Why Is Infinite Degrees of Freedom Important for Thermal States?

[specdude]

Are electrons closed systems?

 

[stevendaryl]

Are electrons closed systems?

What does "closed system" mean? Do you mean in the sense of thermodynamics?

 

[specdude]

What does "closed system" mean? Do you mean in the sense of thermodynamics?

Sorry actually I mean An isolated system. One which cannot exchange any heat, work, or matter with the surroundings.

 

[stevendaryl]

Sorry actually I mean An isolated system. One which cannot exchange any heat, work, or matter with the surroundings.

Well, electrons are viewed as point-particles. "Heat" is something that only applies to a system of many particles. There is no way to make sense of the temperature, for instance, of a single electron.

But electrons can certainly exchange energy with their surroundings, by absorbing or emitting radiation (light, or other electromagnetic waves). So I don't think it makes sense to call them isolated.

 

[specdude]

Thanks. That helps.

 

[Jazzdude]

Well, electrons are viewed as point-particles. "Heat" is something that only applies to a system of many particles. There is no way to make sense of the temperature, for instance, of a single electron.

I find this answer rather misleading. First of all, the number of particles doesn't matter for a system to be statistically describable or in thermal equilibrium. It's the number of degrees of freedom that count, and the electron has infinitely many of them. In quantum theory you can easily construct a thermal system with just one electron and describe it with a density matrix. If the electron is bound you get an incoherent superposition of all bound energy states with the usual exp(-E_n/kT) weights where the temperature of the system *is* defined and given by T.

The second problem I have with your answer is the way you arrive at your conclusion from subscribing to a point particle ontology for the electron. When we usually say the electron is a "point particle", we don't imply that it has the configuration space of a single point (which you seemingly used to imply that it's not a thermal system), but that 1) there is no known inner structure and we can spatially constrain the electron state as accurately as we wish, at least theoretically, and 2) the interaction terms in QED are those of single points, not extended regions. That does in no way however imply a point particle ontology of any sorts, because these point shaped interactions are always in nontrivial superpositions for all valid quantum states. But even if you feel that the Bohmian ontology is the way to go, the conclusion is not that the electron has the thermal properties of a classical point particle, because it is still fully equivalent for statistical physics to viewing the electron as having all the degrees of freedom the wavefunction offers.

Cheers,

Jazz

 

[bhobba]

Well, electrons are viewed as point-particles.

Hmmm.

Not quite so sure about that.

IMHO, in relation to the issue of the 'point particle' nature of quantum objects, its best to view it as an an approximation to QFT that fully melds the field and particle aspect. Basically it means position is an observable.

Thanks
Bill

 

[stevendaryl]

I find this answer rather misleading. First of all, the number of particles doesn't matter for a system to be statistically describable or in thermal equilibrium. It's the number of degrees of freedom that count, and the electron has infinitely many of them. In quantum theory you can easily construct a thermal system with just one electron and describe it with a density matrix. If the electron is bound you get an incoherent superposition of all bound energy states with the usual exp(-E_n/kT) weights where the temperature of the system *is* defined and given by T.

The second problem I have with your answer is the way you arrive at your conclusion from subscribing to a point particle ontology for the electron. When we usually say the electron is a "point particle", we don't imply that it has the configuration space of a single point (which you seemingly used to imply that it's not a thermal system), but that 1) there is no known inner structure and we can spatially constrain the electron state as accurately as we wish, at least theoretically, and 2) the interaction terms in QED are those of single points, not extended regions. That does in no way however imply a point particle ontology of any sorts, because these point shaped interactions are always in nontrivial superpositions for all valid quantum states. But even if you feel that the Bohmian ontology is the way to go, the conclusion is not that the electron has the thermal properties of a classical point particle, because it is still fully equivalent for statistical physics to viewing the electron as having all the degrees of freedom the wavefunction offers.

Cheers,

Jazz

Hmm. But if you are considering an isolated electron, then what does it mean to be in thermal equilibrium? What is the temperature of an isolated electron?

 

[Jazzdude]

Hmm. But if you are considering an isolated electron, then what does it mean to be in thermal equilibrium? What is the temperature of an isolated electron?

This is not really about an isolated electron but one electron singled out of a large collection of electrons (or atoms, or molecules, or clusters, ... ). A single electron is in thermal equilibrium if the reduced stated of the electron is a thermal density matrix. The temperature follows from that state.

Cheers,

Jazz

 

[Jano L.]

It's the number of degrees of freedom that count...

Could you please explain why?

and the electron has infinitely many of them.

Could you please explain that too? What do you mean by degree of freedom?

The second problem I have with your answer is the way you arrive at your conclusion from subscribing to a point particle ontology for the electron. When we usually say the electron is a "point particle", we don't imply that it has the configuration space of a single point

Sometimes some of us do mean that. Especially in non-relativistic theory. I am afraid there is no universal agreement on the meaning of point-like. It depends on context.

But even if you feel that the Bohmian ontology is the way to go, the conclusion is not that the electron has the thermal properties of a classical point particle, because it is still fully equivalent for statistical physics to viewing the electron as having all the degrees of freedom the wavefunction offers.

The function $\psi(\mathbf r)$ is a concept that describes electron. Not all of its characteristics are also characteristics of the electron. For example, the function is complex-valued, but electron is not. Electron has mass, but $\psi$ function has not.

Similarly, the $\psi$ function may belong to Hilbert space with infinite basis, but electron does not. There is no necessity to project all characteristics of mathematical functions onto material bodies.

 

[bhobba]

The function $\psi(\mathbf r)$ is a concept that describes electron. Not all of its characteristics are also characteristics of the electron. For example, the function is complex-valued, but electron is not. Electron has mass, but $\psi$ function has not.

Similarly, the $\psi$ function may belong to Hilbert space with infinite basis, but electron does not. There is no necessity to project all characteristics of mathematical functions onto material bodies.

Our theories are mathematical descriptions - what an 'electron' is apart from that is anyone's guess.

Thanks
Bill

 

[Jano L.]

Our theories are mathematical descriptions - ...

...of electrons. Yes! That's what I'm saying.

 

[Jazzdude]

Could you please explain why?

Because that is what comes out of statistical physics. The theory doesn't care for the number of particles, it even applies to systems without any particles at all!

Sometimes some of us do mean that. Especially in non-relativistic theory. I am afraid there is no universal agreement on the meaning of point-like. It depends on context.

If you mean that then you're in disagreement with the standard meaning. And I very much believe that there is universal agreement on the meaning of "point particle" in quantum theory, namely what I wrote in my first post. Every serious textbook on QT or QFT I've seen uses exactly that meaning. Bohmians may use the term with a different meaning, but that's not relevant here.

The function $\psi(\mathbf r)$ is a concept that describes electron. Not all of its characteristics are also characteristics of the electron. For example, the function is complex-valued, but electron is not. Electron has mass, but $\psi$ function has not.

That sounds like gibberish to me. What is that even supposed to mean: "the electron is not complex"? When we're doing physics we're describing nature with mathematics, and we use the best possible mathematical description that we can up with. And we came up with the wavefunction description because it is required for describing the electron. Insisting on a more "physical" and "simple" description is just ignoring the results of 100 years of quantum theory. It is also the wavefunction that appears as the state of the system in all relevant (quantum) physical theories, especially statistical physics. So if we make a statement about statistical physics we of course use the math and terminology of statistical physics as applied to quantum theory.

Similarly, the $\psi$ function may belong to Hilbert space with infinite basis, but electron does not. There is no necessity to project all characteristics of mathematical functions onto material bodies.

I think your understanding of the relationship between theory, mathematics and nature is a little off. In theoretical physics we don't deal with "what is" but with the best possible mathematical description for it. It doesn't make sense to say "the electron belongs (or not) to a Hilbert space", because an electron is not a mathematical object. But our best theory of nature describes an electron as a mathematical entity that lives in a Hilbert space, and that is the most simple description that agrees with what we observe. Saying that this is not the same as material reality is missing the point entirely. Everything we say is what terms mean and predictions are in context of a physical theory. If you want more than that you might want to consider going into philosophy instead.

Cheers,

Jazz

 

[Jano L.]

It's the number of degrees of freedom that count, and the electron has infinitely many of them.

Let's get back to your point. Could you please explain why is infinite number of degrees of freedom important for the possibility of system being in a thermal state?