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Thermodynamics deals with "equilibrium states of macroscopic matter", that is, considering macroscopic systems there are states which can be characterized fully by a few number of measured degrees of freedom and on such states we are not able, through macroscopic measurements to see the fact that the molecules and atoms are not really in equilibrium states. Those are the equilibrium states and Thermodynamics deals with those states of macroscopic systems.

Now I'm starting to study Statistical Mechanics and I wonder if the situation is the same. This question is because the book talks about the "thermodynamic limit" which is attained when we let $E,V,N\to \infty$ with finite $u = E/N$ and $v = V/N$.

This led me to think that Statistical Mechanics can deal with some more systems than the ones thermodynamics is concerned with. So Statistical Mechanics just drops the "macroscopic matter" part and deals with equilibrium states of general systems or it deals just with the same systems considered in thermodynamics but with another viewpoint?

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Indeed, statistical mechanics in principle deals with completely general systems.

From the man himself who coined the term "statistical mechanics":

The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results. The laws of statistical mechanics apply to conservative systems of any number of degrees of freedom, and are exact.

Moreover, stat mech applies to any states that involve uncertainty, not only equilibrium states. That said, however, the most popular applications of stat mech, and easiest to calculate, tend to involve thermodynamic systems, and so many textbooks will focus on that subcase.

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