# Physics:Coulomb gas

__: Many-body of charged particles__

**Short description**In statistical physics, a **Coulomb gas** is a many-body system of charged particles interacting under the electrostatic force. It is named after Charles-Augustin de Coulomb, as the force by which the particles interact is also known as the Coulomb force.

The system can be defined in any number of dimensions. While the three-dimensional Coulomb gas is the most experimentally realistic, the best understood is the two-dimensional Coulomb gas. The two-dimensional Coulomb gas is known to be equivalent to the continuum XY model of magnets and the sine-Gordon model (upon taking certain limits) in a physical sense, in that physical observables (correlation functions) calculated in one model can be used to calculate physical observables in another model. This aided the understanding of the BKT transition, and the discoverers earned a Nobel prize in physics for their work on this phase transition.^{[1]}

## Formulation

The setup starts with considering [math]\displaystyle{ N }[/math] charged particles in [math]\displaystyle{ \mathbb{R}^d }[/math] with positions [math]\displaystyle{ \mathbf{r}_i }[/math] and charges [math]\displaystyle{ q_i }[/math]. From electrostatics, the pairwise potential energy between particles labelled by indices [math]\displaystyle{ i,j }[/math] is (up to scale factor) [math]\displaystyle{ V_{ij} = q_iq_jg(|\mathbf{r}_i - \mathbf{r}_j|), }[/math]

where [math]\displaystyle{ g(x) }[/math] is the **Coulomb kernel** or Green's function of the Laplace equation in [math]\displaystyle{ d }[/math] dimensions,^{[2]} so
[math]\displaystyle{
\begin{align}
g(x) =
\begin{cases}-\log|x| & \text{ if } d = 2, \\
\frac{1}{(d-2)|x|^{d-2}} & \text{ if }d \gt 2.
\end{cases}
\end{align}
}[/math]
The free energy due to these interactions is then (proportional to) [math]\displaystyle{ F = \sum_{i \neq j} V_{ij} }[/math], and the partition function is given by integrating over different configurations, that is, the positions of the charged particles.

## Coulomb gas in conformal field theory

The two-dimensional Coulomb gas can be used as a framework for describing fields in minimal models. This comes from the similarity of the two-point correlation function of the free boson [math]\displaystyle{ \varphi }[/math],
[math]\displaystyle{ \langle \varphi(z, \bar z) \varphi(w, \bar w) \rangle = - \log|z - w|^2 }[/math]
to the electric potential energy between two unit charges in two dimensions.^{[3]}

## See also

- Sine-Gordon equation
- XY model

## References

- ↑ Kosterlitz, J M; Thouless, D J (12 April 1973). "Ordering, metastability and phase transitions in two-dimensional systems".
*Journal of Physics C: Solid State Physics***6**(7): 1181–1203. doi:10.1088/0022-3719/6/7/010. https://iopscience.iop.org/article/10.1088/0022-3719/6/7/010. Retrieved 28 July 2023. - ↑ Chafaï, Djalil (24 August 2021). "Aspects of Coulomb gases". arXiv:2108.10653.
- ↑ Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997) (in en).
*Conformal Field Theory*. https://link.springer.com/book/10.1007/978-1-4612-2256-9. Retrieved 22 August 2023.

Original source: https://en.wikipedia.org/wiki/Coulomb gas.
Read more |