Glossary of Symbols

Consistency is the last refuge of the unimaginative.
Oscar Wilde

xThe mean value of variable x §
Empty set §
ˆDenotes a unit vector; x̂ = x|x| §
||A vector norm; |x| = i=1dim(x)xi2; also, cardinality of a set §
Denotes symmetric difference of two sets §
Denotes vector direct product, a.k.a. dyadic; a bT = c, ci,j = aibj §
(,)A scalar product of two vectors, a.k.a. dot product; (a,b) = aT b §
{}A set of objects; {xi}1n = {x1,x2,x3,,xn} §
0A matrix or a vector filled with zeros §
AOne of the two superstates in two-state kinetic model §
AApproximation to the inverse Hessian matrix, H1 §
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 §
BOne of the two superstates in two-state kinetic model §
BLJnBinary Lennard-Jones liquid with n atoms in a periodic cell §
CSquare matrix, columns of which are eigenvectors §
CNA chain graph with N nodes §
DEndpoint separation §
Δi(j)ith displacement in magnitude of an atom between structures j 1 and j§
EA set of edges §
αGProbability of escape from G starting from α in a single step §
EEnergy barrier corresponding to the reverse reaction §
E+Energy barrier corresponding to the forward reaction §
FSet of all minima connected to the final endpoint §
Frequency distribution function §
GNAn arbitrary graph with N nodes §
H(x)Hessian matrix evaluated at point x §
IA set containing all the minima that do not belong to A B §
KNA complete graph with N nodes §
LLagrangian function §
LJnn-atom Lennard-Jones cluster §
xThe set of all possible values of the control variable x §
NNumber of atoms; number of nodes in a graph §
NcCooperativity index §
NfNumber of frames or points sampled along a path §
NiNumber of images in a band §
NpParticipation index §
ÑParticipation index evaluated using the endpoints alone §
OkDisplacement overlap evaluated for k atoms using displacements di(j)§
𝒪()f(n) = 𝒪(g(n)) means 0 f(n) cg(n) holds for some constants c > 0 §
PTransition probability matrix §
PieqEquilibrium occupation probability of state i §
Pi(t)Occupation probability of state i at time t §
Pj,iProbability of transition from state i to state j §
𝒫ξPathway probability §
RNA random graph with N nodes §
Rα3N-dimensional rotation matrix about axis α, α {x,y,z} §
The set of all real numbers §
SSet of all minima connected to the starting endpoint §
ΣαGTotal probability of escape from G if started at node α §
𝒮α,βGSum of weights of all pathways connecting α and β and confined to G§
TTemperature §
ΘkDisplacement overlap evaluated for k atoms using displacements Δi(j) §
𝒯iGMean escape time from graph G if started at node i §
USet of all minima that do not belong to S 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 §
𝒱3N-dimensional vector of velocities §
V ̃Spring potential §
W(a,b)Weight of the shortest path ξ = a b; W(a,b) = ln(𝒲ξ) §
𝕎The set of whole numbers; 𝕎 = {0,1,2,} §
𝒲ξProduct of branching probabilities associated with path ξ §
X3N-dimensional vector representing a point in configuration space §
ΞPathway ensemble §
aA state that belongs to a superstate A §
αPathway nonlinearity index §
bA state that belongs to a superstate B §
βEnergy barrier asymmetry index §
cEigenvector §
detMA determinant [33] (a scalar-valued function) of matrix M §
diIntegrated path length for atom i; also, degree of node i §
di(j)Displacement of atom i between structures j 1 and j§
ej,iDirected edge that describes a transition from node i to node j §
𝜖A parameter in LJ potential (the depth of the potential energy well) §
𝜀Small positive parameter §
ηNumber of atomic degrees of freedom §
f(x)Objective function of a vector argument x §
g3N-dimensional gradient vector of the true potential §
γKurtosis of a distribution evaluated using moments about the mean §
γKurtosis of a distribution evaluated using moments about the origin §
g̃Spring gradient vector component parallel to the path §
g̃Spring gradient vector component perpendicular to the path §
g̃3N-dimensional gradient vector of the spring potential §
gTrue gradient vector component parallel to the path §
gTrue gradient vector component perpendicular to the path §
i,j,kIndices; range and meaning may vary depending on the context §
kBBoltzmann’s constant §
kj,iRate constant for transitions from state i to state j §
ksprSpring force constant §
l(ξ)Length of path ξ §
λEigenvalue §
mAtomic mass §
mnnth moment of a distribution function about the mean §
mnnth moment of a distribution function about the origin §
nTime parameter of a discrete-time stochastic process §
o()f(n) = o(g(n)) means 0 f(n) cg(n) holds for all constants c > 0 Otherwise known as an upper bound that is not asymptotically tight. §
pSearch direction vector §
πPath length asymmetry index §
ri(j)Three-dimensional Cartesian coordinates vector of atom i for structure j §
sIntegrated path length §
σA parameter in the LJ potential (216σ is the pair equilibrium separation) §
τ3N-dimensional tangent vector §
τiMean waiting time in state i before escape §
tTime §
δtTime integration step §
T A matrix or vector transpose §
ϖStep size §
viith graph node §
w(u,v)Weight of the undirected edge connecting nodes u and v§
xiith component of vector x §
ξA pathway §
x,y,zVectors; dimensionality and meaning may vary depending on the context §