though in literature \(t_{ex}\) is usually labeled as \(t\).
Summary of probabilities
In our analysis we encountered several different probabilities of which we try a list here:
- \(p(t_{in},t_{ex}|t):=v(t,t_{in},t_{ex})/V(t)\): the joint residence time and life expectancy probability of particles in reactor conditional to clock time, \(t\)
- \(p(t_{ex}|t)\equiv p(t_{ex}-t|t)\): the probability of life expectancy of particle inside the reactor (a snapshot of expected particle lives) conditional to clock time.
- \(p(t_{in}|t)\equiv p(t-t_{in}|t)\): the probability of residence time of particle inside the reactor (a snapshot of particles ages) conditional to clock time.
- \(p_{ex}(t|t_{in})\): the probability of response times (i.e. probability of exit times), i.e. a certain restriction of the joint residence times probability, conditional to injection time.
- \(p_{in}(t_{in}|t)\): the travel time probability, also a restriction on the travel time probabilities (\(p_Q\) in Rigon et al., 2016), conditional to clock time (which, in this case, is equal to exit times).
It is evident that all of these probabilities derive from the joint one, but under different operations. Niemi's identity relates \(p_{ex}\) and \(p_{in}\) .
One interesting goal is to find also a relation, for instance, between \(p_{ex}\left(t|t_{in} \right)\) and \(p(t_{ex}|t_{in})\) (and correspondingly between \(p_{in}(t_{in}|t)\) and \(p(t_{in}|t)\)). In Botter et al., 2011, this relation is set to be a function \(\omega(t,t_{in})\) for which: