5 Gamma Function Applications 5.19 Mathematical Applications 5.21 Methods of Computation

§5.20 Physical Applications

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Keywords:: applications, beta function, gamma function, physical
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:: §5.1
Permalink:: http://dlmf.nist.gov/5.20
See also:: Annotations for Ch.5

Rutherford Scattering

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Keywords:: Coulomb phase shift, Rutherford scattering, gamma function, nonrelativistic, quantum mechanics
See also:: Annotations for §5.20 and Ch.5

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

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Keywords:: solvable models, statistical mechanics
See also:: Annotations for §5.20 and Ch.5

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		$W=\frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln\|x_{\ell% }-x_{j}\|.$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $j$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $W$ : potential energy Permalink: http://dlmf.nist.gov/5.20.E1 Encodings: TeX, pMML, png See also: Annotations for §5.20, §5.20 and Ch.5

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

5.20.2		$P(x_{1},\dots,x_{n})=C\exp\left(-W/(kT)\right),$
ⓘ Symbols: $\exp\NVar{z}$ : exponential function, $n$ : nonnegative integer, $x$ : real variable, $W$ : potential energy, $T$ : temperature and $C$ : constant Permalink: http://dlmf.nist.gov/5.20.E2 Encodings: TeX, pMML, png See also: Annotations for §5.20, §5.20 and Ch.5

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		$\psi_{n}(\beta)=\int_{{\mathbb{R}}^{n}}e^{-\beta W}\,\mathrm{d}x\\ =(2\pi)^{n/2}\beta^{-(n/2)-(\beta n(n-1)/4)}\*\left(\Gamma\left(1+\tfrac{1}{2}% \beta\right)\right)^{-n}\prod_{j=1}^{n}\Gamma\left(1+\tfrac{1}{2}j\beta\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\mathbb{R}$ : real line, $j$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $W$ : potential energy, $\psi_{n}(\beta)$ : partition function and $\beta=1/(kT)$ Permalink: http://dlmf.nist.gov/5.20.E3 Encodings: TeX, pMML, png See also: Annotations for §5.20, §5.20 and Ch.5

See (5.14.6).

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

5.20.4		$W=-\sum_{1\leq\ell<j\leq n}\ln\|e^{i\theta_{\ell}}-e^{i\theta_{j}}\|,$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $j$ : nonnegative integer, $n$ : nonnegative integer and $W$ : potential energy Permalink: http://dlmf.nist.gov/5.20.E4 Encodings: TeX, pMML, png See also: Annotations for §5.20, §5.20 and Ch.5

and the partition function is given by

5.20.5		$\psi_{n}(\beta)=\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}e^{-\beta W}\,\mathrm% {d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=\Gamma\left(1+\tfrac{1}{2}n\beta% \right)(\Gamma\left(1+\tfrac{1}{2}\beta\right))^{-n}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $n$ : nonnegative integer, $W$ : potential energy, $\psi_{n}(\beta)$ : partition function and $\beta=1/(kT)$ Permalink: http://dlmf.nist.gov/5.20.E5 Encodings: TeX, pMML, png See also: Annotations for §5.20, §5.20 and Ch.5

See (5.14.7).

For further information see Mehta (2004).

Elementary Particles

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Keywords:: applications, beta function, gamma function, hadronic matter, partition function, physical, string theory
See also:: Annotations for §5.20 and Ch.5

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.

5.19 Mathematical Applications 5.21 Methods of Computation

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§5.20 Physical Applications

Rutherford Scattering

Solvable Models of Statistical Mechanics

Elementary Particles

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