**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?** 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.