Dirac delta distribution

The Dirac delta distribution (or generalized function) is written as \(\delta(x)\). It is the derivative of the Heaviside step distribution,

\[\frac{d}{dx}[H(x)] = \delta(x)\]

It has the property

\[\int_{- \infty}^{\infty} f(x) \delta (x-a) dx = f(a)\]

Applications in statistical mechanics

• 1-dimensional hard rods