If I have an isolated composite system which is dependent on several extensive properties (say $U_i, V_i$ and $N_i$ for each subsystem $i$) then a quasi-static locus is just a continuous curve through $U_1-V_1-N_1-U_2-V_2-N_2-\cdots-U_n-V_n-N_n$-space. Moving along this curve means infinitesimally slowly changing the parameters $U_i,V_i$ and $N_i$ so the system moves from one point to the next.

If at any point moving along the locus would cause the entropy of the whole system to drop, then we would not be able to traverse that part of the curve. I'm asking about the opposite problem: if the the entropy along a locus is non-decreasing, can it always be traced out by some physical process?

I realise that no quasi-static process is actually physically realisable - I'm talking about following the curve arbitrarily closely by varying boundary conditions alone.

Another way of posing the questions is: from any particular point in phase space, is it physically possible to move in any direction (within phase space) in which the entropy is not decreasing. If this were true then by taking many small steps we could follow any locus arbitrarily closely.

This is a very interesting question whose best answer is probably a very definite maybe... Defining a quasi-static process as one that is very very slow is not very useful as the simplest ferromagnetic hysteresis would show. If instead you define a quasi-static process as one that at any instant of time can be described by being on a path in the n+1 dimension space with coordinates $\theta, x_1,x_2, ...,x_n$ where the $x_i$ are the extensive (non-thermal) parameters and $\theta$ is the empirical temperature then what you are saying is probably correct for all the cases that are discussed in most textbooks. I say probably because it would be true if one also assumed that Joule's experiment holds universally. By this I mean the generalization of Joule's paddle-wheel stirring experiment that showed that adiabatic work without expansion work is potential. In this context the generalization would be that in any state one could fix the configuration coordinates $x_i$ and still perform arbitrary amount of adiabatic work and thereby raise the internal energy and temperature of the system. Now this is not obvious at all, but if you assume it to be true then you can get all of Caratheodory's theory, see Adkins: Equilibrium Thermodynamics. If you now assume this, then you can show that given any two states $J_1,J_2$ there is either an adiabatic process from $J_1$ to $J_2$ or there is one from $J_2$ to $J_1$ (always from lower to higher entropy). Furthermore that adiabatic process has two parts: a reversible one connecting the given configurations (isentropic) followed by an irreversible stirring (a la Joule).

So if you do this in small steps, isentropic & "stirring", then you could approximate arbitrarily closely the quasi-static path you were asking about.

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