# Can someone reconcile the Boltzmann transport equation with the Maxwell equations for photons/light?

Having taking courses in both physics and nuclear engineering, I've noticed that the two fields tend to describe photons/light in two different settings.

In nuclear engineering, the radiative transfer equation (Boltzmann transport equation for photons) is often used. For example, as noted in the Wikipedia article, simulations for radiation therapy treatments solve this equation to model the dose applied to a patient. From what I understand, this equation treats light essentially as particles.

In another more physics/E&M-oriented class that I'm taking, light is described via the Maxwell equations. I'm not a physicist so I don't know that much about those equations, but it seems like it's an entirely unrelated/different approach for describing light. Here, it seems like light is treated more as a wave as opposed to particles. In this class I'm taking, we see that, under certain assumptions, the Maxwell equations eventually lead to a Helmholtz equation, which seems very different from the Boltzmann transport equation. (For starters, there's a second order spatial derivative instead of a first-order derivative.)

Can someone help me reconcile the two approaches? i.e., how is it that we can look at light from such drastically different points of view? I know that there's the whole thing about light being both a particle and a wave, but I don't see how the two approaches are even related. Are there certain frequency ranges where the transport equation is more applicable or something?

Any relevant references or explanations would be appreciated. Thanks!

• Doesn't the RTE describe the time-evolution of the intensity of the wave, not the photons themselves? – Kyle Kanos Feb 5 '15 at 20:16
• Ah hm.. I see. And so, it would be physically incorrect to view the intensity of the wave as composed of individual photons? – nukeguy Feb 5 '15 at 21:31
• Did you ever find any good references relating RTE to Maxwell? – icurays1 Feb 23 '16 at 17:24
• I did not unfortunately. I took a math class by a physicist where the RTE equations are derived from Maxwell's equations using asymptotic expansions and other techniques, and that's the basis for me asking this question here. He did it from the physics perspective, which I'm not really familiar with. A lot of the physical intuition was foreign to me (i.e., I understood the mathematical validity of all the steps in the derivation, but I had difficulty connecting the approximations used to what was going on in the physical world.) I can send you some class notes/slides if you're interested – nukeguy Feb 23 '16 at 20:42

The radiative transfer equation is a simplified model for describing light transfer. Of course it is possible to derive the radiative transfer equation by the Boltzmann equation for a photon density function $f(x,t)$: $$\partial_t f(x,t) + v_x \partial_x f(x,t) = (\partial_t f(x,t))_{coll}.$$
Here, the term $(\partial_t f(x,t))_{coll}$ is the gain and loss of the photon density per time due to light scattering, extinction and illumination. It is not easy to derive this right hand side term (for exact calculations you need quantum mechanics). However, with some suitable assumptions (like in the kinetic theory of gases) basing on Maxwell's equations, the radiative Transfer equation can be obtained.
• To obtin macroscopic quantities from the Boltzmann equation you multiply the Boltzmann equation with an observable that depends on the space and then you integrate over the whole space. The photon density function is normalized, i.e. \int d^3x f(x,t) = N_p = constant. When you have some electrodynamic quantities like the energy density $\frac{E^2 + B^2}{2}$ or the Poynting vector you can multiply the electrodynamic quantities with the Boltzmann equation and integrate over the whole space. When using the linearity rule of Integration, you can obtain the RTE equation. – kryomaxim Feb 5 '15 at 21:45