycle. Suppose V _j <V _i . Then

$$M(C)=\arg\max\{V_{ij}-V_i,V_{ij}-V_j\}=\arg\min\{V_i,V_j\}=j. $$

Hence Lemma 3 holds for first order cycles.

In [6–8], the main state for higher order cycles is defined via the concept of i-graph.

Definition 6

Let S be a set of states and i be a selected state. Then an i-graph is a directed graph G(S,E) satisfying the following properties.

• 1. Every state m∈S∖{i} is the initial point of exactly one arrow.

• 2. For any state m∈S∖{i} there exists a sequence of arrows leading from m to i.

Let C be a kth order cycle where k≥2. Then the main state M(C) is given by

$$ M(C)=\arg\min_{j\in C} \biggl(\min _{g\in G_j(C)}\sum_{(m\rightarrow n)\in g} U_{mn} \biggr). $$

(47)

Here G _j (C) is the collection of all j-graphs defined on the cycle C and ∑_(m→n)∈g is the sum over all edges of the j-graph g. Using Eq. (3) we rewrite Eq. (47) as

(48)

We have split the sum into two sums, the sum over all saddles and the sum over all minima associated with the j-graph g. The sum in Eq. (48) is minimized over all i-graphs that may be defined on the set of states C. Next we make the following the following two observations.

Observation 1

The union

$$\bigcup_{j\in C}G_j(C) $$

can be obtained as follows. Let T(C) be the collection of all trees on C. We remind that a tree is an undirected graph in which any two vertices are connected by exactly one simple path (i.e., a path with no repeated vertices). Then for every tree t∈T(C) and every state j∈C we can define a j-graph. We note that

$$ \sum_{(m\rightarrow n)\in g\in G_j(C)}2V_{mn} = \sum _{(m\rightarrow n)\in t(g)}2V_{mn}, $$

(49)

where the tree t(g) is obtained from the directed graph g by making all its edges undirected. This means that the minimum value of the sum over all saddles associated with a j-graph g is completely determined by the tree t(g) corresponding to this graph. Hence the sum over the saddles does not influence argmin _j∈C in Eq. (48). Figure 10 illustrates this phenomenon for the second order cycle {A,B,C} in the 3-well potential example in Sect. 2.1.

Fig. 10

Illustration for the proof of Lemma 3. The collections of i-graphs for each state of the second order cycle C={A,B,C} in the 3-well potential example in Sect. 2.1. Obviously, there exist three different trees for the set of vertices {A,B,C}. For each of the states A, B, and C, the collection of i-graphs is obtained from these trees

Observation 2

Next we observe that the second sum in Eq. (48) is completely determined by the state j, i.e., it is independent of the j-graph g. Indeed, since g is connected, the second sum in Eq. (48) contains the values of the potential at all states but one: the state j.

Taking into account Observations 1 and 2 we rewrite Eq. (48) as

(50)

Therefore, the main state of a cycle C is its deepest minimum. □

Proof

(Lemma 4) For a first order cycle C, the stationary distribution rate is defined by [7]²

$$ m_C(i)=U_{M(C)J(M(C))}-U_{iJ(i)}, \quad \mathrm{where}\ J(i)=\arg\min_{j}U_{ij}. $$

(51)

By Lemma 1, first order cycles consist of two states: C=i∪j. Let V _j <V _i , i.e., j is the main state of the cycle C. Then Eq. (51) becomes

(52)

Hence the claim holds for the first order cycles.

For higher order cycles, the stationary distribution rate is defined as

$$ m_C(i)=\min_{g\in G_i(C)}\sum _{(m\rightarrow n)\in g} U_{mn}-\min_{g\in G_{M(C)}(C)}\sum _{(m\rightarrow n)\in g} U_{mn}. $$

(53)

Using Eq. (3) we rewrite Eq. (53) as

(54)

Taking into account Observations 1 and 2 made in the proof of Lemma 3 we get

(55)

Thus, Lemma 4 holds for cycles of order k≥2 as well. □

Proof

(Lemma 5) The exit rate [7] of a cycle C is defined by

$$ \mathcal{E}(C)=\min_{i\in C,~l\notin C} \bigl(m_C(i)+U_{il} \bigr). $$

(56)

Using Eq. (3) and Lemma 4 we rewrite Eq. (56) as

(57)

This completes the proof. □