The typical set and entropy in stochastic systems with arbitrary phase space growth

The existence of the typical set is key for the consistence of the ensemble formalization of statistical mechanics. We demonstrate here that the typical set can be defined and characterized for a general class of stochastic processes. This includes processes showing arbitrary path dependence, long range correlations or dynamic sampling spaces. We show how the typical set is characterized from general forms of entropy and how one can transform these general entropic forms into extensive functionals and, in some cases, to Shannon path entropy. The definition of the typical set and generalized forms of entropy for systems with arbitrary phase space growth may help to provide an ensemble picture for the thermodynamic paths of many systems away from equilibrium. In particular, we argue that a theory of expanding/shrinking phase spaces in processes displaying an intrinsic degree of stochasticity may lead to new frameworks for exploring the emergence of complexity and robust properties in open ended evolutionary systems and, in particular, of biological systems.