Glossary of Symbols

$x>$	The mean value of variable $x$	§
$\emptyset$	Empty set	§
$ˆ$	Denotes a unit vector; $\hat{x} = x ∕ \| x \|$	§
$\| \|$	A vector norm; $\| x \| = \sqrt{\sum_{i = 1}^{\dim (x)} x_{i}^{2}}$ ; also, cardinality of a set	§
$⊖$	Denotes symmetric difference of two sets	§
$\otimes$	Denotes vector direct product, a.k.a. dyadic; $a \otimes b^{T} = c$ , $c_{i, j} = a_{i} b_{j}$	§
$(,)$	A scalar product of two vectors, a.k.a. dot product; $(a, b) = a^{T} \cdot b$	§
${}$	A set of objects; ${x_{i}}_{1}^{n} = {x_{1}, x_{2}, x_{3}, \dots, x_{n}}$	§
$0$	A matrix or a vector filled with zeros	§
$A$	One of the two superstates in two-state kinetic model	§
$A$	Approximation to the inverse Hessian matrix, $H^{- 1}$	§
$Adj [i]$	Set of all the nodes adjacent to node $i$	§
$AdjIn [i]$	Set of all the nodes connected to node $i$ via incoming edges	§
$AdjOut [i]$	Set of all the nodes connected to node $i$ via outgoing edges	§
$B$	One of the two superstates in two-state kinetic model	§
${BLJ}_{n}$	Binary Lennard-Jones liquid with $n$ atoms in a periodic cell	§
$C$	Square matrix, columns of which are eigenvectors	§
$C_{N}$	A chain graph with $N$ nodes	§
$D$	Endpoint separation	§
$Δ_{i} (j)$	$i$ th displacement in magnitude of an atom between structures $j - 1$ and $j$	§

$E$	A set of edges	§
$ℰ_{α}^{G}$	Probability of escape from $G$ starting from $α$ in a single step	§
$E_{-}$	Energy barrier corresponding to the reverse reaction	§
$E_{+}$	Energy barrier corresponding to the forward reaction	§
$F$	Set of all minima connected to the final endpoint	§
$ℱ$	Frequency distribution function	§
$G_{N}$	An arbitrary graph with $N$ nodes	§
$H (x)$	Hessian matrix evaluated at point $x$	§
$I$	A set containing all the minima that do not belong to $A \cup B$	§
$K_{N}$	A complete graph with $N$ nodes	§
$L$	Lagrangian function	§
${LJ}_{n}$	$n$ -atom Lennard-Jones cluster	§
$℧ x)$	The set of all possible values of the control variable $x$	§
$N$	Number of atoms; number of nodes in a graph	§
$N_{c}$	Cooperativity index	§
$N_{f}$	Number of frames or points sampled along a path	§
$N_{i}$	Number of images in a band	§
$N_{p}$	Participation index	§
$\tilde{N}$	Participation index evaluated using the endpoints alone	§
$O_{k}$	Displacement overlap evaluated for $k$ atoms using displacements $d_{i} (j)$	§
$𝒪 ()$	$f (n) = 𝒪 (g (n))$ means $0 \leq f (n) \leq cg (n)$ holds for some constants $c > 0$	§
$P$	Transition probability matrix	§
$P_{i}^{eq}$	Equilibrium occupation probability of state $i$	§
$P_{i} (t)$	Occupation probability of state $i$ at time $t$	§
$P_{j, i}$	Probability of transition from state $i$ to state $j$	§
$𝒫_{ξ}$	Pathway probability	§
$R_{N}$	A random graph with $N$ nodes	§
$R_{α}$	$3 N$ -dimensional rotation matrix about axis $α$ , $α \in {x, y, z}$	§
$ℝ$	The set of all real numbers	§
$S$	Set of all minima connected to the starting endpoint	§
$Σ_{α}^{G}$	Total probability of escape from $G$ if started at node $α$	§
$𝒮_{α, β}^{G}$	Sum of weights of all pathways connecting $α$ and $β$ and confined to $G$	§
$T$	Temperature	§
$Θ_{k}$	Displacement overlap evaluated for $k$ atoms using displacements $Δ_{i} (j)$	§
$𝒯_{i}^{G}$	Mean escape time from graph $G$ if started at node $i$	§
$U$	Set of all minima that do not belong to $S \cup F$	§
$ϒ (x, 𝜀)$	A set of feasible points contained in the neighbourhood $𝜀$ of $x$	§
$V$	Potential energy functional; also, a set of graph nodes	§
$𝒱$	$3 N$ -dimensional vector of velocities^∗	§
$\tilde{V}$	Spring potential	§
$W (a, b)$	Weight of the shortest path $ξ = a \leftarrow b$ ; $W (a, b) = - \ln (𝒲_{ξ})$	§
$𝕎$	The set of whole numbers; $𝕎 = {0, 1, 2, \dots}$	§
$𝒲_{ξ}$	Product of branching probabilities associated with path $ξ$	§
$X$	$3 N$ -dimensional vector representing a point in configuration space	§
$Ξ$	Pathway ensemble	§
$a$	A state that belongs to a superstate $A$	§
$α$	Pathway nonlinearity index	§
$b$	A state that belongs to a superstate $B$	§
$β$	Energy barrier asymmetry index	§
$c$	Eigenvector	§
$\det M$	A determinant [33] (a scalar-valued function) of matrix $M$	§
$d_{i}$	Integrated path length for atom $i$ ; also, degree of node $i$	§
$d_{i} (j)$	Displacement of atom $i$ between structures $j - 1$ and $j$	§
$e_{j, i}$	Directed edge that describes a transition from node $i$ to node $j$	§
$𝜖$	A parameter in LJ potential (the depth of the potential energy well)	§
$𝜀$	Small positive parameter	§
$η$	Number of atomic degrees of freedom	§
$f (x)$	Objective function of a vector argument $x$	§
$g$	$3 N$ -dimensional gradient vector of the true potential	§
$γ$	Kurtosis of a distribution evaluated using moments about the mean	§
$γ^{'}$	Kurtosis of a distribution evaluated using moments about the origin	§
$\tilde{g} ∥$	Spring gradient vector component parallel to the path	§
$\tilde{g} ⊥$	Spring gradient vector component perpendicular to the path	§
$\tilde{g}$	$3 N$ -dimensional gradient vector of the spring potential	§
$g ∥$	True gradient vector component parallel to the path	§
$g ⊥$	True gradient vector component perpendicular to the path	§
$i, j, k$	Indices; range and meaning may vary depending on the context	§
$k_{B}$	Boltzmann’s constant	§
$k_{j, i}$	Rate constant for transitions from state $i$ to state $j$	§
$k_{spr}$	Spring force constant	§
$l (ξ)$	Length of path $ξ$	§
$λ$	Eigenvalue	§
$m$	Atomic mass	§
$m_{n}$	$n$ th moment of a distribution function about the mean	§
$m_{n}^{'}$	$n$ th moment of a distribution function about the origin	§
$n$	Time parameter of a discrete-time stochastic process	§
$o ()$	$f (n) = o (g (n))$ means $0 \leq f (n) \leq cg (n)$ holds for all constants $c > 0$ ^∗ ^∗Otherwise known as an upper bound that is not asymptotically tight.	§
$p$	Search direction vector	§
$π$	Path length asymmetry index	§
$r_{i} (j)$	Three-dimensional Cartesian coordinates vector of atom $i$ for structure $j$	§
$s$	Integrated path length	§
$σ$	A parameter in the LJ potential ( $2^{1 ∕ 6} σ$ is the pair equilibrium separation)	§
$τ$	$3 N$ -dimensional tangent vector	§
$τ_{i}$	Mean waiting time in state $i$ before escape	§
$t$	Time	§
$δt$	Time integration step	§
$^{T}$	A matrix or vector transpose	§
$ϖ$	Step size	§
$v_{i}$	$i$ th graph node	§
$w (u, v)$	Weight of the undirected edge connecting nodes $u$ and $v$	§
$x_{i}$	$i$ th component of vector $x$	§
$ξ$	A pathway	§
$x, y, z$	Vectors; dimensionality and meaning may vary depending on the context	§