statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...

, a Coulomb gas is a

many-body system The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...

charged particles In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...

interacting under the

electrostatic force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...

. It is named after

Charles-Augustin de Coulomb Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attrac ...

, 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 The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for . Definition G ...

of magnets and the

sine-Gordon model The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of sur ...

(upon taking certain limits) in a physical sense, in that physical observables (

correlation functions The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms. D ...

) calculated in one model can be used to calculate physical observables in another model. This aided the understanding of the

BKT transition BKT or bkt can refer to: * Bak kut teh, a Malaysian and Singaporean dish of pork ribs in broth * Balkrishna Industries, an Indian tire manufacturer * Banka Kombëtare Tregtare, a commercial bank in Albania * Bayesian knowledge tracing, an algori ...

, and the discoverers earned a

Nobel prize in physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...

for their work on this

phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...

Formulation

The setup starts with considering

N

charged particles in

\mathbb^d

with positions

\mathbf_i

and charges

q_i

. From

electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...

, the pairwise potential energy between particles labelled by indices

i,j

is (up to scale factor)

V_ = q_iq_jg(, \mathbf_i - \mathbf_j, ),

where

g(x)

is the Coulomb kernel or

Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...

of the

Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...

d

dimensions, so

\begin
g(x) = 
\begin-\log, x,  & \text d = 2, \\
 \frac & \textd > 2.
\end
\end

The free energy due to these interactions is then (proportional to)

F = \sum_ V_

, 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 A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...

of the free boson

\varphi

\langle \varphi(z, \bar z) \varphi(w, \bar w) \rangle = - \log, z - w, ^2

to the electric potential energy between two unit charges in two dimensions.

References

{{reflist Statistical mechanics

Formulation

Coulomb gas in conformal field theory

See also

References