# Condition for spontaneity of a non-equillibrium statistical system

As the title suggests: Is there any condition, which can determine the spontaneity of a non-equilibrium statistical system?

If it is a "thermodynamic" system, we can calculate its $$\Delta G$$ (change in Gibb's Free Energy) to determine whether the system is spontaneous or not. Even for any kind of system in "equilibrium", the increase in entropy indicates the spontaneity of that system. Is there any such condition for a general (not necessarily thermodynamic) "non-equilibrium" "statistical" system?

Wikipedia is saying,

According to Kondepudi (2008), and to Grandy (2008),there is no general rule that provides an extremum principle that governs the evolution of a far-from-equilibrium system to a steady state. According to Glansdorff and Prigogine (1971, page 16), irreversible processes usually are not governed by global extremal principles because description of their evolution requires differential equations which are not self-adjoint, but local extremal principles can be used for local solutions. Lebon Jou and Casas-Vásquez (2008) state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997) offers the opinion that "... the extremum principles of thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)."

For a steady state characterized by a distribution $$\rho$$ over configurations, one can always, tautologically, define a 'potential' $$V = -\log(\rho)$$ that plays a similar role to the argument of the Boltzmann distribution in an equilibrium context. The principle problem in the theory of nonequilibrium steady states is that obtaining such a $$\rho$$, or $$V$$, from the dynamics is generally quite hard (read: intractable), as it typically consists of solving an extremely high dimensional Fokker-Planck equation in a regime without any easy simplifications.
• But what is the condition (on $\rho$) for being a spontaneous system?