The Least action Principle and the Maximum entropy principle are surely both corner stones, ... but at quantum level the situation is different.

In fact in the quantum world we can have also negative temperatures. (See, http://arxiv.org/abs/1206.4856.) Furthermore, solutions of variational PDEs are only extremals ...

Agostino,

I've heard about this, but I think this is a step foreward, regarding collective quantum physics and the relation with the vacuum and possibility to violate the Heisenberg uncertainty principle under some special conditions.

According to your opinion the Lorentz Group has chances to be violated at low energy?

What you wrote in your paper is well far beyond my possibilites. I think would not be enough a month for me to understand it.

Heisemberg uncertainty principle cannot be violated in quantum world ...

Poincare' group and Lorentz group are good symmetry groups in quantum (super) Yang-Mills PDEs.

Agostino,

it might be violated with a coerent ensamble whose frequency is faster than the planck frequency. "Collective polinuclear superradiance", but anyway this is not the case of this thread.

Excuse my naive question

Having completed the course of physical chemistry and thermodynamics from pure mathematical point of view 

How can a temperature be negative  being a function of mean square velocity of the particles in the system and all other positive items in the formula:

You probably refer to energy-time uncertainty. But I proved that this is wrongly interpreted in usual way. It must be reformulated for quantum-manifolds to he correct.

Dear Andrew, I understand your astonishment ... By the way in quantum world the situation is different. This has been also proved in some experiments.

I would say that the principle of "least action" (actually stationary action) only survives approximatively into the quantum world --- as a stationary phase approximation. At least when one interpreters quantum mechanics from the path integral point of view.

While the maximum entropy principle has the same origin as why I never win the EuroLotto jackpot.

Hence, I fail to see any deep connection between the two principles. But maybe there are other (better) ways to look at it...

Dear Kåre,

the least action principle (HAMILTON) is an evolution principle,   the maximum entropy (KLAUSIUS) is an evolution principle as well.

They certainly meet in quantum physics in the planck radiator and  at 0k in superconductors or Bose-Einstein condesates, and probably in "superradiance" phenomena.

The least action principle makes  systems evolve the way  local energy density is minimised, and so far it has been suitable for mechanics and quantum physics, Hamiltonians are omnipresent. Systems tend  to reach a minimum level of energy, getting rid of as much energy  as possible . Doesn't Space-time like localized energy?

The maximum entropy principle makes  high frequencies evolve to thermalization,Photons go from higher to lower frequency. It makes particles minimize their average kinetic energy by occupying the degrees of freedom and expand in the 3D space. The quality of the energy gets degradated.

 Is theSpace-time fabric  interested in keeping frequency of phenomena far below the planck frequency??

Both are oriented towards spontaneous phenomena, matter spontaneous phenomena (action) and radiation spontaneous phenomena (entropy), and they both define the Arrow of time.

Noether is the one who defined the properties of the space-time....

I will try to answer your question from a very different perspective than taken by others in the thread.

The principle of least action is a principle of classical mechanics (no-relativistic and relativistic). One only considers the path of minimum action. But quantum mechanics requires all paths and not just the path of least action. The other paths describe quantum fluctuations, which are absent in classical mechanics. Thus, quantum mechanics does not follow from the least action principle. The contrast is the same as between thermodynamics and statistical mechanics (equilibrium or not); the former does not contain any fluctuations while the latter includes them in its description.

Now, entropy is a thermodynamic concept even for Clausius. It is not normally associated with a mechanical system a la Hamiltonian dynamics. (I am not considering dissipative dynamics, which I consider phenomenological and not fundamental.) One can certainly define entropy for a classical system, but if you do so, you will find it to remain constant. It can take any value you want depending on how you define it. It has no unique value. There is no change in it while it can change for a thermodynamic system. It is the change in the entropy that drives thermodynamics. So, the maximum entropy principle plays a very important role in thermodynamics and statistical physics. Again, while thermodynamics does not include in its description any fluctuations in the entropy, statistical physics contains it and relates it to the heat capacity.

Now let me add the following comments:

1a. The action takes its minimum value in classical mechanics.

1b. The negative of entropy takes its minimum value in thermodynamic equilibrium.

2a. Other paths describe fluctuations in quantum mechanics.

2b. Other values of entropy describe fluctuations in equilibrium in statistical physics.

This is all I can think about what is common.

I am not sure if this is what you were asking for?

I agree with Eddington that the principle of least action and Clausius's maximum entropy principles are two fundamental principles. But I am not sure if he was thinking of any similarity between the two. I may be wrong.

Dear  P. D. Gujrati, .

Eddington affirmed that these laws were a sort of "governing dynamics". He didn't infact mention hypotheses of their possible integration as you correctly thought.

I was wondering that after 80 years a step forward could be attempted in order to find a common root.

"The principle of least action is a principle of classical mechanics (no-relativistic and relativistic)."

yes the uncertainty in quantum physics is modeled with the path integrals which converge to one, the real path, the actual path,  because of the Least action principle. The real path is the one only which respects such principle.

"2a. Other paths describe fluctuations in quantum mechanics"

The action creates the infinite alleys of the Quantum Physics

"The action takes its minimum value in classical mechanics."

 The least action principle determines the real alley,

not only classical mechanics but something more.. reality

"The negative of entropy takes its minimum value in thermodynamic Equilibrium."

Negative entropy??? Entropy takes its maximum value at the Equilibrium as far as I know..

 "Other values of entropy describe fluctuations in equilibrium in statistical physics".

yes lower values describe distance from the equilibrium.

Dear Mr. Quattrini:

You are right. The entropy S takes its maximum value at equilibrium. Thus, F=-S takes its minimum value as I wrote. I am not talking about negative entropy but negative of enetropy -S.  I was trying to compare F with the action which also takes its minimum value. If you prefer to think of F as F=-TS with T as a positive temperature like constant, then F represents the free energy for a athermal system (zero energy) and it is known that F takes its minimum value in equilibrium. That is all I was saying.

However, it appears to me that you think that the least action principle also works in quantum mechanics as you think of the corresponding path as real. But there are really no trajectories in quantum mechanics as the momentum is an operator.

Did I misunderstand you?

Puru

Dear Puru,

"However, it appears to me that you think that the least action principle also works in quantum mechanics as you think of the corresponding path as real"

The action principle is used in QM as you said correctly,  the least action principle makes the reality emerge from QM as it seems, since reality responds to the least action principle, or rather the space-time tends to work against increments of the energy density and makes the reality works as a consequence. Space-time acts to reach locally the lowest possible energy state.

Hello,

The Least Action principle can be derived from Maximum Path Entropy Principle (or Maximum Caliber).

Just as in statistical mechanics, maximizing entropy (Shannon Entropy Entropy or Shannnon-Jaynes), with the constraints

< 1 > = 1 = \int dx p(x) ln(p(x))

< H > = E

the form of the probability function is obtained

p(x) = exp(- b H(x)) / Z(b)

In Maximum Path Entropy, the entropy is a functional integral, its name is Caliber

C = \int Dx() p[x()] ln(p[x()])

and the constraint

< 1 > = 1

< A > = A_0 (With A the classical action)

then, the functional of probability is

p[x()] = exp(- b A[x()]) / Z(b)

then, the most probable path is obtained by finding the maximum of p,

( d/dx(t') is the functional derivative )

d/dx(t') p[x()] = 0

d/dx(t') exp(- b A[x()]) / Z(b) = - b ( exp(- b A[x(t')]) / Z(b) ) d/dx(t') A[x()] = 0

- b p[x(t')] d/dx(t') A[x()] = 0

therefore the most probable path follows the Least Action Principle

d/dx(t') A[x()] = 0

(the least action principle is obtained from maximum entropy principle.)

I hope I helped you,

regards.

for: the attachment in Section II contains the clearest equations.

Dear Diego,

"The principle of Maximum Caliber was suggested by E. T. Jaynes [1]. It postulates that the most unbiased distribution of trajectories is the one that maximizes the Shannon entropy associated with them (i.e. maximizes the \Caliber" of the system)."

Very interesting, MAXCAL. Shannon Hartley.... sophisticated information theory... I have to check the paper...

I invite everybody to have a look at this paper.

Dear Stefano:

I am not I sure if I understand you well. For me, the fluctuations due to other geometrical or classical trajectories are just as real. Think of my analogy with statistical mechanics where the fluctuations are related to the heat capacity and are most certainly real. It appears that you are inclined to use reality as the one defined by the least action principle. This to me implies that you do not consider fluctuations to have any physical relevance.

Did i understand you correctly?

Puru

Dear Puru,

according to my opinion something real is something which I can interact with dynamically, energetically.

"the fluctuations are related to the heat capacity and are most certainly real"

The fluctuations are related to the  Vector Potential, according to the Bohm Aharanov effect. According to such effect   there should not be any change in absence of fields. But the phase changes regardless of absence of local fields...implementing the non locality principle.

Non locality deal with phases, which play a role in coherent quantum physics and as such in reality, the quantum entanglement if you prefer.

We can define then two types of interplay one is dynamical the other is non dynamical...if you agree...giving a physical relevance to both.

I agree that inter-Action would not be bad as well including the non locality features, but better extra-Action, what do you think??.

Caro Stefano, I think that this paper is adequate to this thread

Irreversibility, least action principle and causality

http://arxiv.org/abs/0812.3529

Regards

Stefano:

I seem to miscommunicate as we have different perspectives. Let us think of the action in mechanics, which is in general determined by the Lagrangian, which for a free particle is just its kinetic energy. No vector potential. In the path integral formalism, I need to consider all paths; still no vector potential. Now the interference between all these paths is responsible for quantum fluctuations. We can compute these fluctuations in the position and in the momentum. They will satisfy the uncertainty principle. Instead of quantum discussion, let us consider the statistical mechanics of noninteracting particles. We can easily convert the problem to a single particle and I can study its fluctuations in energy in the canonical ensemble. This is related to the heat capacity. We can also calculate the fluctuations in the temperature. So, to this end, fluctuations seem real to me. 

We never measure the entropy or the energy; we only measure their differences. Does it mean that the two notions have no reality?

I only considered a single particle above so there is no entanglement to worry about, which is more complicated.

Puru 

Dear Puru,

I don't think any heat capacity can account for fluctuations.

If you think of energy storage in space-time due to the warpage to the space-time fabric, I can agree, that is certainly true and might be connected to the local vacuum energy fluctuations too.

But heat capacity in itself means that there is a radiator and a photon emission in some spectra. That CMBR exists it is evident, but that such black body radiation is omnipresent and roughly constant is also evident.

Stefano

Stefano> I don't think any heat capacity can account for fluctuations.

But as you (should) know, heat capacity is a measure of fluctuations.

Kåre,

for what I know, heat regards actual fluctuation of matter, moving center of mass of particles. If there is no matter, no fermions, for me it is difficult to define a temperature and define heat.

Stefano> If there is no matter

what carries your entropy?

if there is no matter there is nothing which can get the radiation...

The entropy is something regarding moving particles and radiation.

But I connect heat with real fluctuations, motion of center of mass of fermions.

The quantum fluctuations are something else. Let me understand.

Isn't that you are trying to make  the oscillations of particles, due to the vacuum interactions, be similar to a "heat"? 

The Mossbauer effect would suggest that a bit... The ZPE of nuclei at virtually zero Kelvin, according to Rudolph Mossbauer (nobel lecture), would work as a source of oscillation,  supposed to make the gamma nuclear resonance absorption occur...

Stefano: You do not have to bring all these complex issues to discuss traditional statistical mechanics of fields and matter. Heat is the part of energy that is stochasically distributed among all the particles in a body. Think of the kinetic energy of the center of mass of the body that is lost due to friction by distributing among all the particles in the body. This is classical thermodynamic picture of heat. Now, even in equilibrium, particles have their speeds distributed according to Maxwell's distribution. You can now calculate the root mean square fluctuation in the particles' speed, which is related to the temperature of the gas. The heat capacity of an ideal gas is directly proportional to the temperature. As the root mean square speed is proportional to the energy per particle of an ideal gas, the fluctuations in the energy is related to the specific heat of the gas. This is what I and  Kåre have been trying to say. This is well known in statistical mechanics. Pick up any good undergraduate textbook in this field and you will find it discussed there. 

I hope this helps.

Best wishes,

Puru

Dear Puru,

"You can now calculate the root mean square fluctuation in the particles' speed, which is related to the temperature of the gas. The heat capacity of an ideal gas is directly proportional to the temperature. As the root mean square speed is proportional to the energy per particle of an ideal gas, the fluctuations in the energy is related to the specific heat of the gas. This is what I and  Kåre have been trying to say."

This is basic thermodynamics I've learned 28 years ago for the first time and done it for about 5 other times.

I think we misunderstood a bit eachother then. This is something classical of 1870's.

We were talking about  path integrals in Quantum Physics 1950's, if I don't remember badly. Unfortunately I cannot understand the connection between heat capacity and quantum flactuations. 

There is surely a connection between specific heat and quantum physics given by  Walter Nernts, which is one of the best Chemical Physicist ever. He gave a convincing demonstration which accounts for the non diverging entropy of a system approaching absolute zero, Always heading to a situation where temperature effects are excluded.

 "Now, even in equilibrium, particles have their speeds distributed according to Maxwell's distribution. "

Yes, statistical molecular mechanics, Maxwell-Boltzmann, is one thing. As far as I know quantum mechanics is something a bit different about Bose Einstein and Fermi-Dirac statistics where Kåre worked a lot I think.             

I would like to put under everybody's attention a very interesting paper on the maximum entropy and lagrangians which is in PHYSICAL REVIEW D.

It would be a good idea to have a closed look to the maximum caliber principle suggested by DIEGO

Article Conjugate variables in continuous maximum-entropy inference

Dear Stefano: Let me try to put things, hopefully, in a correct perspective.

 You said: We were talking about path integrals in Quantum Physics 1950's, if I don't remember badly. Unfortunately I cannot understand the connection between heat capacity and quantum flactuations.

Remember, we were talking about the laest action principle (field theory) and maximum entropy principle (thermodynamics) and I had brought about a connection with the quantum field theory and the statistical mechanics.

There is a deep connection between field theory in terms of the action and equilibrium statistical mechanics as is well known. All you need to do is to treat time t in the definition of action (the integral of the Lagrangian over time) as imaginary and identify this as the inverse temperature to get a statistical mechanical description. Now you can see the connection between a trajectory in the field theory and the Boltzmann weight in its statistical version: different trajectories correspond to different energies. Thus what you call quantum fluctuations are nothing but fluctuations in energy in the statistical analog. All this is well known, and this is all I have been saying. I am sorry if my comments were not clear.

Puru

Dear Puru,

"Thus what you call quantum fluctuations are nothing but fluctuations in energy in the statistical analog."

Yes now it is much clearer since it appears a connection with the Heisenberg's Indetermination principle between the energy content and its uncertainty.

"All you need to do is to treat time t in the definition of action (the integral of the Lagrangian over time) as imaginary and identify this as the inverse temperature to get a statistical mechanical description."

This was obscure to me. So it comes out like that: imaginary time as inverse temperature?

Stefano: Yes on both counts.  Thermodynamic fluctuations also obey what seems like an uncertainty relation. Imaginary time is like inverse temperature.

Indeed, there is a beautiful symmetry between the field theory and statistical mechanics, This is all well known.

I hope this all has been beneficial to you in some way.

Puru

Dear Puru,

that is interesting.

Then how do you connect them the 0K situation to statistical mechanics?

Like in the case of the Mossbauer effect ZPE fluctuations of h/2?

Or in the case of the specific heat that at 0k does not make the Entropy diverge??

The "uncertainty relations" in statistical mechanics contain the temperature as a factor and so vanish as T tends to zero. I have written a paper on that but it is long. However, Landau and Lifshitz have a clean discussion of that in their book on Statistical Physics. You should read their chapter on fluctuations. 

In summary, you find that the thermodynamic cross fluctuation =T2 where Delta(.) is the deviation from equilibrium value of (.).

   Dear Puru,

   I have interpreted the question put by Stefano more related with the extremal variational principles of Physics (least action and maximum entropy) than with formalisms as the Euclidean Matsubara formalism. It is true that the Thermo Field Theory is quite appealing, but all that know it is not more than one development in Mathematical Physics. I am wrong? Please could you say what are the actual physical applications?

Dear Puru,

one thing is puzzling me... the Einstein Energy stress tensor is connected with the inernal energy content of the volume containing matter.  If the energy increases in such a volume, for example HEAT increases so that temperature rises, then the stress tensor increses its value and the time slows down... this description between the clock-rate and the HEAT density convinces me. The heat density increases while the clock-rate slows down, time dilation and heat density variation are connected this way...

Dear Daniel: What you are saying is correct. Most I have said is quite well known. I was just trying to bring forth a parallel between the least action principle in classical mechanics and maximum entropy principle in equilibrium thermodynamics. I then brought forth the parallel between quantum theory and statistical mechanics due to fluctuations, which confused Stefano. This made the thread deviate considerably from its original intent.

You ask for the physical applications. I have written a paper long time ago in which I pointed out that absolute temperature going to zero is similar to Planck's constant going to zero, which brings forth, at least in my opinion, a beautiful parallel between the quantum and statistical fluctuations. This kind of discussion is not part of any Mathematical Physics course to the best of my knowledge.

I would love to continue this discussion, but this thread may not be the palce to conduct that.

Puru

Dear Stefano: I must confess I do not see any connection between the energy momentum tensor, a classical mechanics concept, and heat. Heat is a thermodynamic concept, not defined in classical mechanics. Even entropy has no significance in classical mechanics but it is central to thermodynamics. That is why, temperature has no meaning in classical mechanics (by which I mean Hamiltonian mechanics).

One can do thermodynamics by taking a thermodynamic average of the stress tensor, which will bring the idea of entropy, heat, temperature, etc. Then you can do a consistent thermodynamics. This is well established. There are no problems that I am familiar with in this approach.

I am sorry if I have made the thread you initiated too involved. I did not have any such intention.

Puru

Dear Puru,

"I must confess I do not see any connection between the energy momentum tensor, a classical mechanics concept, and heat."

Einstein energy stess tensor is the central parto of GRT . THat it might be a classical theory it can be but to classify GRT  as classical mechanics it is difficult for me to affirm that.

"I have written a paper long time ago in which I pointed out that absolute temperature going to zero is similar to Planck's constant going to zero, which brings forth, at least in my opinion, a beautiful parallel between the quantum and statistical fluctuations."

If your conclusions are similar to set Planck constant to 0, I don't see much Physical meaning on that . Usually when Planck constant is set to 0 in physcs the  classical mechanics approximation is considered. In your case 0k would be reached with a classical approximation while at 0k only quantum physics plays a role.

Dear Stefano:

The energy momentum tensor also appear in special relativity and has a counterpart in non-relativistic mechanics.

Regarding T and h, it is a mathematical conclusion. As T tends to zero, statistical fluctuations disappear in statistical mechanics. As if h tends to zero, quantum fluctuations disappear, that is you retrieve classical mechanics. That is all I have said; nothing more nothing else. We are only talking about analogies and not equivalence. You are attempting to treat my comments as if I am talking about equivalence.

Puru

Dear Puru,

"You are attempting to treat my comments as if I am talking about equivalence"

Far from me to talk about the Einstein Equivalence Principle in this session, if as equivalence you mean that. Equivalence and stress tensor as Synge said do not have anything in common.

I just wanted to say that in my case I understand the link between the temperature and the time through the energy density.. The temperature is certainly linked to the time dilation. The longer the pace (clock-rate slows down) the higher the temperature (energy content rises), which is not easy to deny.

Dear All, I am afraid that this talk is going in some wrong direction ... Really you state that Heisenberg uncertainty relation (HUR) does not work more at zero temperature. This should mean that quantum phenomena disappear at zero temperature ! But this statement is in contradiction with experimental results ... See, for example ones on quantum superfluids ... and negative temperatures just allowed by quantum phenomena. The logic of quantum world is noncommutative, therefore the HUR is a necessity at any temperature there..  In order to understand that HUR is not necessarily directly related to \hbar and to temperature, please look to Theorem 5.41 in my book:

Geometry of PDEs and Mechanics, World Scientific Publishing, River Edge, NJ, 1996, 760 pp. ISBN 9810225202 .

Let me add that you talk about quantum world like a fluctuation around classical world ... This is completely wrong !

It is just this misunderstanding that caused the unification historical failure of GR with QM.

Agostino,

"Let me add that you talk about quantum world like a fluctuation around classical world ... This is completely wrong !"

But fluctuations of something (Action), which converges (with the least action principle), to the actual world for me can be correct.

Gravitation and Quantum physics will be eventually go together. 

GRT and QM for me is impossible.

NO ! It is wrong ! The quantum world is encoded by means of quantum PDEs ... The classic world is encoded by commutative PDEs. To obtain classic PDEs from quantum PDEs it is necessary to restrict the fundamental quantum algebra, underling quantum manifolds, to the commutative subalgebra \mathbb{R} or \mathbb{C} . Unfortunately whether you do not work in PDEs geometry, you cannot understand this technical step ... The idea that the quantum world is a sort of ghost that fluctuates around the classic world is a naive point of view.

Agostino,

I was not talking of classical equations but the reality in itself..

How do you explain the infinite paths which eventually determine the real one?

Sorry but I think that you would like to talk about philosophy, not mathematics or mathematical physics ... Well ! In such a case you can say what you want ... but I am not interested to continue this discussion ...

Agostino,

you are right, so how do you explain the behaviour of the path integrals and the fact that they converge to the actual trajectory given by the least action principle?

"Everything happens... as if ...the force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."                      NEWTON

The two words "as if" contain the essence of physics method.

The (infinite) possible trajectories in Hamilton Principle are just an induced interpretation by calculus of variations (overrated by Feynman).

Classical Mechanics is a deterministic theory.

Dear all, 

there is a paper (link below), where non-equilibrium thermodynamics, which contains both reversible Hamiltonian evolution and irreversible dissipative evolution, is formulated by means of a variational principle (Section 3.1.5) in a geometric setting compatible with thermodynamics. Interestingly, the functional to be extremized is the time integral of entropy production.

In other words, both classical least action principle, which leads to Hamiltonian evolution, and entropy production extremization, which yields dissipative evolution, are contained in a single variational principle.

Article Contact Geometry of Mesoscopic Thermodynamics and Dynamics

Dear Michal,

"both classical least action principle, which leads to Hamiltonian evolution, and entropy production extremization, which yields dissipative evolution, are contained in a single variational principle."

Yes because they both give account to a very good approximation of what really happens in the real world in Agreement with measurements.

Dear Stefano,

  Let me to start saying that your question is very interesting from my humble point of view as a no expert in this subject: is there a relationship between the least action principle used extensively in Physics with the maximum of entropy?.

  For me the difficulty is with the maximum of entropy interpretation. Let me summarize only two points:

   1.  The macroscopic entropy (Clausius) only increase in no reversible processes. In reversible changes acts as a conserved quantity in thermodynamics. Carathéodory was the first to relate the irreversibility  with trajectories and integrability needed for a good least action application but no full solution was found for a general variational principle in all that I know.

   2. The microscopic informational concept of entropy given by Shannon is possible to be applied to stochastic dynamics, where the classical trajectory is changed by some noise amount. In such a case and with the concepts more constrained to given trajectories, then is is possible to try to answer your question. Taking  the path probability as the exponential of action, then the measure of path uncertainty is measured just Shannon information entropy, which obviously is going to increase in no reversible process.

  I attach a paper which tries to answer these points.

       Sorry, the paper is:

      http://arxiv.org/ftp/arxiv/papers/0704/0704.0880.pdf

Thanks a lot Daniel,

that paper is quite interesting and easy to read.

"The macroscopic entropy (Clausius) only increase in no reversible processes. In reversible changes acts as a conserved quantity in thermodynamics."

This is something which always gave me  serious headaches....

As far as I know entropy is defined as a state variable int(dQ/T) (with a non exact differential) between an initial and final state of a thrm. system along a set of infinite reversible (quasi-static) transformations which correspond to an infinite number of states of equilibrium.... it is a quite sophisticated tool...

The entropy increases in irreversible processes...that is ok for me

but it is always a state variable. Did you mean as conserved quantity in reversible processes as "Invariant"? because as such I agree...

The entropy in itself is a weird beast because its "genial" and complicated definition as a state variable in a "complete mess" has a lot of implications... 

Dear Agostino,

in response to Puru I wrote some threads ago.

 "In your case 0k would be reached with a classical approximation while at 0k only quantum physics plays a role."

This was in order to specify that it is only QM that works close to 0k, in agreement with your point of view, it seems.

The Mossbauer effect I cited too, was in the same line: if I get rid of the effects of the temperature setting the system close to 0K, the statistical kinetics is gone, is over, only QM can be considered to have any influence.

This in order to demonstrate that heat fluctuations and quantistic wave function behaviour have really nothing to do with eachother in a direct way.

Is it possible to get the Shannon entropy from the least action principle?

    Dear Bing, 

    Unfortunately there is no a straightforward answer to your question, in all that I know, because the physical system must follow severe conditions:

1. The heat Q of the system only can be associated with Shannon entropy if the probabilities used in the partition function are exponential functions of the action.

2. The entropy gets a maximum for the stationary or equilibrium configuration while the action gets a minimum. In this case you can apply the equipartition theorem and the temperature appears a simple physical magnitude to measure the maximum degrees of freedom reached with equal probability.

3. The random dynamics must follow the Planck-Fokker equation and its conditions.

Entropy is a state variable defined through quasi-static (reversible) transformations while the Action  is the responsible of a generic transition/transformation.

Though we can imagine to make the action proceed through micro-steps, equilibrium states, and that way we can find the entropy as a consequence as a limiting case.

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity