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statistical mechanics: inference for stationary distributions

September 25, 2010

E.T. Jaynes is famous for his (often quite dogmatic) promotion of the idea that the whole of statistical mechanics is nothing more than a statistical inference problem. While I’m not sure if I buy his entire program, these ideas have gotten me interested trying to figure out which parts of statistical mechanics follow from the microscopic laws of physics  and which parts are inference methods in disguise.

This goal — while mostly philosophical — does have some practical importance. Due to its success in physics, statistical mechanics is often viewed as a “model framework” from which to construct other theories of complex systems. However, it is unrealistic to expect that the microscopic dynamics of social or biological systems is dictated by some Hamiltonian dynamics (unless we go all the way down to the microscopic particles that comprise them). If statistical mechanics depends on these mathematical properties in some crucial way, then there is not much hope for extending its scope to systems that drastically differ in their microscopic dynamics.

In this post, I plan to lay out my current understanding of the foundation of classical statistical mechanics — mainly to record some thoughts that have been floating around in my head for the past few years.

We start with some classical system whose microstates are characterized by canonical coordinates (p,q) that are governed by Hamilton’s equations

\dot{q} = \frac{\partial H}{\partial p} \quad \dot{p} = - \frac{\partial H}{\partial q}

for some Hamiltonian H.  More to come…

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