# Reaching equilibrium in a blackbody and light-matter interaction

Suppose we have a metallic cavity maintained at a fixed temperature. Suppose we start with any distribution of radiation that is not in equilibrium with the container. Gradually, when the equilibrium is established, the radiation inside becomes a blackbody radiation at temperature T. Suppose initially the energy density corresponding to a frequency $\omega_1$ was greater than the equilibrium density at temperature T, and that corresponding to $\omega_2$ was lesser than the equilibrium density. How does the matter-radiation system, on its own, goes over to the Blackbody distribution? What tells the system to absorb extra photons of frequency $\omega_1$ and emit those at frequencies near $\omega_2$?

Is it anyhow relared to the Einstein coefficients? If yes, then how?

Is it possible to understand this in terms of interaction between the matter and quantized radiation?

How does the matter-radiation system, on its own, goes over to the Blackbody distribution?

Evolution towards equilibrium (in macroscopic sense) happens when the system matter + radiation is isolated, for example if a piece of matter is inside a cavity that slows down leakage of energy out of the system. For example, a piece of coal in a well reflecting metallic cavity.

There isn't any law in microscopic theory that would imply such system has to evolve to a "microstate" (EM field in the cavity) that corresponds to equilibrium macrostate (where Poynting energy density has the Planck frequency distribution).

But there are mathematical models (sets of equations) that show how a simplified system might approach equilibrium in time, such as the Einstein model of a molecule.

Let us consider a molecule that has 2 possible states - $g,e$ with energies $E_g$, $E_e$ and let the intensity of radiation at frequency $\omega$ be $\rho(\omega)$. Einstein introduced two processes:

• radiation-stimulated process, where the molecule changes its state due to presence of the EM radiation; expected average number of molecules that change their state per unit time is proportional to radiation intensity at frequency $\omega_{eg}=\frac{E_e-E_g}{\hbar}$ - let us denote this intensity as $I$;

• spontaneous process, where the molecule in the state $e$ changes into state $g$ on its own; expected average number of molecules that change their state this way per unit time is independent of the spectral density $\rho$ at any frequency.

Both processes are allowed to be happening at the same time.

Let the probability that the molecule is in state $e$ be $p_e$. Then the probability that the molecule is in state $g$ is $p_g = 1-p_e$. The above two processes allow $p_g$ to change in time and this can be described by the equation

$$\frac{dp_e}{dt} = BI(p_g - p_e) - Ap_e.$$

and when we use $p_g = 1-p_e$, this is

$$\frac{dp_e}{dt} = BI - (A+2BI)p_e.$$

If the intensity $I$ is constant in time, this equation has solution $p_e(t)$ that approaches the value

$$p_{e,eq} = \frac{BI}{A+2BI}$$

following exponential function of time.

This model describes a molecule interacting with prescribed EM radiation; if $I$ is that given by the Planck function, it only describes passage of molecules towards equilibrium, not of radiation.

However, each change of state of a molecule means the radiation gained or lost energy $E_e-E_g$. Since the radiation energy is also a functional of $\rho(\omega)$, this should have impact on the value of $I$. It should therefore be possible to extend the model so that $I$ is also time-dependent and solve the equations to get both $p_e(t)$ and $I(t)$, obtaining a picture of how the combined system approaches mutual equilibrium.

What tells the system to absorb extra photons of frequency ω1ω1 and emit those at frequencies near ω2ω2?

Any system emits complicated radiation that has a range of frequencies, not just one. But considering the spectral function of the molecule to be a sharp peak, the above model implies that decreasing of radiation at frequency $\omega_1$ can happen if the population of molecules with this frequency has more molecules in $g$ state than is the equilibrium value for the current intensity $\rho(\omega_1)$.

Is it anyhow related to the Einstein coefficients? If yes, then how?

Yes, $A$ is the Einstein coefficient of spontaneous emission, $B$ is the coefficient of stimulated emission/absorption.

Is it possible to understand this in terms of interaction between the matter and quantized radiation?

Perhaps some understanding can be gained from the sketched model. It is a very simplistic and schematic one though, as it does not really involve basic laws of physics, but is based on lots of idealized assumptions.