Abstract
In this work we propose a model for open Markov chains that can be interpreted as a system of non-interacting particles evolving according to the rules of a Markov chain. The number of particles in the system is not constant, because we allow the particles to arrive or leave the state space according to prescribed protocols. We describe this system by looking at the population of particles on every state by establishing the rules of time-evolution of the distribution of particles. We show that it is possible to describe the distribution of particles over the state space through the corresponding moment generating function. This description is given through the dynamics ruling the behavior of such a moment generating function and we prove that the system is able to attain the stationarity under some conditions. We also show that it is possible to describe the dynamics of the two first cumulants of the distribution of particles, which in some way is a simpler technique to obtain useful information of the open Markov chain for practical purposes. Finally we also study the behavior of the time-dependent correlation functions of the number of particles present in the system. We give some simple examples of open chains that either, can be fully described through the moment generating function or partially described through the exact solution of the cumulant dynamics.
Highlights
In this work we propose a model for open Markov chains that can be interpreted as a system of non-interacting particles evolving according to the rules of a Markov chain
We have introduced a simple model for open Markov chains by interpreting the state space of a usual Markov chain as physical “sites” where non-interacting particles can be placed and moving throughout it according to “jumping rules” given by a kind of stochastic matrix
We have shown that this model can be treated by means of the moment generating function technique, allowing us to obtain, in a closed form, the moment generating function of the distribution of particles over the state space
Summary
The main idea behind our model for an “open” Markov chain is that we allow the particles to arrive and leave the state space S. The most simple case for the incoming protocol consists of a sequence of constant random variables all these taking the same value, which we call J0 ∈ N0 This means that the number of particles arriving at every time-step is J0. The probability of remaining in the state is q and the probability of jumping to the outside is 1 − q Notice that the above matrix is not stochastic, because some rows does not add to one, but less than one The latter means that not all the states allow the particles to leave to the outside. This shows that both approaches does not math each other in general, it would be interesting if they might coincide in particular cases
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