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

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\delta (x-a)dx=f(a)$

Applications in statistical mechanics

• 1-dimensional hard rods