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In statistical thermodynamics we can prove that the evolution of a system minimises some potential with units of energy (e.g. energy). This can be done purely statistically, by using the first two laws of thermodynamics, and showing that the state where the appropriate potential is minimised is the most likely one. Therefore, minimisation of energy is just an argument about entropy.

Is there a similar (or any other) proof of the principle of least or stationary action? I have seen it referred to as an axiom, but is there at least a possibility that there exists an underlying theory from which it can be derived? Can it be shown that systems where action is stationary are the most likely ones? Also, it is interesting that the Lagrangian has units of energy as well.

Edit: To clarify on the first paragraph. I was referring to quasistatic systems that satisfy a sepcific set of conditions. Energy is minimised in the stable state of an isolated system (where entropy, volume, and the number of particles is conserved). For a system where the temperature, pressure, and the number of particles are constant, the Gibbs free energy is minimised etc.

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    $\begingroup$ Are you aware or the Path Integral Formulation of Quantum Mechanics and QFT? $\endgroup$
    – Jeanbaptiste Roux
    Commented May 28, 2023 at 16:08
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    $\begingroup$ > "In statistical thermodynamics we can prove that the evolution of a system minimises some potential with units of energy" Source? Non-decrease of entropy follows from macroscopic observations, or from special initial conditions; it can't be derived from mechanics and probability theory alone. $\endgroup$
    – Ján Lalinský
    Commented May 28, 2023 at 16:11
  • $\begingroup$ For the opposite question, see physics.stackexchange.com/q/51534/2451 $\endgroup$
    – Qmechanic ♦
    Commented May 28, 2023 at 16:16
  • $\begingroup$ Jeanbaptiste Roux - Is there a derivation that derives the principle of stationary action as a classical limit of the path integral formulation? $\endgroup$
    – pll04
    Commented May 28, 2023 at 16:30

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Unfortunately, the premise of your question is wrong.


I remember witnessing a demonstration of a particularly striking counterexample.

The demonstration involved two beakers, stacked, the openings facing each other, initially a sheet of thin cardboard separated the two.

In the bottom beaker a quantity of Nitrogen dioxide gas had been had been added. The brown color of the gas was clearly visible. The top beaker was filled with plain air. Nitrogen dioxide is denser than air.

When the separator was removed we saw the brown color of the Nitrogen dioxide rise to the top. In less than half a minute the combined space was an even brown color.

And then the teacher explained the significance: in the process of filling the entire space the heavier Nitrogen dioxide molecules had displaced lighter molecules. That is: a significant part of the population of Nitrogen dioxide had moved against the pull of gravity. This move against gravity is probability driven.


So what was demonstrated there was that in moving to a state with higher probability the system did not move to a state with lower potential energy. On the contrary: the system moved to a state of higher potential energy.



Tendency to increase entropy and tendency to go to minimal potential energy can be aligned, but they don't have to be.

Once you are aware of that possibility you will start recognizing instances of the two acting in opposite direction.



As to Hamilton's stationary action: there is no overlap between Hamilton's stationary action and statistical mechanics. The two concepts can be used side-by-side, to cover a subject, but there is no overlap between the concepts of entropy and Hamilton's stationary action.


About relation of Hamilton's stationary action to Classical Mechanics as a whole.

In physics it is often the case that derivations can be run in both directions. See the discussion by stackexchagne contributor Kevin Zhou, in an answer that touches on the general subject of derivation in physics

A common presentation of Hamilton's stationary action is to start with positing it, proceeding to show that Newton's second law can be recovered from it. It is also possible to proceed in the other direction: Hamilton's stationary action (Stackexchange answer submitted in 2021 by me.)

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  • $\begingroup$ I agree with your example - I edited the question to clarify that potential minimisation holds under specific conditions. So if there is no statistical proof of the stationary action principle, is there any other proof? $\endgroup$
    – pll04
    Commented May 28, 2023 at 16:28
  • $\begingroup$ @pll04 I edited the last paragraphs of my answer, to point out that my 2021 post about Hamilton's stationary action goes towards your question. $\endgroup$
    – Cleonis
    Commented May 28, 2023 at 16:45
  • $\begingroup$ Thank you. A related argument is the derivation of the Euler-Lagrange equation (and therefore the Lagrangian formulation) from the d'Alembert principle i.e. from Newton's second law. But not only is this specific to classical mechanics, this also assumes Newton's second law, which is an empirical axiom. I am interested in a general derivation of the stationary action principle. $\endgroup$
    – pll04
    Commented May 28, 2023 at 17:39
  • $\begingroup$ @pll04 Stackexchange has a policy that the comment section is not for threaded discussion. I endorse that policy; I think it is healthy. Contact information for me is available on my stackexchange profile page. As to 'general derivation of the stationary action principle'. I submit that that is attempting too wide a scope. It's like saying: 'I am looking for a general derivation of the principle of differential equations.' (Yeah, that sounds like a quip, but my thoughts do go in that direction.) $\endgroup$
    – Cleonis
    Commented May 28, 2023 at 18:11
  • $\begingroup$ In the example you give, the potential energy of the nitrogen dioxide has increased, but the potential energy of the air has decreased. The potential energy of both gasses combined should still be the same, right? The system did not move to a state of higher potential energy. $\endgroup$
    – fishinear
    Commented May 28, 2023 at 20:05

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