TITLE: Does thermodynamics only deal with homogenous systems?
QUESTION [5 upvotes]: In thermodynamics, quantities like pressure, temperature and entropy are associated with overall states of a macroscopic system. In that case, we do not talk about "the quantity $Q$ at the point $p$ of the system" thinking of it as a continuum, for instance, we rather say "the quantity $Q$ of the system" as if it were the same for all of it.
In that sense, does thermodynamics only deal with homogenous systems? That is, do we always suppose that pressure, temperature and all those quantities are the same on the entire system under study? If so, what is the need and motivation for this?

REPLY [0 votes]: TL; DR: the probability density is homogeneously spread over all the states with the same energy
Thermodynamics vs. Statistical physics
Let me first point out that thermodynamics and statistical physics are not the same thing: these are two descriptions of the same phenomena, one of which is phenomenological (thermodynamics) and the other is microscopic (statistical physics).
Non-equilibrium thermodynamics
Further, the question seems to refer to equilibrium thermodynamics/statistical-physics. Both of these have extensions (in fact multiple extensions) for treating non-equilibrium (and thus non-homogeneous) systems.
So, homogeneous or not?

In that sense, does thermodynamics only deal with homogenous systems?

Directly extending phenomenological thermodynamic description to inhomogeneous systems might be hard. From the point of view of statistical physics, we do make homogeneity assumption, but somewhat indirect: we assume that all the phase space configurations with the same energy (and same value of some other parameters) are equally probable, and that system will visit all of them, so that we can replace the time averaging by ensemble averaging. Tu put it into more "homogeneity" language: we do assume that
the probability density is homogeneously spread over all the states with the same energy. This is known as microcanonical ensemble, and serves to build canonical and grand canonical cases.
Homogenuity of quantities

That is, do we always suppose that pressure, temperature and all those quantities are the same on the entire system under study?

Pressure and temperature are intrinsic variables that characterize the system as a whole. As such, they say nothing about whetehr the system is homogeneous or not. What make them seem like specific to homogeneous system is the interpretation of these quantities as in terms of ideal gas, where pressure comes from the molecules colliding against the container walls, while the temperature is the average kinetic energy of these molecules. These quantities (and many other, such as, e.g., magnetic moment) can be however defined in very general thermodynamic ways, without resorting to their microscopic interpretation (note how thermodynamics here is more convenient than statistical physics):

Temperature as the derivative of the internal energy in respect to entropy $$T=\left(\frac{\partial U }{\partial S}\right)_{V,N}\text{ or } \frac{1}{T}=\left(\frac{\partial S }{\partial U}\right)_{V,N}$$
Pressure as the derivative of the internal energy in respect to volume
$$P=\left(\frac{\partial U}{\partial V}\right)_{S,N}$$

Remark: equating these with the mechanistic notions of pressure and temperature as defined for the ideal gas may sometimes be tricky however - see, e.g., Significance of Stokes hypothesis.
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