What is the statistical boundary for violating the 2nd Law of Thermodynamics?

[BWV]

Is it fair to say that the 2nd Law basically is the law of large numbers, which given the immense numbers of microstates involved in entropy calculations, is inviolable?

With a Boltzmann distribution, one could have arbitrarily small decreases in entropy from time t to t+1 as for a system at equilibrium there would be some fluctuation proportional to the variance of the distribution. In a pool table example, while a return to the original state of the cue ball traveling toward the racked balls would be an incredibly unlikely event, one would expect that elastic collisions from time to time would leave one ball at rest - which would, I guess, be a trivial and temporary reduction of the dispersal of energy in the system. If this is correct, is there some statistical boundary (i.e. x standard deviations of the Bolzmann distribution) that would be have to passed to constitute a violation of the 2nd Law?

 

[Mapes]

If this is correct, is there some statistical boundary (i.e. x standard deviations of the Bolzmann distribution) that would be have to passed to constitute a violation of the 2nd Law?

If you take the Second Law to say that there is only a tendency for entropy to increase (and that counterexamples would become vanishingly rare as system size increases, as you point out), then this deviation wouldn't even be a violation.

 

[Andrew Mason]

It is implicit that thermodynamics is about the behaviour of systems containing at least billions of billions of particles for which a temperature is defined. Such systems will always obey the second law. The second law was never meant to apply to systems of 16 particles, such as balls on a pool table.

AM

 

[A. Neumaier]

Is it fair to say that the 2nd Law basically is the law of large numbers, which given the immense numbers of microstates involved in entropy calculations, is inviolable?

Yes, and statistical thermodynamics let's you even determine the expected variance if a system gets tiny. See Chapter 6 in "http://lanl.arxiv.org/abs/0810.1019