Alternative definitions of energy?

juanrga

I already said you that tracing is a time-reversible operation. If you start from a reversible equation and apply the trace over environmental degrees of freedom, the resulting equation for the open system is time reversible and violates the second law.

Of course, if you apply some of the «mathematical funambulism» so popular in a part of the so-called open systems quantum literature, then you can prove anything that you want...

For this reason, the Brussels school (and others serious guys) now{*} start from an irreversible equation for the isolated system (a generalization of QM) and then obtain the correct irreversible equation for the open subsystem.

I am so tired of the plain nonsense written in part of the literature on irreversibility that I plan to write a paper probably titled «mathematical funambulism on the theories of irreversibility» or something as that. But not now. Now I am with a paper that generalizes the first and second law of thermodynamics to open systems (yes also in this topic many literature in open systems is wrong).

If you read the authors' work you will discover that they are not saying that the irreversible equation is inexact. At contrary they claim that it is the reversible equation which is inexact.

There are several subtle technical issues in the meaning of the TDL in their work that you fail to understand, this limit is not being taken to approximate the equation from other. It is being taken to eliminate some spurious non-Markovian effects related to the evolution of correlations in the multiparticle system (which does not follow the Liouville equation).

The reason which they take this limit is also related to the fact that the exact mathematical nature of the extended space is not still well-understood, and neither them nor any mathematician knows how to obtain the specific spectral decomposition in a pure ab initio fashion. Although in the same volume in Adv. Chem. Phys. a mathematician claims to obtain the spectral decomposition using a new algebra, without appealing to the TDL anymore.

In my own view (sometimes discussed with relevant member of the Brussels school including the Nobel laureate himself) the resulting irreversible equation is the result of bifurcation points in the extended Liouville space, but for LPSs the non-Markovian terms are lost and the irreversibility generated by those points mimics what would obtain from a fictitious TDL.

That is, the TDL is a simple way to introduce the elements lost by the Markovinization. It is a kind of trick to obtain some results, althought you pretend to take it seriously even after being warned to not do it.

This is not very different from starting from Newtonian p=mv and then obtaining the relativistic momentum by doing a trick m→m(v). Evidently, the analogy is not complete, specially because the math behind SR is well-understood and easy and such tricks are not more needed to obtain a relativistic momentum.

It is not very different from the TDL in equilibrium SM. This trick is used to simplify some mathematical derivations otherwise would be very difficult to do or without rigor (or both).

{*} As said they did your same mistake in the past

Studiot

Whilst I am grateful for your reply, I am disapointed with the level of the response, considering the high level of your other posts.

You have in another post commented upon mathematical exactitude, but offer the highly restricted formulae for the First Law since the integration of both dQ and dW is, in general, path dependent.

Secondly none of the time dependent processes you mention apply to my comment. They all apply to energy transport within a system and fail at the interface between systems, which is what I am talking about.

What I am referring to is another facet of the 'action at a distance' problem, which I am sure you are familiar with. This goes much deeper than schoolboy thermodynamics.

juanrga

Then it seems evident that I did not understand your questions. Unfortunately I do not understand them now either.

I do not understand what is the link between what you say about dQ and dW and what I wrote dQ/dt and dW/dt

I do not understand why you say that the time dependent processes I mentioned apply to energy transport within a system and fail at the interface between systems when

dU/dt = d_eU/dt = dQ/dt + dW/dt

the subscript «e» meaning «external». I.e., the above expression gives the changes in internal energy due to flows through the boundary surface that encloses the system volume. The above expression does not apply to energy transports inside the system. The corresponding expression for changes in the energy due to internal processes is

d_iU/dt = 0

which is another way to state conservation of energy.

And, finally, I miserable fail to understand what do you mean by «another facet of the 'action at a distance' problem»

RonL

Studiot, thanks for making reference to 'action at a distance', searching wiki has opened a vast number of related links that have helped me understand much more, in many areas.

Like Juanrga, I do not know your meaning of "fail at the interface between systems". To me this is a boundry for mass, but not thermal energy. If by design, this interface can represent a storage of, and a speed control for energy moving between systems A and B. As mentioned before, any number of sub-systems within a design.

Again, thanks. I am learning a lot from this thread.

Studiot

Here is a quote from Maxwell that nicely sums up my question.