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The fact that a system is in equilibrium
all along does not guarantee that the change it undergoes is
quasistatic in the above sense. We can imagine a system
thermally isolated from HOTTER surroundings but still a thin
wire connects them so that the temperature of the system
increases very slowly. The system is in equilibrium (passes
through a succession of equilibrium states) but....what
type of process is this????
It is extremely important for us to answer this question.
Classically, the entropy is defined through dS=dQrev/T, but the
subscript "rev" may mean, according to textbooks, two things.
Either the system just passes through a succession of
equilibrium states, or the system passes through a succession
of equilibrium states but, in addition, exchanges heat with
the surroundings ONLY WHEN SYSTEM AND SURROUNDINGS ARE OF THE
SAME TEMPERATURE.