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Inspired by the PBS Space time video on Noether's Theorem released the other day, I got to thinking about what can be considered the most fundamental laws in Physics. They mention in the video that The Principle of Least Action can be used to derive the laws of motion for a wide variety of systems, and is "about as close to a fundamental assumption as we can get."

But I struggle to see how the principle of least action could be considered fundamental to a field like, say, Thermodynamics. Are the axioms of entropy (from Callen's Themodynamics/Thermostatistics) based on a princple of least action? Perhaps there is some action defined over the state variables of a thermodynamics system that I am ignorant of.

Some also suggest that a princple of conservation of information may be more fundamental? Is the principle of least action fundamental to Information Theory?

Are there laws (or a law) that can be considered fundamental to all branches of physics? I.e. Electromagnetism, QCD, Standard Model Particle Physics, QM, Thermodynamics, Statistical Mechanics, General Relatovity, etc. Or, perhaps easier, is there one law, axiom, or fundamental assumption that is foundational to the broadest range of theories/models in Physics?

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  • $\begingroup$ For thermodynamics, the Fundamental Assumption of Statistical Mechanics is critical. In astro/cosmology, the cosmological principle is an important assumption. $\endgroup$
    – zh1
    May 20, 2018 at 3:23
  • $\begingroup$ How about the asymptotically local spacetime being Minkowski? $\endgroup$
    – safesphere
    May 20, 2018 at 8:36

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First off, just to set entropy straight, classical (Boltzmann) entropy is an emergent phenomenon in the study of multi-component systems. We may know the basic dynamical principles of each individual component, as well as the specific nature of the interactions between different components, but when you consider a collection of them (and in particular a statistically large collection of them) and relax your knowledge about the precise state of the system, you get entropy. This general concept applies to systems with arbitrary underlying mechanics, i.e. it could be a system governed by a least action principle, or not. Even more, this general concept applies no matter if your system's underlying mechanics are classical, or quantum, or something deeper. For example, we know how to describe pretty accurately the motion of a point-like gas particle (a la Newton), and when you put $10^{23}$ of them together in a box it becomes impossible to track each of them, so you instead track only statistical averages like pressure and temperature, and thus the system now has "entropy". This definition of entropy is encoded in Boltzmann's entropy formula, $S=k_B\log W$ (keeping the $k_B$ to pay respects).

The least action principle is something that has been found more to apply to the underlying dynamical/mechanical laws. The least action principle is so fundamental because all classical theories of physics can be cast into the form of a least$^1$ action principle. The least action principle still does apply to composite/multi-component systems, but when the number of components grows to statistically significant proportions it becomes useless as a calculational tool.

The least action principle is itself though a limit of a more fundamental principle/object, the path integral (I'll call it the "quantum" action principle). The least action principle specifically arises in the stationary phase, or equivalently steepest descent (via a suitable analytic continuation in the time coordinate), approximation of the path integral. You can read more about it in wikipedia article. The standard model is based on a quantum action principle. General relativity, being a classical field theory, is based on a classical action principle. Microscopic electromagnetism, which is the classical limit of QED, is based on a classical action principle.

However, to directly answer your question, there was at one point an axiom for the concept of entropy that was in some way similar to the concept of least action. It was called the H-theorem, by Boltzmann. Now we know it's not true though, and more importantly why it's not true.

I am not an expert though so please take what I say with a grain of salt.


$^1$ Stationary action, not necessarily least action.

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  • $\begingroup$ Its always been fascinating to me the extent to which Fourier theory seems to be fundamental to physics in general. This seemingly obscure bit of mathematical trickery allowing one function to be transformed in to another, by way of decomposing it in to sinusiods of certain frequencies, seems fundamental to the laws of nature. So odd. $\endgroup$
    – D. W.
    May 20, 2018 at 4:15
  • $\begingroup$ @D.W. What exactly do you mean by "fundamental"? The Fourier transform itself does not appear in any of the axioms that define each of the fields. It merely makes calculation of various details easier - in principle, you could do the same sorts of calculations without it (not that you'd want to, but you could get the same observable answers either way). $\endgroup$ May 20, 2018 at 5:41
  • $\begingroup$ Correct me if I'm wrong, but the Fourier transform seems be the the natural way of expressing the action principle for the quantum Hamiltonian. More generally, the Fourier transform shows up all over physics, as a way to express formulations in a more beautiful and meaningful way. This is not to say that the Fourier transform is necessary. It's just, the degree to which it is useful and elegant is kind of astounding to me. $\endgroup$
    – D. W.
    May 20, 2018 at 5:55
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    $\begingroup$ @D.W. Basically, the fact that you keep seeing the Fourier transform everywhere is not because of anything fundamental about nature, but rather based on the way that we've found it easiest to work with as humans. Applying the same sort of logic, you might conclude that nature prefers stones to be rectangular prisms because sidewalk tiles are common in your area. $\endgroup$ May 20, 2018 at 6:12
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    $\begingroup$ @probably_someone thanks for your answers, I appreciate them. Thanks to your thorough explanations of why my question was ill-posed and my understanding is lacking, I've become completely uninterested in this topic for now. Back to the books I suppose. $\endgroup$
    – D. W.
    May 20, 2018 at 7:33
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First of all, the least-action principle is a bit of a misnomer. The path of a classical system will take the path that extremizes the action. That extremum could be a maximum or a minimum, or even something more complicated like a saddle-point. It's not always the path of minimum action; for example, if I had a system which minimized a certain action $S$, then I could define $S'=-S$, and the same classical path would maximize $S'$. (Such an action would not be bounded from below, and therefore we would typically use $S$ rather than $S'$, but there's nothing fundamental about that; it's just a sign convention to keep the notion of energy consistent between problems.) Since multiplying the action by a constant doesn't affect the path I get from applying the calculus of variations, there's no reason to prefer minima over general extrema.

As for the rest of your question, the least-action principle can in principle be used to derive the equations of motion for any system in classical mechanics, classical electrodynamics, quantum mechanics (including quantum electrodynamics and quantum field theory in general), and general relativity (see the Einstein-Hilbert action: https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action). Given infinite time and computing power, we could also use the principle of least action to describe the motion of systems of $10^{23}$ particles, too. In that case, we wouldn't have any need for statistical mechanics or thermodynamics. Thermodynamics is a phenomenological theory that is meant to address the experimentally-observed behavior of macroscopic systems; statistical mechanics is the underlying microscopic explanation for that theory, which is based on the assumption that it is impractical to calculate the equations of motion of a system of $10^{23}$ particles. Statistical mechanics makes a lot of approximations specifically so that we don't have to worry about the trajectories of individual particles, and as such, the principle of least action in the form that it takes in other fields is irrelevant, as it specifically deals with the paths of individual coordinates. That said, there is a similar principle in statistical mechanics: the maximum entropy production principle, which applies the ideas of the calculus of variations to statistical mechanics (see https://www.scripps.edu/baran/images/grpmtgpdf/Renata_Feb_10.pdf).

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  • $\begingroup$ Not that I mind very much, but it seems you've basically copied the bulk of my answer above. $\endgroup$ May 21, 2018 at 1:18

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