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

 

[sardar]

I would say that the 2nd Law of Thermodynamics is not simply the law of large numbers, but rather a fundamental principle that governs the behavior of energy and matter in our universe. While the concept of entropy and its calculations may involve large numbers of microstates, the 2nd Law is not limited to these calculations alone.

The 2nd Law states that the total entropy of a closed system will always increase over time, unless acted upon by an external force. This means that, in general, the amount of disorder or randomness in a system will tend to increase over time. This principle has been observed and verified in countless experiments and is a fundamental aspect of our understanding of the physical world.

In regards to the example of a pool table, while it is true that there may be small decreases in entropy due to fluctuations in a Boltzmann distribution, these fluctuations are expected and do not violate the 2nd Law. It is important to note that the 2nd Law is a statistical law, meaning that it applies to the overall behavior of a system, rather than individual events.

In terms of a statistical boundary for violating the 2nd Law, it is difficult to define a specific number of standard deviations of a Boltzmann distribution. The 2nd Law applies to the overall behavior of a system, so a violation would be seen as a significant and sustained decrease in entropy, rather than a temporary fluctuation. Additionally, the 2nd Law is a macroscopic law, meaning that it applies to systems with a large number of particles. Therefore, it would be difficult to observe violations of the 2nd Law at the level of individual particles.

In conclusion, the 2nd Law of Thermodynamics is a fundamental principle that governs the behavior of energy and matter in our universe. While there may be small fluctuations in entropy due to statistical variations, these do not violate the 2nd Law. It is important to understand that the 2nd Law is a macroscopic law and applies to the overall behavior of a system, rather than individual events or particles.