# Alternative definitions of energy?

by Millenniumf
Tags: alternative, definitions, energy
 P: 11 In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems. Since work is defined as a force acting through a distance , energy is always equivalent to the ability to exert pulls or pushes against the basic forces of nature, along a path of a certain length. The total energy contained in an object is identified with its mass, and energy, cannot be created or destroyed. When matter is changed into energy , the mass of the system does not change through the transformation process. However, there may be mechanistic limits as to how much of the matter in an object may be changed into other types of energy and thus into work, on other systems. Energy, like mass, is a scalar physical quantity. In the International System of Units (SI), energy is measured in joules.
P: 476
 Quote by kinetico juanrga: Alternative definitions of energy? by 'Millenniumf' In Physics Forums > Physics > 'Classical Physics' Reformulation: In classical mechanics, an alternative definition of energy for a simple particle $$\Delta E = \Delta {\textstyle \frac{1}{2}} \, m \, v^2 - \int_A^B m \; \vec{a} \cdot d\vec{r}$$ $$\rightarrow \; \; \Delta E = 0$$ $$\rightarrow \; \; E = constant$$ Since $\vec{a}=\vec{F} / m$ then $$\Delta E = \Delta {\textstyle \frac{1}{2}} \, m \, v^2 - \int_A^B \vec{F} \cdot d\vec{r}$$ $$\rightarrow \; \; \Delta E = 0$$ $$\rightarrow \; \; E = constant$$ where $$\vec{F} = \sum \vec{F} = \sum \vec{F}_{real} + \sum \vec{F}_{fictitious}$$
1) I do not need to repeat what was said in this thread.

2) Moreover, notice that there is two meanings for «Classical Physics». Older meaning as «pre-relativistic» and the more modern meaning as «non-quantum».
 P: 5,462 @juanrga Your posts in this thread have taken me to an unfamiliar area and certainly set me thinking so thanks. However I would appreciate your take on my comment at the end of post#107 about energy transfer.
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Sorry for the delay, juanrga. I'd like to continue the discussion, but in order to make our discussion more comprehensible, I'll first sketch my general ideas and get to your references and statements later.

In QM, dissipation and irreversible dynamics can be derived from the reversible Hamiltonian dynamics of a larger isolated system. This whole system consists of the system of interest and it's environment. So the dissipator of open systems can be derived from unitarian dynamics. [Taking the dynamics to be Hamiltonian is just a restriction of the initial state to be pure]

In classical mechanics, this seems to be impossible. Although one could imagine something like the increase in entropy in one part of the whole system could be compensated by a decrease in another, this is certainly not true for arbitrary initial conditions. So classically, dissipation and irreversible dynamics can only be explained by neglecting correlations (H-Theorem). They do not arise in a fundamental way from reversible dynamics. Quantum mechanically, they do for open systems.

This fundamental difference between classical mechanics and QM leads to the question, how irreversible dynamics in isolated systems could possibly be encountered. One way is to say our current theory is not exactly right, we have to either change the formalism or the interactions. Another way is to question the isolatedness of systems with irreversible dynamics. I definitely prefer the latter. Real "isolated" systems, like a gas in a box, have borders with which they interact. The isolation-idealization (sounds like a big bang theory episode title :D) is probably good enough as long as we talk about energy and particle exchange. But when it comes to correlations (especially entanglement) we have to be much more careful.

If I get your references right (unfortunately I don't have access to all of them), their argument for the approach of isolated systems to equilibrium relies on taking the thermodynamical limit. Doing this is certainly useful in static (equilibrium) situations, where we don't care how we actually got into our state. In the context of dynamics, I think this is unphysical. Adding particles and increasing the box size involves interactions with the system, which have to be included in the dynamics. I think this is the crucial point in many arguments concerning "isolated" systems.

 Quote by juanrga Some of those isolated systems approach equilibrium and others do not.
Well, what is the fundamental difference between these two classes of "isolated" systems? Both consist of interacting particles.

 Quote by juanrga Once again, the expression for the dissipator is exact.
This is not crucial for the discussion and I don't say you are wrong. I'm just interested and want to read about this. So can you please give a reference? Or is this contained in a already given reference?
P: 476
 Quote by Studiot @juanrga Your posts in this thread have taken me to an unfamiliar area and certainly set me thinking so thanks. However I would appreciate your take on my comment at the end of post#107 about energy transfer.
I suppose that you mean this part:

 Quote by Studiot Work is one way for energy possessed by system A to be transferred to System B. The numerical value of the energy leaving system A equals the numerical value of that entering system B and also equals the numerical value of the work done, in consistent units. The immediate questions are: What is the timescale of this transfer? Is there a time when (some of) the transferred energy has left system A and not yet entered system B? If so where is this energy ?
The timescale of the transfer is not universal and depends of each process. In thermodynamics of processes the first law (for closed systems) is generalized to

dU/dt = dQ/dt + dW/dt

but thermodynamics alone cannot say you the rates. The rates are obtained from rate equations as Fourier law, chemical kinetics laws, diffusion laws, etc.

If A and B are contiguous then energy either belong to A or to B. If the systems are not contiguous, then energy could be stored in some intermediate system C before arriving at B.
P: 476
 Quote by kith Sorry for the delay, juanrga. I'd like to continue the discussion, but in order to make our discussion more comprehensible, I'll first sketch my general ideas and get to your references and statements later. In QM, dissipation and irreversible dynamics can be derived from the reversible Hamiltonian dynamics of a larger isolated system. This whole system consists of the system of interest and it's environment. So the dissipator of open systems can be derived from unitarian dynamics. [Taking the dynamics to be Hamiltonian is just a restriction of the initial state to be pure] In classical mechanics, this seems to be impossible. Although one could imagine something like the increase in entropy in one part of the whole system could be compensated by a decrease in another, this is certainly not true for arbitrary initial conditions. So classically, dissipation and irreversible dynamics can only be explained by neglecting correlations (H-Theorem). They do not arise in a fundamental way from reversible dynamics. Quantum mechanically, they do for open systems. This fundamental difference between classical mechanics and QM leads to the question, how irreversible dynamics in isolated systems could possibly be encountered. One way is to say our current theory is not exactly right, we have to either change the formalism or the interactions. Another way is to question the isolatedness of systems with irreversible dynamics. I definitely prefer the latter. Real "isolated" systems, like a gas in a box, have borders with which they interact. The isolation-idealization (sounds like a big bang theory episode title :D) is probably good enough as long as we talk about energy and particle exchange. But when it comes to correlations (especially entanglement) we have to be much more careful.
It is not right that dissipation and irreversible dynamics can be derived from the reversible Hamiltonian dynamics of a larger isolated system. Trace preserves reversibility. And the dynamics of an open system within an isolated reversible system is also reversible.

Most of the literature in the topic of open systems is completely wrong about the origin of irreversibility. They do not derive irreversibility but force irreversibility by mathematically invalid manipulations. Of course the final equations tested in the lab are valid, but are not compatible with unitary and time reversible equations.

 Quote by kith If I get your references right (unfortunately I don't have access to all of them), their argument for the approach of isolated systems to equilibrium relies on taking the thermodynamical limit. Doing this is certainly useful in static (equilibrium) situations, where we don't care how we actually got into our state. In the context of dynamics, I think this is unphysical. Adding particles and increasing the box size involves interactions with the system, which have to be included in the dynamics. I think this is the crucial point in many arguments concerning "isolated" systems.
Since they are searching a microscopic counterpart to the second law of thermodynamics, it is understandable that they are focusing in the thermodynamic limit. As you notice the thermodynamic limit is a standard tool in equilibrium statistical mechanics.

This limit would not be taken seriously, but only operationally. Somehow as obtaining the non-relativistic limit through taking the limit c → ∞ would not be taken literally seriously (c is a constant!).

 Quote by kith Well, what is the fundamental difference between these two classes of "isolated" systems? Both consist of interacting particles.
They use the terminology: stable versus instable or integrable vs non-integrable (in Poincaré sense).

 Quote by kith This is not crucial for the discussion and I don't say you are wrong. I'm just interested and want to read about this. So can you please give a reference? Or is this contained in a already given reference?
The references given, specially the reviews, give first the exact nonlinear equations and then power expansions in terms of the interaction for computational usage.
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 Quote by juanrga It is not right that dissipation and irreversible dynamics can be derived from the reversible Hamiltonian dynamics of a larger isolated system. Trace preserves reversibility.
Not in QM. There, the entropy of subsystems can be larger than the entropy of the whole system.

 Quote by juanrga And the dynamics of an open system within an isolated reversible system is also reversible.
That's not true in QM. Consider a system with a large environment, where the initial state is seperable and the system is in a pure state. Now let it evolve according to reversible dynamics. The whole state gets entangled and the state of the system approaches thermodynamic equilibrium, which is a stationary state. The entropy of the open system increases, so the dynamics is not reversible.

Please elaborate on why you call such dynamics reversible.

 Quote by juanrga Most of the literature in the topic of open systems is completely wrong about the origin of irreversibility.
Well, what is your opinion about it's origin then?

 Quote by juanrga This limit would not be taken seriously, but only operationally.
I agree that modifying systems operationally is useful if I want to learn something about a class of systems. But that's not what we want to do.

We want to learn something about the dynamics of one given system. We can't just replace it with a similar system and say that the statements derived for the new system are true for the system of interest.

 Quote by juanrga They use the terminology: stable versus instable or integrable vs non-integrable (in Poincaré sense).
Yes, that is the mathematical distinction. But what is the physical one? Which physical systems do show irreversible dynamics on a fundamental level and which don't?

In their framework, it seems to me like systems with a finite number of interacting particles don't approach equilibrium, while systems with an infinite number of particles do. That's certainly no useful physical distinction.
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 Quote by kith Not in QM. There, the entropy of subsystems can be larger than the entropy of the whole system.
In QM trace is a time-reversible operation as well.

 Quote by kith That's not true in QM. Consider a system with a large environment, where the initial state is seperable and the system is in a pure state. Now let it evolve according to reversible dynamics. The whole state gets entangled and the state of the system approaches thermodynamic equilibrium, which is a stationary state. The entropy of the open system increases, so the dynamics is not reversible. Please elaborate on why you call such dynamics reversible.
If the dynamics is reversible the system can either approach equilibrium or not and the entropy of the open system increases or not. The naive approaches to QM (most of the literature in irreversibility) use what van Kampen calls «mathematical funambulism» to pretend that they derive irreversibility from reversibility {*}.

Notice that the own Brussels school (leaded by the Nobel laureate) also pretended to derive irreversibility from reversible laws, but after about four decades of futile efforts they finally understood that so one derivation is impossible, aplogized by past mistakes and wrong approaches, and in latter years they propose irreversible generalizations of QM as in the references cited.

 Quote by kith Well, what is your opinion about it's origin then?
That is still open to debate. The Brussels school claims that the origin are Poincaré resonances in LPSs. I have a slightly different opinion.

 Quote by kith I agree that modifying systems operationally is useful if I want to learn something about a class of systems. But that's not what we want to do. We want to learn something about the dynamics of one given system. We can't just replace it with a similar system and say that the statements derived for the new system are true for the system of interest. Yes, that is the mathematical distinction. But what is the physical one? Which physical systems do show irreversible dynamics on a fundamental level and which don't? In their framework, it seems to me like systems with a finite number of interacting particles don't approach equilibrium, while systems with an infinite number of particles do. That's certainly no useful physical distinction.
That is why I said you to not to take the thermodynamic limit seriously. Recall the example of non-relativistic dynamics that I offered before. You would not take seriously that Newtonian mechanics only applies in a fictitious universe with c → ∞. Non-relativistic mechanics apply to our universe (where c is finite) very well in a specific range.

Moreover, recall that most of statistical mechanics of equilbrium is done in the thermodynamic limit. Of course, nobody is saying you that the resulting thermodynamic formulae only apply to «systems with an infinite number of particles».

{*} This is so nonsensical as claiming that the second law of thermodynamics can be derived from the first law.
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 Quote by Millenniumf I had an interesting challenge earlier this year in physics class, and I got a good grade on my answer, but I'd like to see what other people think about this. Energy is defined in the dictionary as being the ability to do work, while work is defined as the application of energy (roughly speaking, of course). This is circular, so we were challenged to redefine the term energy. I was pretty lost on this, being a first-year physics student at a community college with little more than a periphery understanding of QM, so my answer was hardly more than a best guess. I said that energy is perhaps the vibrations of cosmic strings, with vibrations in one string being transferred to another as they come into contact, which we interpret as energy transfer. Yeah I know; not that brilliant and flawed from the beginning because it relies on unproven ideas. But it was the best I had. Once we get into quantum mechanics I'll probably have a better answer. So how would you have answered that question?

i would have answered: Energy is the ability to move (or better said to deform a frame). The ordered movement of microparticles is regarded macro as useful work, the chaotic one is regarded macro as heat. Probably, i would have been expelled from the class :)) (with a sudden lowering of my entropy, of course)
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 Quote by juanrga If the dynamics is reversible the system can either approach equilibrium or not and the entropy of the open system increases or not.
No. In my example, the entropy of the open system always increases. If you think otherwise, please tell me where exactly you disagree with it.

 Quote by juanrga Recall the example of non-relativistic dynamics that I offered before. You would not take seriously that Newtonian mechanics only applies in a fictitious universe with c → ∞. Non-relativistic mechanics apply to our universe (where c is finite) very well in a specific range.
The situations are different. In both cases, we have a system with an exact equation and an approximate equation. In the relativity case, the exact equation predicts all observed facts. Now let's take your viewpoint for the irreversibility case. Then, we have an observed fact (irreversibility), which is not predicted by the exact equation, but only by the approximate one. Now you argue, that the approximate equation can be used to explain this observed fact. But simultaneously, you seem to think that the exact description is given by the "exact" equation. This doesn't make sense to me.

I don't say that the TDL doesn't give the right answers for finite systems. I just say, it can't be used in explaining them.
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 Quote by kith No. In my example, the entropy of the open system always increases. If you think otherwise, please tell me where exactly you disagree with it.
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).

 Quote by kith The situations are different. In both cases, we have a system with an exact equation and an approximate equation. In the relativity case, the exact equation predicts all observed facts. Now let's take your viewpoint for the irreversibility case. Then, we have an observed fact (irreversibility), which is not predicted by the exact equation, but only by the approximate one. Now you argue, that the approximate equation can be used to explain this observed fact. But simultaneously, you seem to think that the exact description is given by the "exact" equation. This doesn't make sense to me. I don't say that the TDL doesn't give the right answers for finite systems. I just say, it can't be used in explaining them.
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
P: 5,462
 The timescale of the transfer is not universal and depends of each process. In thermodynamics of processes the first law (for closed systems) is generalized to dU/dt = dQ/dt + dW/dt but thermodynamics alone cannot say you the rates. The rates are obtained from rate equations as Fourier law, chemical kinetics laws, diffusion laws, etc. If A and B are contiguous then energy either belong to A or to B. If the systems are not contiguous, then energy could be stored in some intermediate system C before arriving at B.
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.
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 Quote by 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.
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 = deU/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

diU/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»
PF Gold
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 Quote by 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.
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.
P: 5,462
 Unfortunately I do not understand them now either.
Here is a quote from Maxwell that nicely sums up my question.

 we are unable to conceive of propagation in time, except either as the flight of a material substance through space, or as the propagation of a condition of motion or stress in a medium already existing in space... If something is transmitted from one particle to another at a distance, what is its condition after it has left the one particle and before it has reached the other?

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