5 Gamma Function Applications 5.19 Mathematical Applications 5.21 Methods of Computation

§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.
Keywords:: beta function, gamma function
Referenced by:: §5.1
Permalink:: http://dlmf.nist.gov/5.20

¶ Rutherford Scattering

Keywords:: Coulomb phase shift, Rutherford scattering, gamma function, quantum mechanics

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

¶ Solvable Models of Statistical Mechanics

Keywords:: statistical mechanics

Suppose the potential energy of a gas of 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 $W=\frac{1}{2}\sum_{{\ell=1}}^{n}x_{\ell}^{2}-\sum_{{1\leq\ell<j\leq n}}\mathop% {\ln\/}\nolimits|x_{\ell}-x_{j}|.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer, : nonnegative integer, : real variable and : potential energy
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 $P(x_{1},\dots,x_{n})=C\mathop{\exp\/}\nolimits\!\left(-W/(\BoltzmannConstant T% )\right),$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, : nonnegative integer, : real variable, : potential energy, : temperature and : constant
Permalink:: http://dlmf.nist.gov/5.20.E2
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where $\BoltzmannConstant$ is the Boltzmann constant, the temperature and a constant. Then the partition function (with $\beta=1/(\BoltzmannConstant T)$ ) is given by

5.20.3 $\psi_{n}(\beta)=\int_{{\Real^{n}}}e^{{-\beta W}}dx\\ =(2\pi)^{{n/2}}\beta^{{-(n/2)-(\beta n(n-1)/4)}}\times(\mathop{\Gamma\/}% \nolimits\!\left(1+\tfrac{1}{2}\beta\right))^{{-n}}\prod_{{j=1}}^{n}\mathop{% \Gamma\/}\nolimits\!\left(1+\tfrac{1}{2}j\beta\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\int$ : integral, $\Real$ : real line (excluding infinity), : nonnegative integer, : nonnegative integer, : real variable, : potential energy, $\psi_{n}(\beta)$ : partition function and $\beta=1/(\BoltzmannConstant T)$
Permalink:: http://dlmf.nist.gov/5.20.E3
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See (5.14.6).

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

5.20.4 $W=-\sum_{{1\leq\ell<j\leq n}}\mathop{\ln\/}\nolimits|e^{{i\theta_{\ell}}}-e^{{% i\theta_{j}}}|,$

Symbols:: : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer, : nonnegative integer and : potential energy
Permalink:: http://dlmf.nist.gov/5.20.E4
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and the partition function is given by

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : closed interval, : differential of , : base of exponential function, $\int$ : integral, : nonnegative integer, : 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

Keywords:: beta function, gamma function, partition function, string theory

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.

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