Restricting RMS Speed of Molecules to Comply with Relativity

In summary: But KE, exactly, = mc^2(γ-1) [m being rest mass] which is completely different from substituting mγ into the Newtonian formula, getting (1/2)mγv^2. The point is that there is only one relativistic analog of Newtonian kinematic formulas for which you can pretend mγ plays the role of mass. So thinking there is a general concept of relativistic mass which is analogous to the m in Newtonian formulas leads only to a large number... confusion.
  • #1
Vodkacannon
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How can we restrict the RMS speed of molecules to comply with relativity?

They obviously can't go at or faster than the speed of light.

If you are dealing with particles inside of very hot stars for example you may get very high erroneous speeds.

[itex]v = \sqrt{\frac{3KT}{m}}[/itex]

How could we manipulate this equation?
 
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  • #2
I am not familiar with the equation you gave. However, increasing energy (heating particles up) close to relativistic speeds leads to increase in m rather than v.
 
  • #3
Yes that may have been a more appropriate question for me to ask in hopes of connecting relativity and kinetic gas theory
 
  • #4
Vodkacannon said:
How could we manipulate this equation?

We don't. :wink:

Instead, we start from scratch and re-derive the Maxwell speed distribution using relativistic kinetic energy instead of non-relativistic kinetic energy. This appears to have what you are looking for:

http://www.marcelhaas.com/docs/maxrel.pdf (PDF)

See equation 21 on page 4.
 
  • #5
jtbell said:
We don't. :wink:

Instead, we start from scratch and re-derive the Maxwell speed distribution using relativistic kinetic energy instead of non-relativistic kinetic energy. This appears to have what you are looking for:

http://www.marcelhaas.com/docs/maxrel.pdf (PDF)

See equation 21 on page 4.

Interesting. thanks. I had known about Maxwells Distribution laws before as it's in my AP textbook. I can see how the formula sort of resembles the plain old relativistic equations.
 
  • #6
jtbell said:
We don't. :wink:

Instead, we start from scratch and re-derive the Maxwell speed distribution using relativistic kinetic energy instead of non-relativistic kinetic energy. This appears to have what you are looking for:

http://www.marcelhaas.com/docs/maxrel.pdf (PDF)

See equation 21 on page 4.

It's interesting that this makes no mention of the Maxwell-Juttner distribution, which should be the applicable distribution. Do you know if this analysis is consistent with that? At a glance, it doesn't look like it would be.
 
  • #7
mathman said:
However, increasing energy (heating particles up) close to relativistic speeds leads to increase in m rather than v.

This is the relativistic mass convention, which is very old-fashioned. Physicists today use the convention that mass is invariant, so, e.g., rather that writing p=mv with an m that is a function of v, they write [itex]p=m\gamma v[/itex], where m is a constant.
 
  • #8
The Maxwell Speed Distribution for Relativistic Speeds (PDF)
I could be wrong, but I think the formula given in this paper is incorrect. Basically he's taking the density of states to be uniform in velocity space with a sharp cutoff imposed at v=c.

The correct approach is to use a uniform density in momentum space. For nonrelativity it doesn't matter since p = mv is linear, but it does matter in our case. In momentum space there is no cutoff, one integrates over all p out to infinity.
 
  • #9
bcrowell said:
This is the relativistic mass convention, which is very old-fashioned. Physicists today use the convention that mass is invariant, so, e.g., rather that writing p=mv with an m that is a function of v, they write [itex]p=m\gamma v[/itex], where m is a constant.

Nitpicker - what would you call [itex]m\gamma [/itex]?
 
  • #10
mathman said:
Nitpicker - what would you call [itex]m\gamma [/itex]?

m[itex]\gamma[/itex]

The reason I discourage calling it a mass is the temptation for those learning SR to substitute it into Newtonian formulas. There is only one common formula (momentum) for which this works.

F = m[itex]\gamma[/itex]a anyone?
KE = (1/2)m[itex]\gamma[/itex]v^2 anyone?

Another observation is that [itex]\gamma[/itex] is really part of the normalization of 4-velocity to be a unit 4-vector. The correct 4-force equation for constant mass particle is simply m * 4-acceleration.
 
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  • #11
PAllen said:
m[itex]\gamma[/itex]

The reason I discourage calling it a mass is the temptation for those learning SR to substitute it into Newtonian formulas. There is only one common formula (momentum) for which this works.

F = m[itex]\gamma[/itex]a anyone?
KE = (1/2)m[itex]\gamma[/itex]v^2 anyone?

Another observation is that [itex]\gamma[/itex] is really part of the normalization of 4-velocity to be a unit 4-vector. The correct 4-force equation for constant mass particle is simply m * 4-acceleration.

It may be old fashioned, but my understanding was that the total energy is mc2, which can be expanded as a series in v to get m0c2 +m0v2/2 + ... = rest mass energy + kinetic energy.
 
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  • #12
mathman said:
It may be old fashioned, but my understanding was that the total energy is mc2, which can be expanded as a series in v to get m0c2 +m0v2/2 + ... = rest mass energy + kinetic energy.

But KE, exactly, = mc^2(γ-1) [m being rest mass] which is completely different from substituting mγ into the Newtonian formula, getting (1/2)mγv^2. The point is that there is only one relativistic analog of Newtonian kinematic formulas for which you can pretend mγ plays the role of mass. So thinking there is a general concept of relativistic mass which is analogous to the m in Newtonian formulas leads only to a large number of common errors

[edit: Thinking more, I see the main reason for the shift is the wide adoption of 4-vectors for SR. In this scheme you have mU. for 4-momentum; m is rest mass, U is 4 velocity. No mγ in sight (it is internal to the 4-velocity). Then, 4-force is naturally mA, using 4-acceleration; no nonsensical transverse and longitudinal relativistic mass as seen in some ancient relativity books.]
 
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  • #13
PAllen said:
But KE, exactly, = mc^2(γ-1) [m being rest mass] which is completely different from substituting mγ into the Newtonian formula, getting (1/2)mγv^2. The point is that there is only one relativistic analog of Newtonian kinematic formulas for which you can pretend mγ plays the role of mass. So thinking there is a general concept of relativistic mass which is analogous to the m in Newtonian formulas leads only to a large number of common errors

[edit: Thinking more, I see the main reason for the shift is the wide adoption of 4-vectors for SR. In this scheme you have mU. for 4-momentum; m is rest mass, U is 4 velocity. No mγ in sight (it is internal to the 4-velocity). Then, 4-force is naturally mA, using 4-acceleration; no nonsensical transverse and longitudinal relativistic mass as seen in some ancient relativity books.]

The main point of the formulation I gave is that it is easy to see where the Newtonian KE comes from, since the correction term is ~ (v/c)2.
 
  • #14
PAllen said:
m[itex]\gamma[/itex]

The reason I discourage calling it a mass is the temptation for those learning SR to substitute it into Newtonian formulas. There is only one common formula (momentum) for which this works.[..]
See http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html for a balanced discussion but note one glitch: the FAQ ignores that the second law of Newton is in fact F~dp/dt. It works fine like that.
PAllen said:
[..] [edit: Thinking more, I see the main reason for the shift is the wide adoption of 4-vectors for SR. [..]
Right. Notice that invariant mass and relativistic mass are alternative solutions to avoid the introduction of "longitudinal" and "transverse" mass concepts.
 
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FAQ: Restricting RMS Speed of Molecules to Comply with Relativity

What is the concept of restricting RMS speed of molecules to comply with relativity?

The concept of restricting root mean square (RMS) speed of molecules is based on the theory of relativity, which states that the speed of light is the maximum speed at which anything can travel. This means that the speed of molecules, which are made up of atoms and subatomic particles, cannot exceed the speed of light. Therefore, scientists have developed methods to restrict the RMS speed of molecules to comply with this principle.

Why is it important to restrict RMS speed of molecules?

Restricting the RMS speed of molecules is important for several reasons. First, it ensures that the laws of physics, particularly the theory of relativity, are upheld. Additionally, it is crucial for accurate measurements and predictions in experiments and theories involving high-speed particles, such as in nuclear reactions and particle accelerators. It also plays a role in the stability and behavior of materials and substances.

How is the RMS speed of molecules restricted?

The RMS speed of molecules can be restricted through various methods, such as cooling techniques and confinement in a small space. Cooling molecules reduces their kinetic energy, which in turn decreases their speed. Confining molecules in a small space also limits their movement and therefore their speed. Other methods involve using strong magnetic or electric fields to slow down the molecules.

What are the challenges in restricting the RMS speed of molecules?

One of the challenges in restricting the RMS speed of molecules is the trade-off between accuracy and efficiency. Cooling techniques can lower the speed of molecules, but they may also affect the molecules' properties, making it difficult to study them. Additionally, it can be challenging to accurately measure the speed of molecules that are moving at extremely high speeds, close to the speed of light. Finally, the cost and complexity of the equipment and methods used to restrict RMS speed can also be a challenge.

What are the potential applications of restricting RMS speed of molecules?

Restricting the RMS speed of molecules has applications in various fields, such as in fundamental research, materials science, and technology. It can help scientists study the properties and behavior of molecules more accurately, leading to a better understanding of the building blocks of matter. It also has implications in developing new materials with specific properties, improving technologies such as semiconductors and superconductors, and advancing fields like quantum computing and nanotechnology.

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