The definition of Quasi-static process? A quasi-static process is often defined as a process "that occurs infinitely slowly such that equilibrium holds at all times."(Harvard, Matthew Schwartz, statistical mechanics Spring 2019). My question is a simple but possibly subtle one which I haven't seen mentioned anywhere.
Simply put, does the system need to maintain equilibrium with the surroundings at all times during the process in order for the process to be classified as a quasi-static process or is the lesser requirement of simply having the system maintain internal equilibrium with itself good enough to classify the process as quasi-static?
As an example, suppose we have a fixed volume system immersed in a heat bath (the surroundings). The system is at temperature $T_{sys}$ while the heat bath is at fixed $T_{bath}$ with $T_{sys}<<T_{bath}$. The walls/boundary of the system are virtually adiabatic but not totally (i.e they posses a very low thermal conductivity) and so heat can and will seep into the system across a large temperature gradient but this will happen very slowly (perhaps even infinitely slowly). After a very long time, the systems temperature will equal the heat baths temperature. Does this count as a quasi-static process? Throughout the process, the system had a well-defined internal equilibrium however it was never in equilibrium with the surroundings and so I am not sure whether it counts as quasi-static or not.
Any help on this issue would be most appreciated!
 A: In branches of physics other than thermodynamics, the word "adiabatic" is used as synonym of "infinitesimally slow." This creates some confusion.
In equilibrium thermodynamics, as you correctly intuit, you need to be more specific. Infinitesimally slow processes are introduced as a useful concept in order that the equation of state be valid throughout. In this way, you can calculate work, specific heats, etc. by using calculus.
As an example of why you need to distinguish "adiabatic" from "quasi static", consider the slow expansion of an ideal gas, in which you wish to keep it at constant pressure (calculation or measurement of $C_p$). You would obviously have to supply some heat so that your gas doesn't lose pressure. But you would have to do it very slowly, so that pressure itself, as well as temperature, are well defined.
This would be in sharp contrast with the adiabatic expansion (also infinitesimally slow, but with no heat supplied, and thereby suffering a decrease in pressure, as well as in temperature).
Why do you need to distinguish "quasi static" at all? Because sometimes you want to discuss changes that occur almost instantaneously. The system immediately readjusts to a new equilibrium condition, so to speak, so that it jumps over all possible intermediate equilibrium conditions.
In a quasi-static process, on the contrary, the system immediately re-adjusts to an intermediate equilibrium condition at every infinitesimal step.
I hope that was helpful.
A: I would consider the process you described as quasi static.  In fact, in that particular process, both the system and the surroundings experience reversible changes.  However, for the low thermal conductivity medium in-between, the process is not reversible, and entropy is generated within this medium. This generated entropy is transferred to the system.  So the increase in entropy of the system is greater than the decrease in entropy of the surroundings, and the net result is an increase in entropy of the "universe."
