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Equilibrium thermodynamics handles non-equilibrium (irreversible) processes,
if they connect well defined initial and final equilibrium states, by noting
that:
1) The initial and final states are equilibrium states in which state
variables are well defined. and collectively define the system state.
2) One can then construct an alternate, reversible process consisting of a
series of equilibrium states which evolve from the initial to the final
state according to the "laws" of reversible, equilibrium thermodynamic
processes. Every state in this reversible process is an equilibrium state
and has well defined state variables.
3) One then uses the physics of equilibrium, reversible processes to
calculate, along the chosen, reversible path of equilibrium states, whatever
system state changes are of interest in the transition from the initial to
the final states.
4) Since "a state is a state is a state", these system changes are identical
to the system changes undergone by the originally considered irreversible
process.
Even something as central and basic as the Carnot heat engine
requires two heat baths, at temperature T1 and temperature T2.
Each heat bath is in equilibrium with itself ... but not with
the other.
My specific comment:
During the Carnot cycle the system under consideration is always in a well
defined equilibrium state. This in no way requires the two heat baths to be
in equilibrium with each other. What are you thinking of?