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If I understand the question, you are wondering how to justify the statement that a (reverible) adiabatic process is isentropic from the point of view of statistical mechanics (the classical thermodynamics definition makes sense to you). Let us then start with the entropic fundamental relationship, S = S (U, V, N), where U stands for energy, V for volume, N ...


By definition a reversible adiabatic system has $dQ = 0$. We also know the following from the Clausius Theorem : $dS = \frac{dQ}{T}$ Then it is easy to see that there can be no change in entropy. Note that irreversible adiabatic systems CAN see a change in entropy because in that case the above equation is no longer an equality but an inequality : $dS ...

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