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There are interesting changes that occur in a sample of interacting objects, such as gas particles, as you approach a statistically significant sample. The position or velocity of any given particle no longer matters as much as the collective behavior of the gas. At what point does this become true? I'm interested in what really happens in this transition, and in others like it. So I ask: Is there a field of mathematics that describes the transition into statistical mechanics?

Edit: Thanks for the comments & for indulging my poorly worded question.

Recommendations thus far have been to look into Non-Eq Thermodynamics, Synergetics, Markov random fields, Gibbs measures, and large deviation theory.

User on Quora suggested the following review on large deviation theory: http://people.math.umass.edu/~rsellis/pdf-files/Touchette-review.pdf

LBM
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  • An interesting question. Possibly this is covered by "emergence" or "emergent behaviour". – sammy gerbil Jun 11 '16 at 19:39
  • Crossposted from http://math.stackexchange.com/q/1822348/11127 – Qmechanic Jun 11 '16 at 19:53
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    The assumption that the dynamics of the individual particle doesn't matter is not correct in general. Statistical mechanics can only be used for systems that are sufficiently homogeneous. If that is not the case, we have to use different approximations (non-equilibrium TD, synergetics etc.) or, worst case, can't use any, at all. For intermediate particle numbers (tens to maybe tens of thousands) one can use molecular dynamics simulations, but those are less useful than one would like. Phase transitions, in particular, have very long range correlations and don't easily show up for $n<<\infty$. – CuriousOne Jun 11 '16 at 20:01
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    Well, this is certainly an important part of probability theory. For equilibrium statistical mechanics (the only part that is well understood, even from a physical point of view), you can have a look at the subfield of probability theory known as (mostly, infinite) Markov random fields, Gibbs measures, etc. There are also deep relations with (aspects of) large deviation theory, also a part of probability theory. – Yvan Velenik Jun 11 '16 at 20:33
  • Also crossposted from Quora https://www.quora.com/What-is-the-field-of-mathematics-that-describes-the-transition-into-statistical-mechanics – LBM Jun 11 '16 at 21:31
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    You probably also want to look at ergodic theory -- it describes the features of systems whose time-averaged properties can be described as an integral over phase space / sum over microstates, which is central to Boltzmann's definition of entropy, etc.. – Mark A Jun 12 '16 at 23:52
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    I think stochastic processes is important here. – DanielSank Jun 13 '16 at 00:42
  • Isn't collective behavior the result of individual particles behavior? Even in statistical mechanics individual particles are considered but mostly approximated to have the same behavior, for example, the energy of the larger system is ~nE where n is the number of particles in the chamber. I think, there is no transition from classical to statistical physics, it is just that statistical physics uses statistics while classical physics uses mathematics. –  Jun 14 '16 at 18:57
  • Look at Chaos Theory, Mandelbrot Fractals, Mitchell Feigenbaum works. – Guill Jun 15 '16 at 05:18
  • @GFG Statistical physics does not use statistics, it uses probability theory, which is part of mathematics (and much of statistics is part of mathematics too, in any case). Moreover, there is a "transition" from mechanics to statistical physics, since the latter's validity requires (well behaved) large systems. – Yvan Velenik Jun 15 '16 at 17:53
  • @YvanVelenik Probability theory is the foundation of statistics. I can't think of statistics without probability theory. Mathematics may contain all of them but I meant that part of certain (as opposed to uncertain) mathematics, i.e., differential equations and algebra. Yes, statistical physics uses statistics (see wikipedia [link] (https://en.wikipedia.org/wiki/Statistical_physics)). That's why we call it statistical physics. –  Jun 15 '16 at 18:43
  • @GFG No, statistical physics does not use statistics. Probability theory and statistics are 2 very different branches and statistics is completely irrelevant to statistical physics (except, marginally, for the analysis of, say, computer simulations). Certainly, it is completely irrelevant to the theoretical aspects. Moreover, probability theory is in no way more "uncertain" than analysis. And the fact that we call it "statistical" physics is an unfortunate historical choice. – Yvan Velenik Jun 15 '16 at 18:47
  • I would like to remind that statistical mechanics is sometimes confused with kinetic theory. Statistical mechanics is just Hamiltonian mechanics with a probability distribution of states, and its foundational equations are exact for all systems, large and small (and most importantly gives us equilibrium statistical mechanics, for describing thermodynamics). Kinetic theory is an approximation to describe large collections of bouncing particles and doesn't really work for other mechanical systems. Statistical physics could be defined as including both but that's not universal. – Nanite Jul 01 '16 at 10:30

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There is no single point where this becomes true - it is a very gradual change. The buzzwords are microscopic $\to$ mesoscopic $\to$ macroscopic.

There is no special kind of mathematics involved; in the mesoscopic domain one uses a mix of quantum mechanics and statistical mechanics.

See https://en.wikipedia.org/wiki/Mesoscopic_physics

Arnold Neumaier
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