Imagine you have an ensemble of 3-level systems whose dynamics is a Markov chain Further, suppose that transition from level 1 to level 3 requires the absorption of a "yellow" photon (high energy), while transitions between the other levels involve the emission or absorption of "infrared" photons (low energy). (the systems are in contact with a thermal reservoir that exchanges energy with the ensemble, mostly infrared but sometimes yellow, and there's also a source of "yellow" light shining on them). My question is this: is there a useful way to calculate the rate at which this ensemble is converting yellow light to infrared light in terms of the transition probabilities and the occupation numbers (how many systems are in a given state at that time)? Even if the ensemble of systems is not in a stationary state?
Thank you.
Edit: Following dan_fulea's suggested notation, say that the current distribution is given by the $ 1 \times 3$ row vector $q$, and that the transition matrix for a particle's state is the $3 \times 3$ matrix $A$. Each system (particle) updates its state every unit of time.
The best I have been able to come up with for the "instantaneous" rate of conversion of yellow light into its energy equivalent in infrared light is something like $r = q_1a_{13}a_{32}a_{21}/3$. The reasoning is that for a particle to contribute to the conversion of energy starting right now, it has to be in state 1, and then get knocked into state 3, and then successively fall back down to state 2 and then state 1. Really, this is a per-particle rate, and would need to be multiplied by $N$ to get the overall rate. The division by 3 is just because the cyclic path considered, 1->3->2->1, takes 3 time steps.
But that's a little cheesy, since the particle could bounce around between states 2 and 3 before it finally got back to state 1. So maybe I need to consider all such paths, and average the time spent in them. (I also forgot that the diagonal elements of $A$ are typically not zero, they may even be dominant. So the particle may remain in the same state for stretches of time).
There are other problems. I don't really like the fact that it depends only on the state 1 occupation number, $q_1$, and not the others. That would seem to indicate that the flow of energy through the system is increased by having all the particles in state 1. But if energy is to be absorbed and then dissipated, they have to spend time in the other two states.
To fix this, I could add two more similar terms proportional to $q_2$ and $q_3$ that account for particles that are at different stages in the cycle at the moment. But then I would get an answer independent of the occupation numbers, depending only on the transition matrix. This also seems dumb. Surely the distribution of the particles among the 3 states affects their overall ability to absorb and dissipate energy.
So that's where my thinking is right now.