Holtsmark distribution

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title: "Holtsmark distribution" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["continuous-distributions", "probability-distributions-with-non-finite-variance", "power-laws", "stable-distributions", "location-scale-family-probability-distributions"] description: "Probability distribution in physics" topic_path: "law" source: "https://en.wikipedia.org/wiki/Holtsmark_distribution" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Probability distribution in physics ::

| name = Holtsmark | type = continuous | pdf_image = [[File:Levy distributionPDF.svg|325px|Symmetric stable distributions]] Symmetric α-stable distributions with unit scale factor; (blue line) represents the Holtsmark distribution | cdf_image = [[File:Levy distributionCDF.svg|325px|CDF's for symmetric α-stable distributions; represents the Holtsmark distribution]] | parameters = c ∈ (0, ∞) — scale parameter

μ ∈ (−∞, ∞) — location parameter | support = x ∈ R | pdf = expressible in terms of hypergeometric functions; see text | cdf = | mean = μ | median = μ | mode = μ | variance = infinite | skewness = undefined | kurtosis = undefined | entropy = | mgf = undefined | char = \exp\left[it\mu!-!|c t|^{3/2}\right]

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 \alpha equal to 3/2 and the skewness parameter \beta of zero. Since \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.{{Cite journal | doi = 10.1002/andp.19193630702 | volume = 363 | issue = 7 | pages = 577–630 | last = Holtsmark | first = J. | title = Uber die Verbreiterung von Spektrallinien | journal = Annalen der Physik | year = 1919 | bibcode = 1919AnP...363..577H | url = https://zenodo.org/record/1424343 | doi = 10.1086/144420 | issn = 0004-637X | volume = 95 | pages = 489 | last = Chandrasekhar | first = S. |author2=J. von Neumann | title = The Statistics of the Gravitational Field Arising from a Random Distribution of Stars. I. The Speed of Fluctuations | journal = The Astrophysical Journal | year = 1942 | bibcode=1942ApJ....95..489C | doi = 10.1103/RevModPhys.15.1 | volume = 15 | issue = 1 | pages = 1–89 | last = Chandrasekhar | first = S. | title = Stochastic Problems in Physics and Astronomy | journal = Reviews of Modern Physics | date = 1943-01-01 | bibcode=1943RvMP...15....1C

Characteristic function

The characteristic function of a symmetric stable distribution is:

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

where \alpha is the shape parameter, or index of stability, \mu is the location parameter, and c is the scale parameter.

Since the Holtsmark distribution has \alpha=3/2, its characteristic function is:

\varphi(t;\mu,c) = \exp\left[~it\mu ! - ! \left|c t\right|^{3/2}\right] .

Since the Holtsmark distribution is a stable distribution with α 1, \mu represents the mean of the distribution. Since , \mu also represents the median and mode of the distribution. And since {{math|α

Probability density function

In general, the probability density function, f(x), of a continuous probability distribution can be derived from its characteristic function by:

f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \varphi(t)e^{-ixt},dt .

Most stable distributions do not have a known closed form expression for their probability density functions. Only the normal, Cauchy and Lévy distributions have known closed form expressions in terms of elementary functions. The Holtsmark distribution is one of two symmetric stable distributions to have a known closed form expression in terms of hypergeometric functions. When \mu is equal to 0 and the scale parameter is equal to 1, the Holtsmark distribution has the probability density function:

\begin{align} f(x; 0, 1) &= \frac{1}{\pi}, \Gamma{\left(\frac{5}{3}\right)} ; {_2F_3}!\left(\frac{5}{12}, \frac{11}{12}; \frac{1}{3}, \frac{1}{2}, \frac{5}{6}; - \frac{4x^6}{729}\right) \ & {} \quad{} - \frac{x^2}{3\pi} ; {_3F_4}!\left(\frac{3}{4}, {1}, \frac{5}{4}; \frac{2}{3}, \frac{5}{6}, \frac{7}{6}, \frac{4}{3}; - \frac{4x^6}{729}\right) \ & {} \quad{} + \frac{7x^4}{81\pi}, \Gamma{\left(\frac{4}{3}\right)} ; {_2F_3}!\left(\frac{13}{12}, \frac{19}{12}; \frac{7}{6}, \frac{3}{2}, \frac{5}{3}; - \frac{4x^6}{729}\right), \end{align}

where {\Gamma(x)} is the gamma function and _mF_n(\cdot) is a hypergeometric function. One has also{{Cite journal | doi = 10.1140/epjp/s13360-020-00248-4 | volume = 135 | pages = 236 | last = Pain | first = Jean-Christophe | title = Expression of the Holtsmark function in terms of hypergeometric _2F_2 and Airy \mathrm{Bi} functions | journal = Eur. Phys. J. Plus | year = 2020 | s2cid = 211030564 | arxiv = 2001.11893

\begin{align} f(x; 0, 1) &= - \frac{x^2}{6\pi}\left[~_2F_2{\left(1, \frac{3}{2}; \frac{4}{3}, \frac{5}{3}; -\frac{4ix^3}{27}\right)} + ~{_2F_2}{\left(1, \frac{3}{2}; \frac{4}{3}, \frac{5}{3}; \frac{4ix^3}{27}\right)}\right]\[2pt] &\quad + \frac{4}{3^{5/3}} \left[\operatorname{Bi}'\left(-\frac{x^2}{3^{4/3}}\right) \cos\left(\frac{2x^3}{27}\right)+\frac{x}{3^{2/3}} ~ \operatorname{Bi}\left(-\frac{x^2}{3^{4/3}}\right)\sin\left(\frac{2x^3}{27}\right)\right], \end{align}

where \mathrm{Bi} is the Airy function of the second kind and \mathrm{Bi}' its derivative. The arguments of the 2F_2 functions are pure imaginary complex numbers, but the sum of the two functions is real. For x positive, the function \mathrm{Bi}(-x) is related to the Bessel functions of fractional order J{-1/3} and J_{1/3} and its derivative to the Bessel functions of fractional order J_{-2/3} and J_{2/3} . Therefore, one can write

\begin{align} f(x; 0, 1) &= \frac{4x^2}{3^{7/2}} \cos\left(\frac{2x^3}{27}\right)\left[J_{-2/3}{\left(\frac{2x^3}{27}\right)} + J_{2/3}{\left(\frac{2x^3}{27}\right)}\right] \[1ex] &+ \frac{4x^2}{3^{7/2}} \sin\left(\frac{2x^3}{27}\right)\left[J_{-1/3}{\left(\frac{2x^3}{27}\right)} - J_{1/3}{\left(\frac{2x^3}{27}\right)}\right] \[1ex] &-\frac{x^2}{6\pi}\left[~_2F_2{\left(1, \frac{3}{2}; \frac{4}{3}, \frac{5}{3}; -\frac{4ix^3}{27}\right)} + ~_2F_2{\left(1, \frac{3}{2}; \frac{4}{3}, \frac{5}{3}; \frac{4ix^3}{27}\right)}\right]. \end{align}

References

References

1. Zolotarev, V. M.. (1986). "One-Dimensional Stable Distributions". [[American Mathematical Society]].
2. Nolan, J. P.. (2008). "Stable Distributions: Models for Heavy Tailed Data".
3. Nolan, J. P.. (2003). "Handbook of Heavy Tailed Distributions in Finance". [[Elsevier]].
4. Lee, W. H.. (August 2024). ["Continuous and Discrete Properties of Stochastic Processes"](http://etheses.nottingham.ac.uk/11194/1/Thesis_Wai_Ha_Lee.pdf}}{{Dead link).

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