1549380323-Statistical Mechanics Theory and Molecular Simulation

Appendix A Properties of the Dirac delta-function

The Dirac delta-function is informally defined as having infinite height,zero width,
and unit area. When centered aroundx= 0, it can be heuristically represented as

δ(x) =

{

∞ x= 0

0 x 6 = 0

. (A.1)

Theδ-function has two important properties. First, its integral is unity:

∫∞

−∞

dx δ(x) = 1. (A.2)

Second, the integral of aδ-function times any arbitrary functionf(x) is

∫∞

−∞

dx δ(x)f(x) =f(0). (A.3)

Eqn. (A.1) is not rigorous and, therefore, is not especially useful when we wish to
derive the properties of theδ-function. For this reason, we replace eqn. (A.1) with a
definition involving a limit of a sequence of functionsδσ(x) known asδsequences:

δ(x) = lim

σ→ 0

δσ(x). (A.4)

The choice of theδ-sequence is not unique. Several possible choices are:

1. Normalized Gaussian function

δσ(x) =

1

√

2 πσ^2

e−x

(^2) / 2 σ 2
, (A.5)

2. Fourier integral

δσ(x) =

1

π

σ

σ^2 +x^2

=

1

2 π

∫∞

−∞

dkeikx−|σ|x, (A.6)

3. Scaledsincfunction

δσ(x) =

1

πσ

sinc

(x

σ

)

=

1

πx

sin

(x

σ

)

. (A.7)