Relativistic Effects on Particles and Gases

[Sabertooth]

TL;DR Summary

Identifying Relativistic Effects in Thermodynamic Systems

Hello everyone, I was doing some calculations recently regarding particle velocities for different elements at different temperatures and I have a few questions for the experts in here.
Usual gas laws in my school book provides information about the velocity of particles in gases, it provides the V_rms, V_avg & V_mp of particles as a function of temperature in Kelvin.

My question is whether we need to treat these particles under the terms dictated by special relativity when they start approaching the speed of light.
Is there relativistic in-variance under these circumstances or is the kinetic energy needed to accelerate particles determined by the "relative mass" of the objects

It seems information about this aspect of high temperature gases is lacking in a lot of resources online.
Also, a side question: Do we have any way of calculating the max velocity for a distribution of gas particles at specific temperatures?

 

[Vanadium 50]

What makes you think gas molecules move at relativistic speeds at any reasonable conditions? (Reasonable defined as one can make a container of them) Room temperature is about 1/40 of an eV. A hydrogen molecule ways 2 GeV: so you need to be at > 10 trillion kelvins for this to be a substantial effect.

 

[Sabertooth]

What makes you think gas molecules move at relativistic speeds at any reasonable conditions? (Reasonable defined as one can make a container of them) Room temperature is about 1/40 of an eV. A hydrogen molecule ways 2 GeV: so you need to be at > 10 trillion kelvins for this to be a substantial effect.

We learn some pretty cut and dry gas formulaes in school which help us get the RMS velocity and Avg velocity o. Any gas particle velocity can be calculated as a function of its temperature: V_avg = 8RT/Pi*M.

Temperature is defined as avg kinetic energy of molecules and kinetic energy is defined in terms of mass and velocity.

 

[Sabertooth]

It's part of a deeper inquiry into the nature of Quark_Gluon Plasm and the Hagedorn temp.

 

[Drakkith]

It seems information about this aspect of high temperature gases is lacking in a lot of resources online.

By the time you would need to start accounting for relativistic effects, you're very likely well into the plasma state of matter and dealing with high-energy, high-density plasmas.

Using our ideal gas laws, even at 100 billion kelvin the speed of a hydrogen atom would be roughly 57 million m/s, or 0.19c (non-relativistic calculation). This temperature is more than 6,000 times higher than the core of the Sun and is far, far above the ionization temperature of a gas, making the material a plasma. A cubic meter volume with one mol of this substance would require over 800 billion pascals of pressure, or over 8 million atmospheres, to contain it.

It's part of a deeper inquiry into the nature of Quark_Gluon Plasm and the Hagedorn temp.

At these temperatures and pressures the ideal gas law is almost certainly not used for calculations requiring any real degree of accuracy. You'd need to look into the physics of high-temp, high-density plasmas, high-energy physics, and the details of quantum chromodynamics.

Also, a side question: Do we have any way of calculating the max velocity for a distribution of gas particles at specific temperatures?

There is no maximum velocity. There is instead a range of velocities that you can find the average, peak, rms velocity, etc. This is because multi-body interactions in a gas or plasma can accelerate particles to velocities that are much higher than the average.

 

[Sabertooth]

"By the time you would need to start accounting for relativistic effects, you're very likely well into the plasma state of matter and dealing with high-energy, high-density plasmas.

Using our ideal gas laws, even at 100 billion kelvin the speed of a hydrogen atom would be roughly 57 million m/s, or 0.19c (non-relativistic calculation). This temperature is more than 6,000 times higher than the core of the Sun and is far, far above the ionization temperature of a gas, making the material a plasma. A cubic meter volume with one mol of this substance would require over 800 billion pascals of pressure, or over 8 million atmospheres, to contain it. "

What kind of Plasma would that be? The Hagedorn temp specifically defines the point at which matter start evaporating into Quark Gluon Plasma. For the Proton that temperature should be quite specific and it's very imprecisely defined as being between 145 Mev to 175 Mev, which is around 1.6E12 to 2.2 E12 Kelvin.

I¨'m doing some calculation to derive a more accurate estimate based on the kinetic energy of the indivdual particles. I have gotten some interesting figures, but I'm unsure if my assumptions are correct, but the values line up with current data.

How does relativity influence the motion of high velocity particles. Does the energy expenditure to accelerate particles increase if we account for relativistic effect, or could there be some type of zero-sum cancelling causing the system to have relativistic invariance. There seems to be little information or agreement on this question from scientists. "There is no maximum velocity. There is instead a range of velocities that you can find the average, peak, rms velocity, etc. This is because multi-body interactions in a gas or plasma can accelerate particles to velocities that are much higher than the average."

There must be a reasonable statistical distribution that you can apply. Surely a 100 000 degree kelvin gas cannot ever reach anything close to Speed of light, and we know light speed is max speed.

 

[Sabertooth]

What makes you think gas molecules move at relativistic speeds at any reasonable conditions? (Reasonable defined as one can make a container of them) Room temperature is about 1/40 of an eV. A hydrogen molecule ways 2 GeV: so you need to be at > 10 trillion kelvins for this to be a substantial effect.

Not quite 10 trillion. Relativistic effect should be clearly present already at 1 trillion kelvin if they behave in the usual Lorentzian fashion.

 

[phinds]

@Sabertooth please learn to use the quote feature rather than putting quotes in BOLD like that, as it appears to be screaming.

 

[Sabertooth]

@Sabertooth please learn to use the quote feature rather than putting quotes in BOLD like that, as it appears to be screaming.

Sure thing. My first day here, so hopefully I'm excused.

 

[Vanadium 50]

So, despite saying "elements" you expected us to figure out you were talking about a QGP. You aslo decided β = 0.4 is too relativistic and you want β = 0.04.

How about asking us what you have in mind rather than having us guess and you saying "no, that's not it". That game gets old quick.

 

[Drakkith]

What kind of Plasma would that be?

I don't know if there are different 'types' of plasmas as the temperature and pressure increase, so I can't answer this question.

How does relativity influence the motion of high velocity particles. Does the energy expenditure to accelerate particles increase if we account for relativistic effect, or could there be some type of zero-sum cancelling causing the system to have relativistic invariance. There seems to be little information or agreement on this question from scientists.

I'm not sure what you mean. The energy required to accelerate an object by a unit of velocity only increases as the velocity increases. So accelerating from 500,000 to 500,001 m/s takes more energy than accelerating from 0 to 1 m/s. This is a well agreed upon fact of relativity and is often the explanation given to explain why you can't reach or exceed the speed of light.

There must be a reasonable statistical distribution that you can apply. Surely a 100 000 degree kelvin gas cannot ever reach anything close to Speed of light, and we know light speed is max speed.

Well, I guess if you know the total energy of the gas you can calculate the maximum conceivable energy a single particle of that gas could ever be given in some incredibly remote multi-body interaction event, so I guess you could calculate a maximum speed that way. Other than that it's a smooth distribution of velocities, with the distribution curve changing as the temperature changes.

 

[Sabertooth]

I don't know if there are different 'types' of plasmas as the temperature and pressure increase, so I can't answer this question.

I'm not sure what you mean. The energy required to accelerate an object by a unit of velocity only increases as the velocity increases. So accelerating from 500,000 to 500,001 m/s takes more energy than accelerating from 0 to 1 m/s. This is a well agreed upon fact of relativity and is often the explanation given to explain why you can't reach or exceed the speed of light.

It's a well agreed upon fact in non-thermodynamic systems.
Mosengeil's Formula also dictates that temperature changes with the Lorentz factor as you increase velocity. That is to say a moving body with respect to a stationary observer, gets colder as it increases velocity.

That is not the issue at hand though. Relativistic thermodynamics is much less agreed upon and there seems to be a starvation of information regarding the topic online.
It seems to me that there could be a major difference in the way relativity treats masses in thermodynamic systems as opposed to the way it deals with observer-variant special relativity.

 

[Sabertooth]

So, despite saying "elements" you expected us to figure out you were talking about a QGP. You aslo decided β = 0.4 is too relativistic and you want β = 0.04.

How about asking us what you have in mind rather than having us guess and you saying "no, that's not it". That game gets old quick.

I think you messed up your calculations. At 1 trillions Kelvin avg velocity will be approx. 0.5 β.
10 trillion K would make no sense as the avg would be above the speed of light. It is also beyond the Hagedown Temperature thus making hadrons ustable.

I already told you what I was looking for. A coherent breakdown of the relativistic effects on thermodynamic systems, particularly particle energies.

 

[Sabertooth]

Well, I guess if you know the total energy of the gas you can calculate the maximum conceivable energy a single particle of that gas could ever be given in some incredibly remote multi-body interaction event, so I guess you could calculate a maximum speed that way. Other than that it's a smooth distribution of velocities, with the distribution curve changing as the temperature changes.

That's the exact thing I'm looking for, the maximal velocity to be achieved by one single particle in a distribution of particles at a given temperature.

 

[Drakkith]

That is not the issue at hand though. Relativistic thermodynamics is much less agreed upon and there seems to be a starvation of information regarding the topic online.
It seems to me that there could be a major difference in the way relativity treats masses in thermodynamic systems as opposed to the way it deals with observer-variant special relativity.

Well, I doubt there's any serious disagreements in this area, but I wouldn't expect to find too much easily digestible information online since this is a fairly advanced and specialized topic. I'm not an expert in either relativity or thermodynamics, so I unfortunately can't offer you much advice. Best of luck to you though.

 

[Sabertooth]

Well, I doubt there's any serious disagreements in this area, but I wouldn't expect to find too much easily digestible information online since this is a fairly advanced and specialized topic. I'm not an expert in either relativity or thermodynamics, so I unfortunately can't offer you much advice. Best of luck to you though.

Apparently, it's shrouded in much historical controversy. Here's an excerpt from a 2002 article:

Relativistic thermodynamics is a highly controversial discipline on issues such as the variance of temperature or the heat; the results concerning these two quantities (T and Q), of Planck-Einstein (1907) were reversed in 1963 by the physicist Ott in moreover following a suggestion formulated by Einstein himself in 1952. Since then there have been two main schools of thermodynamics relativistic with a series of variants.
The somewhat strange status of relativistic thermodynamics invites questions about somewhat “special” relationships that maintain the ethereal RR and thermodynamics. Existence of several schools within this discipline is due to the fact that there is no of unequivocal definition, in particular of temperature in a system moving

 

[Drakkith]

Well, there we go then. You learn something new every day.

 

[Cthugha]

I have little to no idea about the topic at hand, but I know that Hänggi and Dunkel worked on this topic quite a bit and I remember attending a talk on that topic eternities ago and surprisingly the slides still seem to be online:
Link to the talk

One of the main publications they discussed was Dunkel et al., "Non-local observables and lightcone-averaging in relativistic thermodynamics", Nature Physics 5, 741 (2009).

I have no idea whether that is useful to you, but maybe it is helpful to check the papers cited in that paper and also newer publications citing that paper to find what you are looking for.

 

[Sabertooth]

I have little to no idea about the topic at hand, but I know that Hänggi and Dunkel worked on this topic quite a bit and I remember attending a talk on that topic eternities ago and surprisingly the slides still seem to be online:
Link to the talk

One of the main publications they discussed was Dunkel et al., "Non-local observables and lightcone-averaging in relativistic thermodynamics", Nature Physics 5, 741 (2009).

I have no idea whether that is useful to you, but maybe it is helpful to check the papers cited in that paper and also newer publications citing that paper to find what you are looking for.

Definitely a lot of useful information contained in those slides. Thank you.