A derivation of the Grand Canonical Partition Function for systems with a finite number of binding sites using a Markov chain model for the dynamics of single molecules

摘要

We use a Markov chain to model the ligand binding dynamics of a single molecule and show that its stationary distribution coincides with the laws of the Grand Canonical Ensemble. This way of deriving the equilibrium laws has the following advantages: Firstly, the derivation is short and does not require the knowledge of theMicrocanonical, Canonical or Grand Canonical Ensemble. Secondly, it provides a descriptive interpretation of the factors that contribute to the probability of amicrostate. In this regard, it also shows that the chemical activity, which cannot be regarded as a probability (since it is not necessarily bounded by one), can be interpreted as the ratio of two probabilities. Thirdly, our approach allows modeling how the system reaches equilibrium. This can be a useful tool for the study of non-equilibrium states.