How Boltzmann's Entropy Applies to Markov Chains
The Entropy of a Markov Chain
The author explores how to define entropy for Markov chains, motivated by Dyson's toy model of a cell. They first review Clausius's thermodynamic entropy and Boltzmann's statistical definition, using Curie's magnet model to illustrate counting microstates. Then they show how to compute entropy for an equilibrium distribution of a Markov chain via multinomial coefficients, and hint at future work on entropy evolution.
Boltzmann showed that entropy can be written as a function of the number of states that a system can be in, if we fixed what the macro state variables were.
- abetusk
So how does one calculate the entropy of a Markov Chain? Is it actually specified? If so, it seems buried.
The Markov chain provided as an example has the edge labels swapped (np should be qp and qp should be np). Regardless, what is the entropy of the example provided?
The problem with Markov chains is that states are dependent, so simply cataloguing states now violates the basic entropy calculation as neighboring states are now dependent on each other.
If the Markov chain is ergodic then maybe you can talk about the entropy of the stationary distribution? Then it's just $-\sum p_i lg(p_i)$ of the stationary distribution probabilities?
The article alludes to how entropy evolves. In the context of ergodic Markov chains, this is related to the size of the second eigenvalue?
- niklasbuschmann
Stochastic thermodynamics covers this