§ 5.20. Physical Applications

Notes:: See Andrews et al. (1999, pp. 406–407) for proofs of the integral formulas and Mehta (2004, Chapters 4 and 11) for physical applications.
Referenced by:: Tab.5.1.1
Permalink:: http://dlmf.nist.gov/5.20

¶ Rutherford Scattering

In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift $\mathrm{ph}\Gamma\!\left(\ell+1+i\eta\right)$ ; see (Ch.33) 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 $-\infty<x<\infty$ , is given by

5.20.1 $W=\frac{1}{2}\sum _{{\ell=1}}^{n}x_{\ell}^{2}-\sum _{{1\le\ell<j\le n}}\ln|x_{\ell}-x_{j}|.$

Defines:: $W$ : potential energy
Symbols:: $j$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/5.20.E1
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The probability density of the positions when the gas is in thermodynamic equilibrium is:

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

Defines:: $W$ : potential energy, $T$ : temperature and $C$ : constant
Symbols:: $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/5.20.E2
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where $\BoltzmannConstant$ is the Boltzmann constant, $T$ the temperature and $C$ a constant. Then the partition function (with $\beta=1/(\BoltzmannConstant T)$ ) is given by

5.20.3 $\psi _{n}(\beta)=\int _{{\Real^{n}}}e^{{-\beta W}}dx\\ =(2\pi)^{{n/2}}\beta^{{-(n/2)-(\beta n(n-1)/4)}}\times(\Gamma\!\left(1+\tfrac{1}{2}\beta\right))^{{-n}}\prod _{{j=1}}^{n}\Gamma\!\left(1+\tfrac{1}{2}j\beta\right).$

Defines:: $W$ : potential energy, $\psi _{n}(\beta)$ : partition function and $\beta=1/(\BoltzmannConstant T)$
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $j$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/5.20.E3
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See (5.14.6).

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

5.20.4 $W=-\sum _{{1\le\ell<j\le n}}\ln|e^{{i\theta _{\ell}}}-e^{{i\theta _{j}}}|,$

Symbols:: $j$ : nonnegative integer, $n$ : nonnegative integer and $W$ : potential energy
Permalink:: http://dlmf.nist.gov/5.20.E4
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and the partition function is given by

5.20.5 $\psi _{n}(\beta)=\frac{1}{(2\pi)^{n}}\int _{{[-\pi,\pi]^{n}}}e^{{-\beta W}}d\theta _{1}\cdots d\theta _{n}=\Gamma\!\left(1+\tfrac{1}{2}n\beta\right)(\Gamma\!\left(1+\tfrac{1}{2}\beta\right))^{{-n}}.$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $W$ : potential energy, $\psi _{n}(\beta)$ : partition function and $\beta=1/(\BoltzmannConstant T)$
Permalink:: http://dlmf.nist.gov/5.20.E5
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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.