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A colleague of mine is working with single (classical) particle systems. These systems can be modeled very accurately with Ito and Wiener Processes and the formalism of stochastic differential equations, and the predictions seem to agree very well with the experimental measurements.

Now, at some point he starts using the equipartition theorem and the Gibbs-Boltzmann distribution, and here I get a bit uneasy, because I'm not sure how to glue both formalisms. My knowledge of statistical mechanics is constrained to very ideal ensembles, and in the single particle case I guess somehow ergodicity must be assumed. But not only that: in these stochastic systems the energies are usually quite complex, and interactions with the outside world are not only of the thermal kind.

So what can and cannot be done, rigorously, in these cases, and how can the two fields be linked? Could you also provide any good book references?

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closed as too broad by DanielSank, sammy gerbil, Rory Alsop, Gert, heather Jan 16 '17 at 17:29

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Partially related (though sadly unanswered) physics.stackexchange.com/q/301579 $\endgroup$ – Kyle Kanos Jan 13 '17 at 11:15
  • $\begingroup$ I didn't see that post, and maybe I can answer it. But I don't see very clearly the statistical mechanics question part there... $\endgroup$ – gerd Jan 13 '17 at 12:13
  • $\begingroup$ That's why I said partially related ;) $\endgroup$ – Kyle Kanos Jan 13 '17 at 12:14