# Review of Equilibria

## 1 Equilibria Triangle

Particles in an ensemble over time equilibrate into a steady state, which can be described by various distributions. The resulting ensemble of particles is dependent on temperature, where this temperature which may differ between the various equilibria. The following list describes the main equilibria for:

• ${\displaystyle T_{B}}$ The brightness temperature describes the energy distribution of particles as a function of frequency
• ${\displaystyle T_{*}}$ The excitation temperature controls the number density of particles in the various excitation levels
• ${\displaystyle T_{K}}$ The kinetic temperature controls the velocity distribution of an ensemble of particles

### 1.1 Statistical equilibrium

Starting in the bottom right of the triangle, we have the ${\displaystyle T_{*}}$.

${\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {g_{2}}{g_{1}}}e^{-\Delta E/kT_{*}}\,\!}$

This distribution describes the most probable number density distribution for various excitation levels, as a function of the degeneracies of the states and the excitation temperature. Whereas the number of degeneracies increases at higher states (${\displaystyle g\sim {\tilde {n}}^{2}}$), the energy required for transition increases as well, causing the distribution to peak at intermediate state,${\displaystyle {\tilde {n}}}$. The exact peak can be calculated by differentiating the Boltzmann distribution with respect to the state number

${\displaystyle {\tilde {n_{max}}}\sim {\sqrt {\frac {kT}{2\hbar }}}\,\!}$

### 1.2 Planck distribution

For a perfect blackbody, the energy distribution (spectrum) of the emitted particles is described by Planck’s equation. This distribution determines the energy distribution of particles as a function of frequency for a given brightness temperature, ${\displaystyle T_{B}}$ ( top of the triangle).

${\displaystyle I_{\nu }={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu }{kT_{B}}-1}}\,\!}$

The term brightness temperature is often used in radio astronomy in the Rayleigh-Jeans limit (${\displaystyle h\nu <) to express the flux from an object in units of K.

${\displaystyle I_{\nu }={\frac {2kT_{B}}{\lambda ^{2}}}\,\!}$

### 1.3 Maxwellian distribution

A cloud of particles at the same velocity eventually assumes a Maxwellian velocity distribution due to collisions between particles. The kinetic temperature, ${\displaystyle T_{K}}$, controls both the peak of the velocity distribution and the width of the distribution following:

${\displaystyle f(v)=4\pi v^{2}({\frac {m}{2\pi kT_{K}}})^{\frac {3}{2}}e^{-{\frac {1}{2}}mv^{2}}{kT}\,\!}$

### 1.4 Local Thermodynamic Equilibrium

In Local Thermodynamic Equilibrium (LTE) the three temperatures are equal. Depending on the energy imbalance, the processes as shown along the vertices of the triangle allow for energy transfer from one distribution to another. Since these distributions are frequency dependent, the local in LTE does not only imply a spatial equilibrium but also in a spectral sense. \

The processes assure this equilibrium is given by the vertices of the triangles, where each process has an equivalent inverse process that allows for the reverse direction. These processes are as follows :

• Photoemission ${\displaystyle \longleftrightarrow }$ Photoabsorption: The emission of particles requires an electron in an excited state, dropping to a lower energy level and emitting the excess energy. The rate of this is controlled by the Einstein coefficients, ${\displaystyle A_{2}1}$ and ${\displaystyle B_{2}1}$. On the contrary, the absorption of an electron is given controlled by Einstein ${\displaystyle B_{1}2}$.
• Collisional excitation ${\displaystyle \longleftrightarrow }$ Collisional de-excitation: The transfer from kinetic to excitation energy is controlled by collisions, where the rate ${\displaystyle q}$ is dependent on the number density of the particles, the velocity distribution of the particles and the interaction cross-section.
• Compton ${\displaystyle \longleftrightarrow }$ Inverse Compton: The transfer of a photon’s energy to an atom is given by the Compton scattering if the energies involved are large enough. In the low energy equivalent, the transfer follows from Thomson scattering and Doppler shift. The inverse process, that is transfer from electron’s kinetic energy into the photon is given by the inverse Bremsstrahlung.
• Doppler shift: The velocity of particles can Doppler shift the incoming photons to the appropriate frequencies.

## 2 Further equilibria

There are further equilibria which however are only defined in LTE.

### 2.1 Photoionization

In LTE, the ratio of number density at various ionization states (${\displaystyle He^{+},He^{++}}$) with respect to the ground state is described by Saha equation (since we are in LTE all three temperatures are equal and thus any temperature is applicable here). The partition function ${\displaystyle U(T)}$ describes how many different degeneracies are available, which scales with excitation levels of the neutral.

${\displaystyle {\frac {n_{+}n_{e}}{n_{0}}}=\left[{\frac {2\pi m_{e}kT}{h^{2}}}\right]^{\frac {3}{2}}{\frac {2U_{+}(T)}{U(T)}}e^{-\chi /kT}\,\!}$

### 2.2 Bolometric equilibrium

Bolometric equilibrium is used for calculating the equilibrium temperature of a medium. In this case ingoing flux heats up the temperature of a medium, however, the re-radiation occurs at a different wavelength. A typical example is the equilibrium temperature of Earth, that receive the majority of the energy in the optical but re-radiates in the infrared. The energy balance dictates that the absorbed energy over the applicable solid angles must equal to the emitted energy:

${\displaystyle \sigma T_{infrared}^{4}=\int I_{\nu }\phi (\nu )d\nu d\Omega \,\!}$