# Is there a connection between Boltzmann distribution and Uncertainty Principle?

Before asking the question, I should clarify that I am a college student majoring in mathematics and CS, so my understanding on physics is quite shaky. So please understand if this question turns out to be ill-defined.

I was reading a book on quantum mechanics and entropy. In the book, the author says that by the second law of thermodynamics, microstates of a system evolve to produce maximum Boltzmann entropy. Consequently, it turns out that the microstates eventually form an ensemble of Boltzmann distribution (aka canonical ensemble), i.e.

$$P_n = \frac{1}{Z} e^{-\epsilon_n/k_B T}$$

The author then notes that the only information contained in the above distribution is the energy of the states (since temperature is also determined by energy). This is paradoxical, as it seems to imply that all the other information about the initial state has been lost.

To explain how this is possible, the author introduces ergodic hypothesis. And to explain how ergodic hypothesis can hold, the author employs chaos theory and Heisenberg's uncertainty principle. So to summarize:

1. Boltzmann distribution is only dependent on the energy of the states.
2. Such information loss can be explained by chaos theory and uncertainty principle.
3. Thus, quantum mechanics and chaos theory is what drives the second law of thermodynamics.

But I can't agree with this conclusion, since I'm pretty sure that statistical thermodynamics is independent from quantum thermodynamics. Already I have found many comments on PSE saying that the second law of thermodynamics doesn't prove QM. On the other hand, the author is a working theoretical physicist and I don't think the book contains any errors. So in the end, what is the relationship between Boltzmann distribution and the uncertainty principle?

• Which book, edition chapter etc...? Feb 16, 2022 at 9:06

The uncertainty principle solves one difficulty that appears in classical statistical mechanics by giving a natural resolution for the system.

Suppose there wasn't any uncertainty principle, then you can define in principle a state of the system by a point in $$6N$$-dimensional phase space. As system time passes, the system moves around on a constant energy shell. But a point will never cover the whole surface, no matter how much time you wait.

The uncertainty principle gives you the highest resolution that sets the minimum area that you can define on phase space and cover the whole space in sufficient time.

Recall $$d\Gamma_N=\frac{1}{h^{3N} N!} \prod_i d^3\vec{q}_i d^3\vec{p}_i$$ set the measure of phase space.

• There is no difficulty in classical Statistical mechanics (remember that only differences of thermodynamic potentials are observable). The uncertainty principle provides a "natural scale" for defining a microscopic state in phase space. Nothing more. Feb 16, 2022 at 11:52
• @GiorgioP Wouldn't it cause a problem if you don't have this scale? Feb 16, 2022 at 13:07
• If you use $h$,$2h$ or $100h$, in the classical regime there would be no difference. It would change only an additive constant, irrelevant for differences of thermodynamic potentials. Feb 16, 2022 at 18:10

You ask a very good question that many peer of a respectable age and solid understanding of physics don't even dare (or understand) to ask. The short answer is yes. Boltzman explained what was Entropy (introduced by Clausius but ill defined) by considering the world as being an assembly of atoms. However, in doing so, he (and others) came to a difficult situation where you had all those little atoms everywhere bouncing into each other. How could you describe such a mess? He found a turn around to this problem by using statistics, or probability if you will. And there my friend, we lost certainty. It is interesting to note that everything that goes down from there assume, implicitly, this incertitude principle. The incertitude principle isn't a fancy tool as stated above; it is intricate inside this view of the word and is thus part of modern physics and is not a choice you can make to consider it or not. It is there, period. Unless you find a new solution to the same problem Clausius saw with S and that Boltzman solved that can fit observations. One of the main beauty in Boltzman work was it's good fitting of calculation with observation. Sad this man had such a tragic life.

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