# 10. Postulates of statistical mechanics

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Thermodynamics puts constraints on the behavior of macroscopic systems without referencing the underlying microscopic properties. In particular, it does not provide a quantitative connection to the origin of its fundamental quantities \(U\) and \(S\). For \(U\), this is less of a problem because we know from mechanics that

\[ U=\dfrac{1}{2} \sum m_i v_i^2 + V(x_i),\]

and the macroscopic formula arises by integrating over most coordinates and velocities. Somehow the thermal motions end up as \(TS\), and the mechanical and electrical motions end up as terms such as \(–PV+\mu n\).

Statistical mechanics makes the macro-micro connection and provides a quantitative description of U and S is terms of microscopic quantities. For large systems (except near the critical point), its results are in agreement with thermodynamics: one can derive thermodynamic postulates 0 – 3 from statistical mechanics. For systems undergoing large fluctuations (small systems or those systems near a critical point), its prediction are different and more accurate.

In addition as the ‘mechanics’ implies, statistical mechanics can deal with time-varying systems and systems out of equilibrium. Averages over x(t) and p(t)=mv(t) of the microscopic particles are done, but not in such a way that all time-dependent information is lost, as in thermodynamics.

Unlike mechanics, statistical mechanics is not intended to discuss the time-dependence of an isolated particle. Rather, the time-dependent (e.g. diffusion coefficient) and time independent properties of whole systems of particles, and the averaged properties of whole ensembles of such systems, are of interest.

We begin with an introduction to important facts from mechanics and statistics, then proceed to the postulates of statistical mechanics, consider in detail equilibrium systems, and finally non-equilibrium systems.