GLOSSARY OF SYMBOLS

GLOSSARY OF SYMBOLS

$\left< x \right>$	The mean value of variable $x$
$\emptyset$	Empty set
$\,\,\hat{}$	Denotes a unit vector; $\hat{\bf x}={\bf x}/\vert{\bf x}\vert$
$\vert\,\vert$	A vector norm; $\vert{\bf x}\vert=\sqrt{\sum_{i=1}^{dim({\bf x})} x_i^2}$; also, cardinality of a set
$\ominus$	Denotes symmetric difference of two sets
$\otimes$	Denotes vector direct product, a.k.a. dyadic; ${\bf a} \otimes {\bf b}^T = {\bf c}$, $c_{i,j}=a_i b_j$
$(\,,\,)$	A scalar product of two vectors, a.k.a. dot product; $({\bf a},{\bf b})={\bf a}^T\cdot{\bf b}$
$\{\,\}$	A set of objects; $\{{\bf x}_i\}_{1}^n = \{{\bf x}_1,{\bf x}_2,{\bf x}_3,\dots,{\bf x}_n\}$
${\bf0}$	A matrix or a vector filled with zeros
$A$	One of the two superstates in two-state kinetic model
${\bf A}$	Approximation to the inverse Hessian matrix, ${\bf H}^{-1}$
$Adj[i]$	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
${\rm BLJ}_n$	Binary Lennard-Jones liquid with $n$ atoms in a periodic cell
${\bf C}$	Square matrix, columns of which are eigenvectors
$C_N$	A chain graph with $N$ nodes
$D$	Endpoint separation
$\Delta_i(j)$	$i$th displacement in magnitude of an atom between structures $j-1$ and $j$
$E$	A set of edges
$\mE _\alpha^G$	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
$\mF$	Frequency distribution function
$G_N$	An arbitrary graph with $N$ nodes
${\bf H}({\bf x})$	Hessian matrix evaluated at point ${\bf x}$
$I$	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
${\rm LJ}_n$	$n$-atom Lennard-Jones cluster
$\mho \left({\bf x}\right)$	The set of all possible values of the control variable ${\bf x}$
$N$	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
$\widetilde{N}$	Participation index evaluated using the endpoints alone
$O_k$	Displacement overlap evaluated for $k$ atoms using displacements $d_i(j)$
$\mO (\,)$	$f(n)=\mO \Bigl(g(n)\Bigr)$ means $0 \leq f(n) \leq c g(n)$ holds for some constants $c > 0$
${\bf P}$	Transition probability matrix
$P_{i}^{\rm eq}$	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 $j$
$\mP _{\xi}$	Pathway probability
$R_N$	A random graph with $N$ nodes
${\bf R}_\alpha$	$3N$-dimensional rotation matrix about axis $\alpha$, $\alpha\in\{x,y,z\}$
$\mathbb{R}$	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 $\alpha$
$\mS _{\alpha,\beta}^G$	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 $\Delta_i(j)$
$\mT _{i}^G$	Mean escape time from graph $G$ if started at node $i$
$U$	Set of all minima that do not belong to $S\cup F$
$\Upsilon({\bf x},\varepsilon)$	A set of feasible points contained in the neighbourhood $\varepsilon$ of ${\bf x}$
$V$	Potential energy functional; also, a set of graph nodes
$\mathcal{V}$	$3N$-dimensional vector of velocities[1]This vector and the other vectors defined here are column vectors.
$\widetilde{V}$	Spring potential
$W(a,b)$	Weight of the shortest path $\xi=a\leftarrow b$; $W(a,b)=-\ln(\mW _\xi)$
$\mathbb{W}$	The set of whole numbers; $\mathbb{W}=\{0,1,2,\dots\}$
$\mW _\xi$	Product of branching probabilities associated with path $\xi$
${\bf X}$	$3N$-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
${\bf c}$	Eigenvector
$\det{\bf M}$	A determinant [33] (a scalar-valued function) of matrix ${\bf M}$
$d_i$	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 $j$
$\epsilon$	A parameter in LJ potential (the depth of the potential energy well)
$\varepsilon$	Small positive parameter
$\eta$	Number of atomic degrees of freedom
$f({\bf x})$	Objective function of a vector argument ${\bf x}$
${\bf g}$	$3N$-dimensional gradient vector of the true potential
$\gamma$	Kurtosis of a distribution evaluated using moments about the mean
$\gamma'$	Kurtosis of a distribution evaluated using moments about the origin
$\gspar$	Spring gradient vector component parallel to the path
$\gsper$	Spring gradient vector component perpendicular to the path
$\widetilde{\bf g}$	$3N$-dimensional gradient vector of the spring potential
$\gtpar$	True gradient vector component parallel to the path
$\gtper$	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
$l(\xi)$	Length of path $\xi$
$\lambda$	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
$o(\,)$	$f(n)=o\Bigl(g(n)\Bigr)$ means $0 \leq f(n) \leq c g(n)$ holds for all constants $c > 0$[1]Otherwise known as an upper bound that is not asymptotically tight.
${\bf p}$	Search direction vector
$\pi$	Path length asymmetry index
${\bf r}_i(j)$	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)
$\tau$	$3N$-dimensional tangent vector
$\tau_i$	Mean waiting time in state $i$ before escape
$t$	Time
$\delta t$	Time integration step
$\,^T$	A matrix or vector transpose
$\varpi$	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 ${\bf x}$
$\xi$	A pathway
${\bf x},{\bf y},{\bf z}$	Vectors; dimensionality and meaning may vary depending on the context