The Markov process is used repeatedly in Monte Carlo simulations in order to generate new random states.
The reason for choosing a Markov process is that when it is run for a long enough time starting with a random state, we will eventually reach the most likely state of the system.
In thermodynamics, this means that after a certain number of Markov processes we reach an equilibrium distribution.
This mimicks the way a real system reaches its most likely state at a given temperature of the surroundings.