## Glossary of Symbols

Consistency is the last refuge of the unimaginative.
Oscar Wilde

 The mean value of variable $x$ § $\varnothing$ Empty set § $\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}ˆ$ Denotes a unit vector; $\stackrel{̂}{x}=x∕|x|$ § $|\phantom{\rule{0.3em}{0ex}}|$ A vector norm; $|x|=\sqrt{{\sum }_{i=1}^{\mathit{dim}\left(x\right)}{x}_{i}^{2}}$; also, cardinality of a set § $\ominus$ Denotes symmetric difference of two sets § $\otimes$ Denotes vector direct product, a.k.a. dyadic; $a\otimes {b}^{T}=c$, ${c}_{i,j}={a}_{i}{b}_{j}$ § $\left(\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}\right)$ A scalar product of two vectors, a.k.a. dot product; $\left(a,b\right)={a}^{T}\cdot b$ § $\left\{\phantom{\rule{0.3em}{0ex}}\right\}$ A set of objects; ${\left\{{x}_{i}\right\}}_{1}^{n}=\left\{{x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}\right\}$ § $0$ 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, ${H}^{-1}$ § $\mathit{Adj}\left[i\right]$ Set of all the nodes adjacent to node $i$ § $\mathit{AdjIn}\left[i\right]$ Set of all the nodes connected to node $i$ via incoming edges § $\mathit{AdjOut}\left[i\right]$ Set of all the nodes connected to node $i$ via outgoing edges § $B$ One of the two superstates in two-state kinetic model § ${\mathit{BLJ}}_{n}$ 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 § ${\Delta }_{i}\left(j\right)$ $i$th displacement in magnitude of an atom between structures $j-1$ and $j$ § $E$ A set of edges § ${\mathsc{ℰ}}_{\alpha }^{G}$ Probability of escape from $G$ starting from $\alpha$ 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 § $\mathsc{ℱ}$ Frequency distribution function § ${G}_{N}$ An arbitrary graph with $N$ nodes § $H\left(x\right)$ 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 § ${\mathit{LJ}}_{n}$ $n$-atom Lennard-Jones cluster § 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 § $\stackrel{̃}{N}$ Participation index evaluated using the endpoints alone § ${O}_{k}$ Displacement overlap evaluated for $k$ atoms using displacements ${d}_{i}\left(j\right)$ § $\mathsc{𝒪}\left(\phantom{\rule{0.3em}{0ex}}\right)$ $f\left(n\right)=\mathsc{𝒪}\left(g\left(n\right)\right)$ means $0\le f\left(n\right)\le \mathit{cg}\left(n\right)$ holds for some constants $c>0$ § $P$ Transition probability matrix § ${P}_{i}^{\mathit{eq}}$ Equilibrium occupation probability of state $i$ § ${P}_{i}\left(t\right)$ Occupation probability of state $i$ at time $t$ § ${P}_{j,i}$ Probability of transition from state $i$ to state $j$ § ${\mathsc{𝒫}}_{\xi }$ Pathway probability § ${R}_{N}$ A random graph with $N$ nodes § ${R}_{\alpha }$ $3N$-dimensional rotation matrix about axis $\alpha$, $\alpha \in \left\{x,y,z\right\}$ § $ℝ$ The set of all real numbers § $S$ Set of all minima connected to the starting endpoint § ${\Sigma }_{\alpha }^{G}$ Total probability of escape from $G$ if started at node $\alpha$ § ${\mathsc{𝒮}}_{\alpha ,\beta }^{G}$ Sum of weights of all pathways connecting $\alpha$ and $\beta$ and confined to $G$ § $T$ Temperature § ${\Theta }_{k}$ Displacement overlap evaluated for $k$ atoms using displacements ${\Delta }_{i}\left(j\right)$ § ${\mathsc{𝒯}}_{i}^{G}$ Mean escape time from graph $G$ if started at node $i$ § $U$ Set of all minima that do not belong to $S\cup F$ § $\Upsilon \left(x,𝜀\right)$ A set of feasible points contained in the neighbourhood $𝜀$ of $x$ § $V$ Potential energy functional; also, a set of graph nodes § $\text{}\mathsc{𝒱}\text{}$ $3N$-dimensional vector of velocities∗ § $\stackrel{̃}{V}$ Spring potential § $W\left(a,b\right)$ Weight of the shortest path $\xi =a←b$; $W\left(a,b\right)=-\mathrm{ln}\left({\mathsc{𝒲}}_{\xi }\right)$ § $𝕎$ The set of whole numbers; $𝕎=\left\{0,1,2,\dots \phantom{\rule{0.3em}{0ex}}\right\}$ § ${\mathsc{𝒲}}_{\xi }$ Product of branching probabilities associated with path $\xi$ § $X$ $3N$-dimensional vector representing a point in configuration space § $\Xi$ Pathway ensemble § $a$ A state that belongs to a superstate $A$ § $\alpha$ Pathway nonlinearity index § $b$ A state that belongs to a superstate $B$ § $\beta$ Energy barrier asymmetry index § $c$ Eigenvector § $\mathrm{det}M$ A determinant [33] (a scalar-valued function) of matrix $M$ § ${d}_{i}$ Integrated path length for atom $i$; also, degree of node $i$ § ${d}_{i}\left(j\right)$ 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 § $\eta$ Number of atomic degrees of freedom § $f\left(x\right)$ Objective function of a vector argument $x$ § $g$ $3N$-dimensional gradient vector of the true potential § $\gamma$ Kurtosis of a distribution evaluated using moments about the mean § ${\gamma }^{\prime }$ Kurtosis of a distribution evaluated using moments about the origin § $\stackrel{̃}{g}\parallel$ Spring gradient vector component parallel to the path § $\stackrel{̃}{g}\perp$ Spring gradient vector component perpendicular to the path § $\stackrel{̃}{g}$ $3N$-dimensional gradient vector of the spring potential § $g\parallel$ True gradient vector component parallel to the path § $g\perp$ 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 $j$ § ${k}_{\mathit{spr}}$ Spring force constant § $l\left(\xi \right)$ Length of path $\xi$ § $\lambda$ Eigenvalue § $m$ Atomic mass § ${m}_{n}$ $n$th moment of a distribution function about the mean § ${m}_{n}^{\prime }$ $n$th moment of a distribution function about the origin § $n$ Time parameter of a discrete-time stochastic process § $o\left(\phantom{\rule{0.3em}{0ex}}\right)$ $f\left(n\right)=o\left(g\left(n\right)\right)$ means $0\le f\left(n\right)\le \mathit{cg}\left(n\right)$ holds for all constants $c>0$∗ ∗Otherwise known as an upper bound that is not asymptotically tight. § $p$ Search direction vector § $\pi$ Path length asymmetry index § ${r}_{i}\left(j\right)$ Three-dimensional Cartesian coordinates vector of atom $i$ for structure $j$ § $s$ Integrated path length § $\sigma$ A parameter in the LJ potential (${2}^{1∕6}\sigma$ is the pair equilibrium separation) § $\text{}\tau \text{}$ $3N$-dimensional tangent vector § ${\tau }_{i}$ Mean waiting time in state $i$ before escape § $t$ Time § $\mathit{\delta t}$ Time integration step § ${\phantom{\rule{0.3em}{0ex}}}^{T}$ A matrix or vector transpose § $\varpi$ Step size § ${v}_{i}$ $i$th graph node § $w\left(u,v\right)$ Weight of the undirected edge connecting nodes $u$ and $v$ § ${x}_{i}$ $i$th component of vector $x$ § $\xi$ A pathway § $x,y,z$ Vectors; dimensionality and meaning may vary depending on the context §