# §5.20 Physical Applications

## Rutherford Scattering

In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift $\operatorname{ph}\Gamma\left(\ell+1+\mathrm{i}\eta\right)$; see (33.2.10) and Clark (1979).

## Solvable Models of Statistical Mechanics

Suppose the potential energy of a gas of $n$ point charges with positions $x_{1},x_{2},\dots,x_{n}$ and free to move on the infinite line $-\infty, is given by

 5.20.1 $W=\frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell

The probability density of the positions when the gas is in thermodynamic equilibrium is:

 5.20.2 $P(x_{1},\dots,x_{n})=C\exp\left(-W/(kT)\right),$

where $k$ is the Boltzmann constant, $T$ the temperature and $C$ a constant. Then the partition function (with $\beta=1/(kT)$) is given by

 5.20.3 $\psi_{n}(\beta)=\int_{{\mathbb{R}}^{n}}e^{-\beta W}\mathrm{d}x\\ =(2\pi)^{n/2}\beta^{-(n/2)-(\beta n(n-1)/4)}\*\left(\Gamma\left(1+\tfrac{1}{2}% \beta\right)\right)^{-n}\prod_{j=1}^{n}\Gamma\left(1+\tfrac{1}{2}j\beta\right).$

See (5.14.6).

For $n$ charges free to move on a circular wire of radius $1$,

 5.20.4 $W=-\sum_{1\leq\ell

and the partition function is given by

 5.20.5 $\psi_{n}(\beta)=\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}e^{-\beta W}\mathrm{d% }\theta_{1}\cdots\mathrm{d}\theta_{n}=\Gamma\left(1+\tfrac{1}{2}n\beta\right)(% \Gamma\left(1+\tfrac{1}{2}\beta\right))^{-n}.$

See (5.14.7).

For further information see Mehta (2004).

## Elementary Particles

Veneziano (1968) identifies relationships between particle scattering amplitudes described by the beta function, an important early development in string theory. Carlitz (1972) describes the partition function of dense hadronic matter in terms of a gamma function.