Two state model

Question

Suppose a particle can only have two states - either it is in the open state (o) or it is in the closed state (c). The exact transition probabilities from open to closed and vice versa are not that important.

Answer 1

This is a continuous-time Markov chain. You can think of it this way. Each particle, independent of the others, constitutes a two-state continuous-time Markov chain with rate $Prob(o \to c)$ when it is in the "open" state and $Prob(c \to o)$ when it is in the "closed" state. Despite the notation, these are rates, not probabilities. Each particle's probability distribution approaches the equilibrium distribution where by the principle of "detailed balance", if $p_o$ and $p_c$ are the probabilities of open and closed for a particle, $p_o \; Prob(o \to c) = p_c\; Prob(c \to o)$. The equilibrium distribution for the system as a whole is the product measure of these Bernoulli distributions for the individual particles, so the number of open particles is Binomial with parameters $p_o$ and $N$.