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The Wikipedia page for Boltzmann distribution describes it as a distribution of microstates as a function of energy and temperature. From my understanding, the 2nd law of thermodynamics is based on the idea that there are a greater proportion of microstates for some given macrostate (in a combinatorial sense), and this macrostate is thus what is most likely.

Is there an intuitive way to directly use the concept of Boltzmann distribution to explain the 2nd law (i.e. the H-theorem)?

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  • $\begingroup$ The H-theorem is not equivalent to the 2nd law of thermodynamics, so using i.e. above is not correct. $\endgroup$ – Ján Lalinský Feb 4 '19 at 1:08
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The Boltzmann distribution is the equilibrium distribution of a system in contact with a thermal reservoir. It is therefore derived assuming maximum entropy of the joint system+reservoir. Obviously, the second law cannot be derived from this distribution.

In order to ‘derive’ the second law, one requires some form of kinetic theory that describes how the system, which is not in equilibrium with the reservoir initially, equilibrates over time. The Boltzmann equation can describe such a process in specific cases.

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  • $\begingroup$ Second law of thermodynamics is not about systems approaching equilibrium in time. What you're probably thinking of is the so-called zeroth/minus first law of thermodynamics: given sufficient time, any isolated system approaches some equilibrium state. physik.uni-augsburg.de/theo1/hanggi/Minus.pdf $\endgroup$ – Ján Lalinský Feb 4 '19 at 1:07
  • $\begingroup$ @JánLalinský: Thanks for your comment, I agree with your statement that the second law is not about systems approaching equilibrium in time. This is, why I put the word `derive' in quotation marks. However, I felt that Tejas Ramdas was hoping for an intuitive way to gain insight into the second law. In my opinion, this is what kinetic theories can provide to some extent making them a great didactic tool. $\endgroup$ – flaudemus Feb 4 '19 at 20:11
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It is not clear why you think Boltzmann's probability distribution should play a role in derivation of 2nd law. 2nd law states something about possible values of macroscopic constraints like $V,T$ at the end of processes that begin and end at some equilibrium state. Boltzmann's distribution depends on such variables as $V,T$, but does not place any constraint on them. So the relation is more likely the opposite: from the 2nd law of thermodynamics, or some equivalent (maximum information entropy for isolated system), one can derive the Boltzmann distribution.

If you want to derive 2nd law, this can be done in several ways, but one has to assume some equivalent of it. For example, there is the method of information entropy, introduced to thermodynamics by E.T. Jaynes. In this method, one assumes that thermodynamic entropy is maximum possible information entropy. See my answer here:

Understanding Gibbs $H$-theorem: where does Jaynes' "blurring" argument come from?

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