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

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Discussion Overview

The discussion revolves around the statistical interpretation of the Second Law of Thermodynamics, particularly in relation to the behavior of systems with varying numbers of particles. Participants explore the implications of statistical fluctuations in entropy and the conditions under which these might be considered violations of the law.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants propose that the Second Law can be viewed as a manifestation of the law of large numbers, suggesting that its inviolability stems from the vast number of microstates involved in entropy calculations.
  • One participant discusses the possibility of small decreases in entropy occurring due to statistical fluctuations, questioning whether a specific statistical boundary (e.g., x standard deviations) could define a violation of the Second Law.
  • Another participant emphasizes that the Second Law applies to systems with a large number of particles, arguing that it was not intended to apply to small systems, such as those with only 16 particles.
  • A later reply supports the idea that statistical thermodynamics can provide insights into expected variances, even in smaller systems, referencing specific literature for further exploration.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the Second Law to small systems and the interpretation of statistical fluctuations. There is no consensus on whether a specific statistical boundary exists for defining a violation of the law.

Contextual Notes

The discussion highlights limitations in applying the Second Law to small particle systems and the dependence on statistical interpretations, which may vary based on system size and conditions.

BWV
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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?
 
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BWV said:
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.
 
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
 
BWV said:
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
 

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