I am studying the works of Alenka Luzar on the hydrogen bonding kinetics in liquid water. Luzar defines an autocorrelation function c(t) that gives the probability that an hydrogen bond is intact at time t, given it was intact at time zero. She defines the rate of relaxation to equilibrium as:
$k(t) = -dc/dt = - \langle \dot{h(0)}[1 - h(t)] \rangle / \langle h \rangle$
Where h(t) equals 1 if the hydrogen bond exists between a pair of atomes and 0 otherwise. She also defines a function, similar to $k(t)$, to further compute $n(t)$, which is the probability at time t that a pair of initially bonded molecules are now unbonded but remain separated by less than a cut-off distance:
$k_{in}(t) = -dc/dt = - \langle \dot{h(0)}[1 - h(t)]H(t) \rangle / \langle h \rangle$
H(t) is unity if the oxygen-oxygen distance of the tagged pair is not larger than a cut-off distance.
Assuming I have the values of c(t), I'd like first to compute k(t) as $k(t) = - \langle \dot{h(0)}[1 - h(t)] \rangle / \langle h \rangle$ in order to verify if the values are equal to $-dc/dt$ and then to compute $k_{in}$. For that, I have an array of arrays for my simulation, which contains the atom ids that form an hydrogen bond for each frame. However, the definition of $\dot{h(0)}$ bugs me. To me, if h(0) is equal to 0 or 1, the derivative should be zero. So I don't understand exactly how to compute these functions.
I'd appreciate your ideas on the matter.