Alternative definitions of energy?

[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?

 

[cbetanco]

Work done on a system is defined as the change in Kinetic Energy (KE) of that system. While The total energy of a system is the potential energy (PE) plus the kinetic energy, E=PE+KE.

 

[lustrog]

I would have answered it the way Feynman explained it (see: http://student.fizika.org/~jsisko/Knjige/Opca%20Fizika/Feynman%20Lectures%20on%20Physics/Vol%201%20Ch%2004%20-%20Conservation%20of%20Energy.pdf). That is to say, energy is just something we have discovered in nature that is always conserved. There are many forms, so when it seems to disappear, we discover we can simply define a new form of energy to account for it. Then, the energy that appears to disappear, we see, can in fact be recovered by converting it back into another form. It doesn't "exist" in some form in the same sense that substances exist. It's just a way of accounting for the various capabilities of the various systems in nature.

A little vague. Feynman of course explains it infinitely better.

 

[Andrew Mason]

Work done on a system is defined as the change in Kinetic Energy (KE) of that system. While The total energy of a system is the potential energy (PE) plus the kinetic energy, E=PE+KE.

I am not sure if you are trying to state the definition or energy or come up with a new one. How does this definition apply when, for example, a car engine does work in making the car go up a hill? Are you saying that the work done is equal to the change in Kinetic energy of the car?

Energy means the ability to do work and work means $\int \vec{F}\cdot d\vec{s}$.You cannot change this definition of energy to something that is not mathematically equivalent without changing the meaning of energy.

AM

 

[cbetanco]

I am not sure if you are trying to state the definition or energy or come up with a new one. How does this definition apply when, for example, a car engine does work in making the car go up a hill? Are you saying that the work done is equal to the change in Kinetic energy of the car?

Energy means the ability to do work and work means $\int \vec{F}\cdot d\vec{s}$.You cannot change this definition of energy to something that is not mathematically equivalent without changing the meaning of energy.

AM

No, I am using the standard definition. The work done between the initial configuration 1, and the final configuration 2 is $W_{12} =\sum_i \int_1^2 \mathbf{F}_i \cdot d\mathbf{s}_i = \sum_i \int_1^2 m_i \dot{\mathbf{v}}_i \cdot \mathbf{v}_i dt = \sum_i \int_1^2 d \left( \frac{1}{2} m_i v_i^2\right)=T_2-T_1$ where $T=\frac{1}{2} \sum_i m_i v_i^2$ is the kinetic energy of the system.

 

[Andrew Mason]

No, I am using the standard definition. The work done between the initial configuration 1, and the final configuration 2 is $W_{12} =\sum_i \int_1^2 \mathbf{F}_i \cdot d\mathbf{s}_i = \sum_i \int_1^2 m_i \dot{\mathbf{v}}_i \cdot \mathbf{v}_i dt = \sum_i \int_1^2 d \left( \frac{1}{2} m_i v_i^2\right)=T_2-T_1$ where $T=\frac{1}{2} \sum_i m_i v_i^2$ is the kinetic energy of the system.

This is not generally true. It is only true if the applied force F is the only force acting on the body. If the force pushes the body up a hill, for example, the work done does not equal the change in kinetic energy of the body. If it pushes the body against a friction force, the work done does not equal the change in kinetic energy of the body. etc. So work and kinetic energy are not the same.

AM

 

[Curl]

Energy is NOT the ability to do work. People who say this know nothing about physics.

The ability to do work can be destroyed. Energy cannot.

 

[Matterwave]

Energy is NOT the ability to do work. People who say this know nothing about physics.

The ability to do work can be destroyed. Energy cannot.

If you are referring to the second law of thermodynamics, I believe you have to preface it with something like "usable work", or "useful work". Even if you dissipate all your energy into Heat - heat can still do "work", just not in any bulk fashion that would create some kind of work you could use (i.e., it can still make other molecules speed up via collisions).

 

[Andrew Mason]

Matterwave is correct. Curl is confusing "Work" with "Mechanical Work" or "Useful Work".

It takes work to add kinetic energy to molecules. So when a Carnot heat engine adds kinetic energy to the molecules in the cold reservoir of a Carnot heat engine by delivering heat flow to the cold reservoir, work is done at the molecular level. However, that "Work" is not useful mechanical work at the macroscopic level.

AM

 

[juanrga]

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?

Energy is not the ability to do work {*}. Energy can be defined simply as the property

<E> = Tr{H ρ}

where H is the Hamiltonian of the system and ρ its state. Tr denotes the quantum or classical trace.

{*} And cosmic strings are a fantasy...

 

[Andrew Mason]

Energy is not the ability to do work.

I can understand why you might want to come up with another equivalent definition of energy, but it has to be mathematically equivalent to the term "energy" as it is used in physics. If it is not, it is a different quantity. In physics, energy is defined as the ability to do work and work is defined as the application of force over a distance. That is how these terms are defined. Now if you want to try to define it in another equivalent way, that is fine. It may be that the conventional definition is not the best way to think of energy in certain circumstances (eg, where the concept of force and distance are not so easy to quantify). But that does not mean that definition is wrong.

AM

 

[juanrga]

I can understand why you might want to come up with another equivalent definition of energy, but it has to be mathematically equivalent to the term "energy" as it is used in physics. If it is not, it is a different quantity. In physics, energy is defined as the ability to do work and work is defined as the application of force over a distance. That is how these terms are defined. Now if you want to try to define it in another equivalent way, that is fine. It may be that the conventional definition is not the best way to think of energy in certain circumstances (eg, where the concept of force and distance are not so easy to quantify). But that does not mean that definition is wrong.

AM

The definition given is the usual expression to compute the energy of a given system. When the system is in a pure state it reduces to the well-known claim (mechanics) that the Hamiltonian is just the energy of the system.

Energy is not the ability to do work {*}. There is kind of energies that cannot be used to do work (ask an engineer or take a course in thermodynamics if you do not trust me).

{*} This, maybe, could be an acceptable definition in a general physics course, but not beyond.

 

[mrspeedybob]

Energy is that which curves spacetime.

 

[Andrew Mason]

The definition given is the usual expression to compute the energy of a given system. When the system is in a pure state it reduces to the well-known claim (mechanics) that the Hamiltonian is just the energy of the system.

Energy is not the ability to do work {*}. There is kind of energies that cannot be used to do work (ask an engineer or take a course in thermodynamics if you do not trust me).

{*} This, maybe, could be an acceptable definition in a general physics course, but not beyond.

I have studied a bit of thermodynamics. The word "useful" is used to qualify "work" in thermodynamics for a reason. You are confusing "useful work" with "work".

AM

 

[Matterwave]

Energy is not the ability to do work {*}. Energy can be defined simply as the property

<E> = Tr{H ρ}

where H is the Hamiltonian of the system and ρ its state. Tr denotes the quantum or classical trace.

{*} And cosmic strings are a fantasy...

Defining the energy to be the expectation value of the Hamiltonian in your mixed state is overly narrow I think. It would also be very round-about and not practical. How do you define the Hamiltonian? If you don't want circular reasoning (e.g. H=T+V) to come into play, you have to define the Hamiltonian as the Legendre transform of your Lagrangian. How do you define the Lagrangian? Again, you can't use circular reasoning (e.g. L=T-V), so you would have to define it as the quantity which, if you integrate with respect to time between two endpoints gives you the Action. You would then have to go on to define the Action as that which, when you take the variation of to be zero, gives you the Euler Lagrange equations. And you would have to define the Euler Lagrange equations as those equations which give you the correct equations of motion for your system. You would then, finally, have to describe the correct equations of motion of your system, and thereby completely render the whole concept of "energy" to be infinitely less useful since "energy" is basically a first constant of integration of the equations of motion which we use to help us solve for the dynamics of a system. By the way, ρ there should be the density matrix, or the statistical operator, and is not the state itself.

I also ask you to give me, from that definition, a way for me to obtain a useful definition of energy in a macroscopic system, e.g. a ball rolling on the ground.

Energy is that which curves spacetime.

Are you saying the energy is the full stress-energy tensor, or the energies are the components of that tensor? If you take this view, then pressure and stress are also defined as "energy", even though they are conventionally not. Also, because the stress-energy tensor in Einstein's relativity is manifestly the stress-energy tensor of the matter fields (and, one should stress NOT the gravitational field itself), this definition would preclude you from being able to define any energy in the curvature itself. I.e. you would not be able to define any sort of gravitational potential energy using that definition. Granted, it is, in general, not advisable to try to attribute an energy "of the curvature", but the notion of energy can still sometimes be useful. For example, the major evidence we have for gravitational waves is due to the speeding up of the orbital period of a binary neutron star, and in that scenario we look at the "gravitational energy" carried off by the gravitational waves and confirm that this is coincident with theory.

My whole point with this post, is that physicists should not act "smug" once they've learned more sophisticated methods and definitions. The basic definition we are given in basic physics "Energy is the ability to do work" may not be the best definition of energy, and, in many cases, it is inadequate; however, to come up with a fully general definition of energy is painfully difficult. Most of the time, we just take working definitions based on which field we are studying. For basic physics, the definition "energy is the ability to do work", I think, is a nice working definition.

 

[Curl]

How about this:

Energy is something which can give rise to motion of a free particle (pick an elementary particle). The more energy, the quicker is the motion of this particle.

 

[Vanadium 50]

Why do we need a new definition? What's wrong with the one we have? Until you can answer that question, it will be rather difficult to improve on things.

 

[Andrew Mason]

Why do we need a new definition? What's wrong with the one we have? Until you can answer that question, it will be rather difficult to improve on things.

There is nothing wrong with the definition of energy or work. The premise of the OP is not correct:

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.

Energy is the ability to do work. But work is not defined as the application of energy. Work is the application of force through a distance: $W = \int \vec{F}\cdot d\vec{s}$. I see nothing circular in that.

AM

 

[juanrga]

I have studied a bit of thermodynamics. The word "useful" is used to qualify "work" in thermodynamics for a reason. You are confusing "useful work" with "work".

AM

I have said something different.

Work ('useful' or not) is only one mechanism of interchange of energy.

The concept of energy is much more fundamental. You say that you studied thermodynamics, then you would know that work is not a state function as energy. energy is a property of a system, work is not.

There are systems, with nonzero energy, whose work is zero.

By all of the above defining energy as the ability of do work must be enough for a introductory course in physics... but is not a serious def.

 

[juanrga]

Defining the energy to be the expectation value of the Hamiltonian in your mixed state is overly narrow I think. It would also be very round-about and not practical. How do you define the Hamiltonian? If you don't want circular reasoning (e.g. H=T+V) to come into play, you have to define the Hamiltonian as the Legendre transform of your Lagrangian. How do you define the Lagrangian? Again, you can't use circular reasoning (e.g. L=T-V), so you would have to define it as the quantity which, if you integrate with respect to time between two endpoints gives you the Action. You would then have to go on to define the Action as that which, when you take the variation of to be zero, gives you the Euler Lagrange equations. And you would have to define the Euler Lagrange equations as those equations which give you the correct equations of motion for your system. You would then, finally, have to describe the correct equations of motion of your system, and thereby completely render the whole concept of "energy" to be infinitely less useful since "energy" is basically a first constant of integration of the equations of motion which we use to help us solve for the dynamics of a system. By the way, ρ there should be the density matrix, or the statistical operator, and is not the state itself.

I also ask you to give me, from that definition, a way for me to obtain a useful definition of energy in a macroscopic system, e.g. a ball rolling on the ground.

The definition is totally practical and emphasizes that energy is an observable with associated fluctuations (which is not said by the other 'definitions' given here).

The best way to introduce the Hamiltonian is as a postulate. There is not guarantee that the Hamiltonian can be derived from a given Lagrangian (already Weinberg discuss about this). Moreover Lagrangians are usually postulated. The Hamiltonian is more fundamental (specially in quantum mechanics) because is the generator of time translations.

Same about the action. The definition above is valid for dissipative and non-deterministic systems for the which its motion is not given by any action (in fact an action is not defined for this kind of systems) nor by Euler-Lagrange equations.

The more general equations of motion can be derived from the Hamiltonian, but not from the other methods that you say (valid only under certain approximations).

ρ is the state itself. The energy of a macroscopic system is easy to obtain, the definition is the same than before

<E> = Tr{H ρ}

If your macroscopic system is classical then Tr denotes the classical trace. If your macroscopic system is in a pure state and there is not instabilitites (e.g., Poincaré resonances) then

<E> = E = Tr{H ρ} = H(p,q)

And effectively as textbooks in classical mechanics emphasize the Hamiltonian is the energy of the system.

 

[Andrew Mason]

I have said something different.

Work ('useful' or not) is only one mechanism of interchange of energy.

Fine. But measure of energy is still the measure of its ability to do work. The dimensions of energy are force x distance (mass x (distance/time^2) x distance). A photon has energy $h\nu$ because it is capable of doing $h\nu$ amount of work on some element of matter. Applying a force through a distance may not be what a photon does, (and it may not be the most appropriate way to model what it does), but the measure of its energy is its ability to do work on some element of matter.

The concept of energy is much more fundamental. You say that you studied thermodynamics, then you would know that work is not a state function as energy. energy is a property of a system, work is not.

The energy of a system is still its ability to do work - that is to apply a force through a distance (at some level). This is not to be confused with its ability to do useful work, W.

Thermodynamics was developed before anyone understood that molecules existed. Heat was thought to be some kind of substance that flowed through matter. So the terminology used in thermodynamics is a bit archaic.

Heat flow, ΔQ, is actually a transfer of energy at the molecular level (molecules doing work on other molecules), W (useful work) is a transfer of energy at the macroscopic level, and ΔU is the change in the ability of the system to do both kinds of work (ie. to generate heat flow or to do macroscopic work).

There are systems, with nonzero energy, whose work is zero.

W is not a property of a system so I am not sure what this means.

By all of the above defining energy as the ability of do work must be enough for a introductory course in physics... but is not a serious def.

Can you provide a better one?

AM

 

[juanrga]

Fine. But measure of energy is still the measure of its ability to do work. The dimensions of energy are force x distance (mass x (distance/time^2) x distance). A photon has energy $h\nu$ because it is capable of doing $h\nu$ amount of work on some element of matter. Applying a force through a distance may not be what a photon does, (and it may not be the most appropriate way to model what it does), but the measure of its energy is its ability to do work on some element of matter.

The energy of a system is still its ability to do work - that is to apply a force through a distance (at some level). This is not to be confused with its ability to do useful work, W.

All of this was answered before. Moreover, I only want remark that by W I am denoting work (as everyone does) not useful work, which is another concept.

Thermodynamics was developed before anyone understood that molecules existed. Heat was thought to be some kind of substance that flowed through matter. So the terminology used in thermodynamics is a bit archaic.

Heat flow, ΔQ, is actually a transfer of energy at the molecular level (molecules doing work on other molecules), W (useful work) is a transfer of energy at the macroscopic level, and ΔU is the change in the ability of the system to do both kinds of work (ie. to generate heat flow or to do macroscopic work).

Apart from the fact of that the concept of molecule had been introduced and used in science even before Carnot started the science of thermodynamics, the rest is plain nonsense, including your repetitive confusion between work and useful work, that your «Heat flow, ΔQ» is not a flow, neither is heat because heat is not a state function in thermodynamic space, and of course, that heat is not a «kind of work» as you seem to believe...

W is not a property of a system so I am not sure what this means.

Reading what was written

Work ('useful' or not) is only one mechanism of interchange of energy.

[...] work is not a state function as energy. energy is a property of a system, work is not.

It would be clear that I mean situation with a system with nonzero E that cannot do work (W=0). For instance, it would be funny to try to understand what is energy (E) from Work (W) for a system verifying ΔE = Q.

Can you provide a better one?

I think that I already gave a general definition beyond the limits of your pseudo-definition.

 

[Andrew Mason]

There is kind of energies that cannot be used to do work

I am having difficulty making sense of your posts. Perhaps you could give us an example of the kind of energy that cannot be used to do work (either at the macroscopic level or at the microscopic level).

AM

 

[Matterwave]

I have said something different.

Work ('useful' or not) is only one mechanism of interchange of energy.

The concept of energy is much more fundamental. You say that you studied thermodynamics, then you would know that work is not a state function as energy. energy is a property of a system, work is not.

There are systems, with nonzero energy, whose work is zero.

By all of the above defining energy as the ability of do work must be enough for a introductory course in physics... but is not a serious def.

The work defined in Thermodynamics is useful work (the work to move a piston). That's why you have U=W+Q. Q is the heat, and you don't consider that work simply because you can't account for all the F*r terms in it.

The definition is totally practical and emphasizes that energy is an observable with associated fluctuations (which is not said by the other 'definitions' given here).

The best way to introduce the Hamiltonian is as a postulate. There is not guarantee that the Hamiltonian can be derived from a given Lagrangian (already Weinberg discuss about this). Moreover Lagrangians are usually postulated. The Hamiltonian is more fundamental (specially in quantum mechanics) because is the generator of time translations.

Same about the action. The definition above is valid for dissipative and non-deterministic systems for the which its motion is not given by any action (in fact an action is not defined for this kind of systems) nor by Euler-Lagrange equations.

The more general equations of motion can be derived from the Hamiltonian, but not from the other methods that you say (valid only under certain approximations).

ρ is the state itself. The energy of a macroscopic system is easy to obtain, the definition is the same than before

<E> = Tr{H ρ}

If your macroscopic system is classical then Tr denotes the classical trace. If your macroscopic system is in a pure state and there is not instabilitites (e.g., Poincaré resonances) then

<E> = E = Tr{H ρ} = H(p,q)

And effectively as textbooks in classical mechanics emphasize the Hamiltonian is the energy of the system.

1) Can you calculate for me, using this definition, the energy of a ball falling in a gravitational field? Let's say it starts at height h, in a uniform gravitational field, and free-falls.

2) If you just "postulate" the Hamiltonian, then all you've done is shift the question from "What is energy?" to "What is the Hamiltonian?". The Hamiltonian formalism is inherently difficult to work with whenever there are gauge invariances of a theory. It is even more difficult to work with in the context of general relativity where the split between time and space should not be made so artificial.

Additionally, there are explicit proofs of "when" the Hamiltonian "is" the energy of the system or not and "when" it is conserved or not. These are 2 separate questions. Goldstein goes into some detail about this. If I recall correctly, the Hamiltonian only corresponds with our usual definition of energy if the kinetic energy is dependent quadratically on the speeds, and the potential is not dependent on speeds.

3) If ρ was the state of the particle, it wouldn't make sense to take the trace of it. The trace is only good for operators (i.e. matrices), a state is a vector (more formally, a ray) in the Hilbert space. How do you take the trace of a vector? You can see this explicitly from the form of ρ:

$$\rho=\sum_i P_i |\Psi_i\rangle \langle \Psi_i |$$

That's manifestly an operator and not a state.

 

[juanrga]

The work defined in Thermodynamics is useful work (the work to move a piston). That's why you have U=W+Q. Q is the heat, and you don't consider that work simply because you can't account for all the F*r terms in it.

The same recommendation that for Andrew Mason: open a textbook on thermodynamics and learn the subject first. I would recommend the section «7.5 USEFUL WORK AND THE GIBBS AND HELMHOLTZ FUNCTIONS» of Klotz & Rosnberg well-known textbook (I have seventh ed.) to understand the difference between work and useful work.

E.g. the Helmholtz free energy measures the useful work obtainable from systems at a constant temperature, volume, and composition. This is not the same than work W.

1) Can you calculate for me, using this definition, the energy of a ball falling in a gravitational field? Let's say it starts at height h, in a uniform gravitational field, and free-falls.

From the expression obtained above $E = H(p,q)$, and as first approximation this is $(p^2/2m + V(q))$

2) If you just "postulate" the Hamiltonian, then all you've done is shift the question from "What is energy?" to "What is the Hamiltonian?". The Hamiltonian formalism is inherently difficult to work with whenever there are gauge invariances of a theory. It is even more difficult to work with in the context of general relativity where the split between time and space should not be made so artificial.

Giving a definition of something is always shifting the question from the definiendum to the definiens. Evidently, this process cannot be repeated forever. Therein that formal systems contain a set of primitive elements which are not defined.

As already said, the Hamiltonian is the generator of the time-translations. All of QFT is based in obtaining a Hamiltonian, from which one obtain the S-matrix, which is tested in experiments. Weinberg has a delicious discussion about those topics.

Regarding GR, the 3+1 formalism is fundamental for an deep (and practical) understanding of dynamics. Indeed the 3+1 formalism is the foundation of most modern numerical relativity.

The problems with the usual Hamiltonian formalism of GR are more related to certain geometric deficiencies of GR than to the Hamiltonian formalism.

As said as well, the Hamiltonian formalism is fundamental when studying more general dynamics beyond QFT and GR.

Additionally, there are explicit proofs of "when" the Hamiltonian "is" the energy of the system or not and "when" it is conserved or not. These are 2 separate questions. Goldstein goes into some detail about this. If I recall correctly, the Hamiltonian only corresponds with our usual definition of energy if the kinetic energy is dependent quadratically on the speeds, and the potential is not dependent on speeds.

I already wrote above that E=H(p,q) is valid as approximation. The rest is wrong.

3) If ρ was the state of the particle, it wouldn't make sense to take the trace of it. The trace is only good for operators (i.e. matrices), a state is a vector (more formally, a ray) in the Hilbert space. How do you take the trace of a vector? You can see this explicitly from the form of ρ:

$$\rho=\sum_i P_i |\Psi_i\rangle \langle \Psi_i |$$

That's manifestly an operator and not a state.

If you open a textbook on QM, you will discover that the operator rho describes the general state of a quantum system (beyond the limits of |ψ>). Recall that my goal was to give a general definition of E, not one valid only in special situations.

Moreover, you are replying to a part where I said that Tr was denoting the classical trace, which means that you do not read my posts. The classical trace is an phase space integration and rho is not an operator therein but the phase space state that correspond to the classical limit.

 

[kmarinas86]

If you are referring to the second law of thermodynamics, I believe you have to preface it with something like "usable work", or "useful work". Even if you dissipate all your energy into Heat - heat can still do "work", just not in any bulk fashion that would create some kind of work you could use (i.e., it can still make other molecules speed up via collisions).

Matterwave is correct. Curl is confusing "Work" with "Mechanical Work" or "Useful Work".

It takes work to add kinetic energy to molecules. So when a Carnot heat engine adds kinetic energy to the molecules in the cold reservoir of a Carnot heat engine by delivering heat flow to the cold reservoir, work is done at the molecular level. However, that "Work" is not useful mechanical work at the macroscopic level.

AM

To Andrew Mason: Matterwave wasn't correct, but what you said after was correct. Matterwave is incorrect in saying that that, "Heat can still do 'work', just not in any bulk fashion that would create some kind of work you could use (i.e., it can still make other molecules speed up via collisions)." The easiest counterexample is a hot air balloon. The result of heating air is the expansion of the balloon, which causes the air to be displaced in such a manner as to be forced underneath the balloon. That expansion constitutes useful work done to the surrounding atmosphere. That expansion is indeed caused by molecules speeding up via collisions, and force*displacement (and thus work) does indeed occur.

To matterwave: Wrong. Adiabatic expansion can convert thermal energy into non-thermal forms of energy. The result is adiabatic expansion work, and this occurs in a bulk fashion. The fact that a gas duster gets quickly cold by means other than radiating heat to the environment proves this.

http://en.wikipedia.org/wiki/Adiabatic

In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time, so that there is no opportunity for significant heat exchange[1]. The term "adiabatic" literally means impassable[2], coming from the Greek roots ἀ- ("not"), διὰ- ("through"), and βαῖνειν ("to pass"); this etymology corresponds here to an absence of heat transfer.

http://www.ehow.com/how-does_5157676_canned-air-cold.html

Thermodynamics in Action

The reason the can gets cold after being used is due to a process known as adiabatic cooling. Adiabatic cooling is a property of thermodynamics that states that when a gas is placed under a high amount of pressure, a significant drop in temperature will occur when that pressure is released. The high level of compression necessary to convert a gas into a liquid allows a massive amount of gas to be stored in a relatively small space, and when that gas is released to a large space, it rapidly expands to fill the space. This results in a drop in its internal energy, and it absorbs a large amount of heat from the surrounding air. This heat absorption results in the cooling effect. This is one of the reasons that every can is labeled with a warning stating that it should never be sprayed onto skin due to the fact that this rapid absorption of heat can easily cause frostbite. The frost that forms on the can is a result of the condensation of water vapor in the surrounding air.

Conductive thermal agitation can transform into convection cells which carry bulk flows of kinetic energy such as that found in wind and oceanic currents.

[E]nergy that appears to disappear, we see, can in fact be recovered by converting it back into another form. It doesn't "exist" in some form in the same sense that substances exist. It's just a way of accounting for the various capabilities of the various systems in nature.

Energy does exist in the same sense that substances exist. Energy is transported by force carriers known as gauge bosons. These are the "substances" which explain the interaction of other particles.

There is kind of energies that cannot be used to do work (ask an engineer or take a course in thermodynamics if you do not trust me).

I am having difficulty making sense of your posts. Perhaps you could give us an example of the kind of energy that cannot be used to do work (either at the macroscopic level or at the microscopic level).

AM

He cannot do that. Juanrga's statement makes no sense.

The problem is that not only is $T\Delta S_{total}$ (i.e. the so called "unavailable" energy) not constant, it can even decrease in an isolated system. Although $\Delta S_{total}$ can only increase in an isolated system, $T$ can spontaneously decrease while maintaining this isolation. This is because $T$ is based upon kinetic energy per degree of freedom. The increase in degrees of freedom can occur as gas expands, for example. (Note: Degrees of freedom do not have to be integers.) Of course, expansion of a gas in isolation implies outward kinetic energy having an internal source, and because cooling results from expansion of a gas, even if that gas isolated by an ideal insulator, then some of the energy in providing that outward kinetic energy is the random kinetic energy that was originally contained as thermal energy in that gas. In other words, energy can spontaneously become "available" or "unavailable" as the number degrees of freedom spontaneously changes.

 

[juanrga]

He cannot do that. Juanrga's statement makes no sense.

The problem is that not only is $T\Delta S_{total}$ (i.e. the so called "unavailable" energy) not constant, it can even decrease in an isolated system. Although $\Delta S_{total}$ can only increase in an isolated system, $T$ can spontaneously decrease while maintaining this isolation. This is because $T$ is based upon kinetic energy per degree of freedom. The increase in degrees of freedom can occur as gas expands, for example. (Note: Degrees of freedom do not have to be integers.) Of course, expansion of a gas in isolation implies outward kinetic energy having an internal source, and because cooling results from expansion of a gas, even if that gas isolated by an ideal insulator, then some of the energy in providing that outward kinetic energy is the random kinetic energy that was originally contained as thermal energy in that gas. In other words, energy can spontaneously become "available" or "unavailable" as the number degrees of freedom spontaneously changes.

As said, any engineer knows that cannot construct an engine that does nothing but convert heat (a kind of energy) to work. Also the so called «Heat death of the universe» implies a final state where «no work can be done, and all the energy will end up as the random motion of atoms and molecules» (http://physics.bu.edu/~duffy/py105/Secondlaw.html)

There are more ways to interpret my original statement about energy and work, (for example I have not discussed the delicate issue of open systems, which introduces an extension of the usual laws of thermodynamics presented here {*}), but none of them include the several misunderstandings that you write above about $T\Delta S_{total}$, kinetic energy, and others {**}.

{*} I wrote a paper, now under review, explaining how ΔU = Q+W, which is only valid for closed systems, must be generalized to deal with open systems.

{**} For instance, $T\Delta S_{total}$ is not $Q = T\Delta_e S$ for closed systems (see {*}). It is false that $\Delta S_{total}$ can only increase in an isolated system, moreover the probability for a fluctuation in temperature $\delta T$ can be obtained from the corresponding decreasing in entropy (usually the thermodynamic theory of fluctuations uses $\delta^2 S$ for the computation of the probability). You seem to confound thermodynamic temperature with kinetic temperature...

An introductory discussion of the limits of the concept of kinetic temperature is given next

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html

The textbook cited is Schroeder, Daniel V., An Introduction to Thermal Physics, Addison Wesley, 2000.

 

[kmarinas86]

As said, any engineer knows that cannot construct an engine that does nothing but convert heat (a kind of energy) to work. Also the so called «Heat death of the universe» implies a final state where «no work can be done, and all the energy will end up as the random motion of atoms and molecules» (http://physics.bu.edu/~duffy/py105/Secondlaw.html)

There are more ways to interpret my original statement about energy and work, (for example I have not discussed the delicate issue of open systems, which introduces an extension of the usual laws of thermodynamics presented here {*}), but none of them include the several misunderstandings that you write above about $T\Delta S_{total}$, kinetic energy, and others {**}.

{*} I wrote a paper, now under review, explaining how ΔU = Q+W, which is only valid for closed systems, must be generalized to deal with open systems.

{**} For instance, $T\Delta S_{total}$ is not $Q = T\Delta_e S$ for closed systems (see {*}). It is false that $\Delta S_{total}$ can only increase in an isolated system, moreover the probability for a fluctuation in temperature $\delta T$ can be obtained from the corresponding decreasing in entropy (usually the thermodynamic theory of fluctuations uses $\delta^2 S$ for the computation of the probability). You seem to confound thermodynamic temperature with kinetic temperature...

An introductory discussion of the limits of the concept of kinetic temperature is given next

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html

The textbook cited is Schroeder, Daniel V., An Introduction to Thermal Physics, Addison Wesley, 2000.

The link you gave (http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html) said that:

Complications such as these have led to the adoption of a different approach to the concept of temperature in the study of thermodynamics. Schroeder's proposal for a theoretical definition of temperature is: "Temperature is a measure of the tendency of an object to spontaneously give up energy to its surroundings. When two objects are in thermal contact, the one that tends to spontaneously lose energy is at the higher temperature."(Thermal Physics, Ch 1.)

That doesn't seem right. I can have a thermal conductor at 1000C and a thermal insulator at 1000C. Yet the definition above would imply that the thermal conductor and the thermal insulator would give up energy with comparable spontaneity, which is obviously false.

 

[Dale]

That doesn't seem right. I can have a thermal conductor at 1000C and a thermal insulator at 1000C. Yet the definition above would imply that the thermal conductor and the thermal insulator would give up energy with comparable spontaneity, which is obviously false.

It is correct, not obviously false. If you have a hot insulator it will spontaneously give up energy to a cold conductor, and if you have a cold insulator it will spontaneously receive energy from a hot conductor, and if they are equal temperature then there will be no net spontaneous energy transfer.

 

[Dale]

I can understand why you might want to come up with another equivalent definition of energy, but it has to be mathematically equivalent to the term "energy" as it is used in physics. If it is not, it is a different quantity. In physics, energy is defined as the ability to do work and work is defined as the application of force over a distance. That is how these terms are defined.

I agree with you on this point. In Hamiltonian mechanics juanrga's expression is certainly more convenient and natural, but the whole reason that we call that expression "energy" is because, in any system where you can compute both, you find that they are equivalent.

 

[Matterwave]

The same recommendation that for Andrew Mason: open a textbook on thermodynamics and learn the subject first. I would recommend the section «7.5 USEFUL WORK AND THE GIBBS AND HELMHOLTZ FUNCTIONS» of Klotz & Rosnberg well-known textbook (I have seventh ed.) to understand the difference between work and useful work.

E.g. the Helmholtz free energy measures the useful work obtainable from systems at a constant temperature, volume, and composition. This is not the same than work W.



From the expression obtained above $E = H(p,q)$, and as first approximation this is $(p^2/2m + V(q))$



Giving a definition of something is always shifting the question from the definiendum to the definiens. Evidently, this process cannot be repeated forever. Therein that formal systems contain a set of primitive elements which are not defined.

As already said, the Hamiltonian is the generator of the time-translations. All of QFT is based in obtaining a Hamiltonian, from which one obtain the S-matrix, which is tested in experiments. Weinberg has a delicious discussion about those topics.

Regarding GR, the 3+1 formalism is fundamental for an deep (and practical) understanding of dynamics. Indeed the 3+1 formalism is the foundation of most modern numerical relativity.

The problems with the usual Hamiltonian formalism of GR are more related to certain geometric deficiencies of GR than to the Hamiltonian formalism.

As said as well, the Hamiltonian formalism is fundamental when studying more general dynamics beyond QFT and GR.



I already wrote above that E=H(p,q) is valid as approximation. The rest is wrong.



If you open a textbook on QM, you will discover that the operator rho describes the general state of a quantum system (beyond the limits of |ψ>). Recall that my goal was to give a general definition of E, not one valid only in special situations.

Moreover, you are replying to a part where I said that Tr was denoting the classical trace, which means that you do not read my posts. The classical trace is an phase space integration and rho is not an operator therein but the phase space state that correspond to the classical limit.

I'm too tired to keep arguing about this. I'll just concede that I was wrong. Obviously I made a few wrong points, but even so, I still don't see your definition as a useful definition in any sense.

Especially since your answer to my question for calculating the energy for a simple system was H=p^2/2m+V(q). V(q) is the potential energy...this has got to be the most circular definition I can think of. "Energy is equal to kinetic energy plus potential energy". That is simply a tautology.

Lastly, a 3+1 formalism of GR is impossible if the space-time is not globally hyperbolic.

 

[bbbeard]

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.

So how would you have answered that question?

I would leave out the part about string theory.

I see after 30 replies that no one has given the "conventional" response. The conventional response is that energy is the conserved charge associated with symmetry of the Lagrangian under time transations in accordance with Noether's theorem.

Other commenters have pointed out that the Hamiltonian is the generator of time translations. In my view that role is subsidiary to the relationship between energy and the Noether current. But we're risking a slide into metaphysics if we start arguing about what axioms are really fundamental.

BBB

 

[bbbeard]

I am having difficulty making sense of your posts. Perhaps you could give us an example of the kind of energy that cannot be used to do work (either at the macroscopic level or at the microscopic level).

Heat transfer is not work. But you knew that.

If you're asking, "can internal energy u always be used to do work?" I would say the answer is contingent on the circumstances of the system and its environment. The second law places strict limits on how much internal energy can be converted to work transfer, and as you know, the conversion is always less than 100%. The other consideration is that zero-point energy of quantum systems (e.g. $\frac{1}{2}h\nu$ for a harmonic oscillator) cannot be converted to a work transfer.

Thermodynamics was developed before anyone understood that molecules existed. Heat was thought to be some kind of substance that flowed through matter. So the terminology used in thermodynamics is a bit archaic.

I sharply disagree with that. I would say thermodynamics replaced caloric theory just as chemistry replaced alchemy and astronomy replaced astrology. The terminology in thermodynamics uses common words that have a precise and unambiguous meaning within thermodynamics. The fundamental ideas of thermodynamics are enhanced and extended by kinetic theory, but by no means require the existence of molecules. The efficiency of a Carnot cycle engine is what it is, regardless of atomic theory. And since none of the laws of thermodynamics have been overturned by modern physics, I find the terminology is anything but archaic.

Heat flow, ΔQ, is actually a transfer of energy at the molecular level (molecules doing work on other molecules)...

First, heat transfer does not require molecular motion. Heat can be transferred from one box of photons to another, for example. What is the "essence" of heat transfer? What makes an energy transfer a heat transfer and not a work transfer? One textbook I have uses an operational definition that an energy transfer is "work" if in principle all the energy can go into the lifting of a weight in a uniform gravitational field -- that is, they define a paradigmatic work transfer, and assert that other forms of work transfer can be converted to that paradigm "in principle" with 100% efficiency. This definition is not entirely satisfactory, of course. Energy transfers that cannot, even in principle, result in the lifting of a weight are deemed heat transfers.

I prefer to think of heat transfers as energy transfers that necessarily involve the transfer of entropy. That is, it is the random aspect of heat transfer that makes it heat transfer. Your formulation "molecules doing work on other molecules" doesn't really incorporate the essence of heat transfer, does it? Here are some thought experiments that show the problems with this view.

1) Consider how a vibrating wall transfers energy to a gas. Suppose the wall is the y-z plane (at t=0) and is oscillating in the x-direction. With a gas on the +x side, energy is transferred by molecular collisions between the wall molecules and the gas molecules. Does this mean that the transfer is "heat"? No, not at all. The fact that the motion of the wall is coherent means that the F dot dx applied by the wall to the gas is a work transfer.

2) Consider a wall in the same orientation as before, except that the wall is not oscillating, but is at some temperature T_w. If the gas is at some temperature T_g < T_w, then there will be a net transfer of energy to the gas. In this case the energy transfer is "heat transfer" because the "motion" of the wall is incoherent.

3) Consider the thermal radiation of a hot black body A to a cold black body B. I assume you will agree that this is a paradigmatic heat transfer.

4) Now suppose that we consider a laser beaming the same amount of power from body A to body B. Is this heat transfer? No, it is actually work transfer! Why? Because the photons are coherent.

A photon has energy hν because it is capable of doing h$\nu$ amount of work on some element of matter. Applying a force through a distance may not be what a photon does, (and it may not be the most appropriate way to model what it does), but the measure of its energy is its ability to do work on some element of matter.

I'd say yes and no. Certainly the paradigmatic experiment showing the quantum nature of light is the photoelectric effect, in which it was shown that the energy of emitted electrons was equal to h$\nu$-W, where W is the so-called "work function" of the metal (which we now understand to be an ionization energy). Clearly this fits your statement.

I would add, however, that Planck's discovery of quantization didn't really involve work transfers per se. E=h$\nu$ was initially an ad hoc assumption that made the mathematics of the Boltzmann factor work out correctly. Prior to Planck, it was believed that all degrees of freedom were continuous, so the expectation value of the energy for each degree of freedom was exactly the same, and proportional to temperature. This meant that a collection of photons in thermal equilibrium would have an infinite energy. It turns out that by restricting the energy of photons of frequency $\nu$ to multiples of h$\nu$, the expectation value of energy for low-energy photons contributed less and less (instead of a constant) to the sum in the infrared regime. To me this is a more natural, simpler interpretation of the meaning of photon quantization. But again, this is starting to drift into metaphysics, when we argue about which evidence and which paradigms are more compelling.

 

[juanrga]

The link you gave (http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html) said that:



That doesn't seem right. I can have a thermal conductor at 1000C and a thermal insulator at 1000C. Yet the definition above would imply that the thermal conductor and the thermal insulator would give up energy with comparable spontaneity, which is obviously false.

Nope, textbook is rigth.

 

[juanrga]

I'm too tired to keep arguing about this. I'll just concede that I was wrong. Obviously I made a few wrong points, but even so, I still don't see your definition as a useful definition in any sense.

Especially since your answer to my question for calculating the energy for a simple system was H=p^2/2m+V(q). V(q) is the potential energy...this has got to be the most circular definition I can think of. "Energy is equal to kinetic energy plus potential energy". That is simply a tautology.

Lastly, a 3+1 formalism of GR is impossible if the space-time is not globally hyperbolic.

In your own message you quote me saying: «I already wrote above that E=H(p,q) is valid as approximation.» Therefore, it is a derived result not a tautology...