Dear Omar

It seems to me that You are very much right.

Applying the least action principle to the electromechanical phenomena, You have always reverible energy copnersion, as long as dissipation is neglected. So the time could be also reversed.

And applying the enthropy increase principle to all thermodynamical enery coversion processes, You can not reverse these changes, and in consequence the time can only increase.

So in the classical approach, there are to main priciples governing the electromechanical conversion and thermodynamical conversion, that are formally separate and not coherent in respect of time.

That is my point of view in reaction to Your excellent question.

With kind regards - Piotr Wach, Opole, Poland

I think as a simple layman like me! One should distinguish the physical law of conservation from the geometric symmetry properties, which can be uniquely described by finite Group theoretical arguments, and easily may be visualized using Euclidean geometry. For the conservation laws one used the LIE Group to show that if one assumes that the space is isotropic and homogeneous one prove that one has conservation laws for the

linear and angular momentum etc.

Actually in Truesdall formulation of general thermodynamics one uses the balance of; linear momentum, moment of momentum, energy and growth of entropy.

When ıt comes to the time symmetry: That means inversion symmetry by just changing the sign of the time variable in the PDE what happens? If you have any differential operator with respect to time variable has an odd order then you can be sure that you are dealing with the irreversible process since that implies dissipation of energy. On the other hand if all the differential operators with respect to time is even order such as Newton and Hamiltonian equations without odd order terms implies perfect symmetry with respect to time inversion since nothing change in the solution by such re labeling the time variable.

Irreversible thermodynamics for evolution phenomenon uses the rate of entropy production which is odd order in time and time inversion changes the sign of the evolution process and it becomes regenerative system. It means transfer of heat from the cold body to hot body spontaneously. Actually the mathematical proof of Second Law relies on the fact that this is an unnatural process and it never takes place in the nature... Caratheodory Theory. BUT ADIABATIC COOLING IS AN EXCEPTION WHICH HAS A VERY SHORT LIFE TIME WHERE ONE REACHES THE NEGATIVE ABSOLUTE SPIN TEMPERATURE

Perhaps I am another layman in this field.

In classical science, the fundamental dimensions like , mass, length, time temperature are applied to systems with dissipative forces and these systems always move ahead with time. In true sense they are not reversible at all because at each step, ahead in time, entropy is created which is irreversible. In fact equations of conservation of mass, momentum, and energy used in engineering are only approximate, valid only for small steps of time where we are prepared to accept a limited error when balances are done. So the question of symmetry (or going back in time dopes not arise).

Therefore, cornerstone principles like energy conservation are approximate! energy transformation is never 100 %. Some energy will be loss because of entropy increase. Hence, the Noether theorem should read something like " If you have a pseudo-symmetry (this is given that time is not quite symmetric) and you apply the principle of least action, you will get an approximate conservation law"?

Perhaps yes. I am thinking about classical systems only. When we talk of quantum level, people are still discovering new particles to find the balance? What do you think?

Time is a physical illusion. It does not exist. Time is a result of the human cognitive activity!!!

Invariance under reversal of motion, also known as time reversal invariance (TRI), is a symmetry associated with time. A Hamiltonian is a generator of a system's time evolution (see Noether). Not all Hamiltonians are invariant under time reversal. Often, time reversal symmetry is imposed as a condition on an Hamiltonian, when there is experimental evidence that the system does not violate TRI. For absorptive, creative, or dissipative systems non-hermitian Hamiltonians are invoked, which yields non-unitary time evolution and entropy changes.

A formulation of QM that goes beyond unitary evolution and includes entropy increases , associated dissipation and friction, and loss of coherence, is provided by the Lindblad equations and its extensions. That formulation is based on the idea of a quantum density matrix. There is an extensive literature on combining QM with thermodynamics.

Hi.

This is very interesting, but I think it resolves in two ways that are different but not exclusive.

As a variable, time is not symmetric, which we know from thermodynamics. However, the basic phenomenon with regard to Noether's theorem is time reversal symmetry or invariance, not the symmetry of time itself.

From a mathematical point of view, the context of time reversal symmetry applies when you have the variable time to even powers, which can lead to time reversal symmetry. Systems that follow such symmetry can, in some circumstances, also be invariant in the context of a Lagrangian, leading to conservation laws via Noether's theorem. This would occur whether or not time is symmetric because of the form of the mechanical equations.

Another point of view is more physically motivated. When modeling mechanical systems, they are closed and described by equations that treat the elements as "discrete" particles or "discrete" systems. For instance, angular momentum conservation arises from the time symmetry of a "particle" or discrete system like a ring that can be thought of as a fixed collection of particles. In both cases, internal structure (or changes in time) is somewhat ignored, as is the interaction with the rest of the "universe". This allows classical mechanical equations to be applied, and tested by Neother's theorem.

Thermodynamics is very different because it inherently deals with the entire "universe" of interacting systems. Even if you have a discrete system, it changes its internal or free energy (or anything else) because it interacts with other systems. Further, thermodynamics allows the entropy of one system to go down as long as the entropy somewhere else (or the total of everywhere else) goes up by that amount or more, so the total entropy change is positive. This progression of systems is not covered by Noether's theorem, nor does it even lead to a "definition" of time any more specific than being some nebulous progress variable that always marches forward. (Even if you apply mechanics, for instance via kinetic theory or statistical mechanics, the thermodynamic properties come from averaging. Again, not covered by Noether's theorem for an average of identical systems.)

I guess, as I view it, thermodynamics deals with the evolution of the universe, as a collection of all kinds of systems and "stuff" in communication. Mechanics (and Noether's applications) deal with more discrete systems that are isolated from other effects. Thus, the role of time in these two applications is different... In thermodynamics, it is a progress variable (in the structure of equilibrium thermo theory, time does not even exist), while in mechanics it is an independent mathematical variable that corresponds to something that can be measured or observed (or at least time intervals can be measured or observed).

I hope this brings some interesting discussion ideas.

Thanks!

Being a layman, I am able to understand well the explanation given by Prof. Bellantone.

Thank you so much for putting it so clearly.

Interesting points from Prof. Bellantone. Thank you all for constructive opinions.

Entropy: A concept that is not a physical quantity

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

Please see the following discussion

https://www.quora.com/What-can-be-thought-of-as-a-time-symmetry

Best Regards Omar J. Yepez

Please see the following link

https://www.quora.com/What-can-be-thought-of-as-a-time-symmetry

Regards Omar J. Yepez