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5 Gamma FunctionApplications

§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 phΓ(+1+iη); 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 x1,x2,,xn and free to move on the infinite line -<x<, is given by

5.20.1 W=12=1nx2-1<jnln|x-xj|.

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

5.20.2 P(x1,,xn)=Cexp(-W/(kT)),

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

5.20.3 ψn(β)=ne-βWdx=(2π)n/2β-(n/2)-(βn(n-1)/4)(Γ(1+12β))-nj=1nΓ(1+12jβ).

See (5.14.6).

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

5.20.4 W=-1<jnln|eiθ-eiθj|,

and the partition function is given by

5.20.5 ψn(β)=1(2π)n[-π,π]ne-βWdθ1dθn=Γ(1+12nβ)(Γ(1+12β))-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.