Bernouilli (ideal gas) method on Photongas? Components of light?

[nonequilibrium]

Hello,

A friend of mine (we're both first-years) has just done his herexam of Thermodynamics & Intr. to Statistical Mechanics. He had to get a relationship between pressure P and energy density E/V for a photongas. As a simplification $$E = \sum |p| c$$ was given. He couldn't do it with the ensemble method, so he decided to try and use the Bernouilli method. Apparently he got it right at the exam, although I don't see the mathematical validity/physical sense of his method:

So basically we use Bernouilli's method for an ideal gas. (I didn't think this was possible at first, but apparently the professor said he was right) Imagine a box with N photons bouncing around without interacting. We focus on one photon and we look at its force (and thus its pressure) on the right wall (of course for one photon this force is statistical and not physical):
$$F_1 = \frac{\Delta p_x}{\Delta t} = \frac{2 |p_x|}{2 L/v_x} = \frac{|p_x| v_x}{L}$$ with L the length of the box and v_x the component for the direction in question (so we're averaging the momentum change for one bounce on the right wall over the time it takes to get back to where it was, at least with respect to the x-component).
Now to get the total and thus physical force, we have $$F = \sum F_i = \sum \frac{|p_x| v_x}{L}$$ (I'm taking the liberty of not specifying the i-dependence as it's pretty obvious).

Now I would personally be stuck at this moment with the product of p_x and v_x, yet apparently he assumed v_x = c... He does admit he can't really justify it (at the moment it seemed obvious), but apparently it is allowed? Or did the professor skip the derivation and just look at the result, which did happen to be right (which implies there might be a reason for v_x = c).

Also, as a side-Q he got "isn't the energy normally dependent on v²?" and as an answer he gave it might have to do with the constancy of light to which my professor said "indeed". How does this all tie together?

PS: Of course once you have F, you can solve the initial question by P = F/A and V = AL

 

[Ambitwistor]

Hello,

Thank you for bringing this up! The Bernouilli method is a valid approach for calculating the pressure of an ideal gas, including a photon gas. However, there are a few things to keep in mind when using this method.

Firstly, as you mentioned, the force on a single photon is a statistical quantity and not a physical force. This means that the pressure calculated using this method is also a statistical quantity and not a physical pressure. It is an average value that represents the overall behavior of the gas.

Secondly, the assumption of v_x = c is not necessarily correct. This assumption is based on the idea that the photons are moving at the speed of light, which is a reasonable assumption for a photon gas. However, it is important to note that this is not always the case and may not be true for all systems.

Regarding the side question about the energy dependence on v^2, this is a reference to the kinetic energy of a particle, which is typically given by 1/2 mv^2. However, for photons, the energy is given by E = pc, where p is the momentum and c is the speed of light. This is a result of the special theory of relativity and the constancy of the speed of light. So, the energy of a photon is not dependent on its velocity, but rather its momentum.

In conclusion, the Bernouilli method is a valid approach for calculating the pressure of an ideal gas, but it is important to keep in mind the assumptions and limitations of this method. It is always a good idea to double check with the professor or consult other resources to ensure the validity of your calculations. Good luck with your studies!