Digital Library of Mathematical Functions
The Gamma Function -- Physical Applications

### Physical Applications

Suppose the potential energy of a gas of n point charges with positions x 1 , x 2 , … , x n and free to move on the infinite line - ∞ < x < ∞ , is given by

W = 1 2 ∑ ℓ = 1 n x ℓ 2 - ∑ 1 ≤ ℓ < j ≤ n ln | x ℓ - x j |

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

P ( x 1 , … , x n ) = C exp ( - W ( k T ) )

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

ψ n ( β ) = ∫ ℝ n ⅇ - β W ⅆ x = ( 2 π ) n 2 β - ( n 2 ) - ( β n ( n - 1 ) 4 ) × ( Γ ( 1 + 1 2 β ) ) - n ∏ j = 1 n Γ ( 1 + 1 2 j β )

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

W = - ∑ 1 ≤ ℓ < j ≤ n ln | ⅇ ⅈ θ ℓ - ⅇ ⅈ θ j |

and the partition function is given by

ψ n ( β ) = 1 ( 2 π ) n ∫ [ - π , π ] n ⅇ - β W ⅆ θ 1 ⋯ ⅆ θ n = Γ ( 1 + 1 2 n β ) Γ ( 1 + 1 2 β ) ) - n