I was thinking of a random process as: a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions.
Given, it is non-deterministic it can be achieved only through processes such as symmetry breaking, collapse of the wave function and hence involve an increase in entropy.
Related: Does the scientific community consider the Loschmidt paradox resolved? If so what is the resolution?
My question is what is the minimum amount of heat involved in creating a random sequence of a particular length?