Holtsmark distribution

The (one-dimensional) Holtsmark distribution is a continuous probability distribution. The Holtsmark distribution is a special case of a stable distribution with the index of stability or shape parameter

α

{\displaystyle \alpha }

equal to 3/2 and the skewness parameter

β

{\displaystyle \beta }

of zero. Since

β

{\displaystyle \beta }

equals zero, the distribution is symmetric, and thus an example of a symmetric alpha-stable distribution. The Holtsmark distribution is one of the few examples of a stable distribution for which a closed form expression of the probability density function is known. However, its probability density function is not expressible in terms of elementary functions; rather, the probability density function is expressed in terms of hypergeometric functions.

The Holtsmark distribution has applications in plasma physics and astrophysics. In 1919, Norwegian physicist Johan Peter Holtsmark proposed the distribution as a model for the fluctuating fields in plasma due to the motion of charged particles. It is also applicable to other types of Coulomb forces, in particular to modeling of gravitating bodies, and thus is important in astrophysics.

Characteristic function

The characteristic function of a symmetric stable distribution is:

φ

(

t

;

μ

,

c

)

=

exp

⁡

[

i

t

μ

−

|

c

t

|

α

]

,

{\displaystyle \varphi (t;\mu ,c)=\exp \left[~it\mu \!-\!\left|ct\right|^{\alpha }\right],}

where

α

{\displaystyle \alpha }

is the shape parameter, or index of stability,

μ

{\displaystyle \mu }

is the location parameter, and c is the scale parameter.

Since the Holtsmark distribution has

α

=

3

/

2

,

{\displaystyle \alpha =3/2,}

its characteristic function is:

φ

(

t

;

μ

,

c

)

=

exp

⁡

[

i

t

μ

−

|

c

t

|

3

/

2

]

.