1. Fundamentals

The fractional Poisson probability distribution captures the long-memory effect which results in the non-exponential waiting time probability distribution function empirically observed in complex classical and quantum systems. Thus, fractional Poisson process and fractional Poisson probability distribution function can be considered as natural generalization of the famous Poisson process and the Poisson probability distribution.

The idea behind the fractional Poisson process was to design counting process with non-exponential waiting time probability distribution. Mathematically the idea was realized by substitution the first-order time derivative in the Kolmogorov–Feller equation for the Poisson probability distribution function with the time derivative of fractional order.^[1][2]

The main outcomes are new stochastic non-Markov process – fractional Poisson process and new probability distribution function – fractional Poisson probability distribution function.

2. Fractional Poisson Probability Distribution Function

The probability distribution function of the fractional Poisson process has been found for the first time by Nick Laskin (see, Ref.[1])

[math]\displaystyle{ P_\mu (n,t)=\frac{(\nu t^\mu )^n}{n!}\sum\limits_{k=0}^\infty \frac{(k+n)!}{ k!}\frac{(-\nu t^\mu )^k}{\Gamma (\mu (k+n)+1)},\qquad 0\lt \mu \leq 1, }[/math]

where parameter [math]\displaystyle{ \nu }[/math] has physical dimension [math]\displaystyle{ [\nu ]=\sec ^{-\mu } }[/math] and [math]\displaystyle{ {\Gamma (\mu (k+n)+1)} }[/math] is the Gamma function.

The [math]\displaystyle{ P_\mu (n,t) }[/math] gives us the probability that in the time interval [math]\displaystyle{ [0,t] }[/math] we observe n events governed by fractional Poisson stream.

The probability distribution of the fractional Poisson process [math]\displaystyle{ P_{\mu }(n,t) }[/math] can be represented in terms of the Mittag-Leffler function [math]\displaystyle{ E_{\mu }(z) }[/math] in the following compact way (see, Ref.[1]),

[math]\displaystyle{ P_{\mu }(n,t)=(\frac{(-z)^{n}}{n!}\frac{d^{n}}{dz^{n}}E_{\mu }(z))|_{z=-\nu t^{\mu }}, }[/math]

[math]\displaystyle{ P_{\mu }(n=0,t)=E_{\mu }(-\nu t^{\mu }). }[/math]

It follows from the above equations that when [math]\displaystyle{ \mu =1 }[/math] the [math]\displaystyle{ P_\mu (n,t) }[/math] is transformed into the well known probability distribution function of the Poisson process, [math]\displaystyle{ P(n,t)=P_1(n,t) }[/math], [math]\displaystyle{ P(n,t)=\frac{(\overline{\nu}t )^n}{n!}\exp(-\overline{\nu} t), }[/math]

[math]\displaystyle{ P(n=0,t)=\exp(-\overline{\nu} t), }[/math]

where [math]\displaystyle{ \overline{\nu} }[/math] is the rate of arrivals with physical dimension [math]\displaystyle{ [\overline{\nu}]=\sec ^{-1 } }[/math].

Thus, [math]\displaystyle{ P_\mu (n,t) }[/math] can be considered as fractional generalization of the standard Poisson probability distribution. The presence of additional parameter [math]\displaystyle{ \mu }[/math] brings new features in comparison with the standard Poisson distribution.

3. Mean

The mean [math]\displaystyle{ \overline{n}_\mu }[/math] of the fractional Poisson process has been found in Ref.[1].

[math]\displaystyle{ \overline{n}_\mu =\sum\limits_{n=0}^\infty nP_\mu (n,t)=\frac{\nu t^\mu }{ \Gamma (\mu +1)}. }[/math]

4. The Second Order Moment

The second order moment of the fractional Poisson process [math]\displaystyle{ \overline{n^2}_\mu }[/math] has been found for the first time by Nick Laskin (see, Ref.[1])

[math]\displaystyle{ \overline{n_\mu ^2}=\sum\limits_{n=0}^\infty n^2P_\mu (n,t)=\overline{n}_\mu +\overline{n}_\mu ^2\frac{\sqrt{\pi }\Gamma (\mu +1)}{2^{2\mu -1}\Gamma (\mu +\frac 12)}. }[/math]

5. Variance

The variance of the fractional Poisson process is (see, Ref.[1])

[math]\displaystyle{ \sigma _\mu =\overline{n_\mu ^2}-\overline{n}_\mu ^2=\overline{n}_\mu + \overline{n}_\mu ^2\left\{ \frac{\mu B(\mu ,\frac 12)}{2^{2\mu -1}} - 1\right\}, }[/math]

where [math]\displaystyle{ B(\mu ,\frac 12) }[/math] is the Beta-function.

6. Characteristic Function

The characteristic function of the fractional Poisson process has been found for the first time in Ref.[1],

[math]\displaystyle{ C_\mu (s,t)=\sum\limits_{n=0}^\infty e^{isn}P_\mu (n,t)=E_\mu (\nu t^\mu (e^{is}-1)). }[/math]

or in a series form

[math]\displaystyle{ C_\mu (s,t)=\sum\limits_{m=0}^\infty \frac 1{\Gamma (m\mu +1)}\left( \nu t^\mu (e^{is}-1)\right) ^m, }[/math]

with the help of the Mittag-Leffler function series representation.

Then, for the moment of [math]\displaystyle{ k^{{\rm th}} }[/math] order we have

[math]\displaystyle{ \overline{n_\mu ^k}= (1/i^k)\frac{\partial ^kC_\mu (s,t)}{\partial s^k}|_{s=0}. }[/math]

7. Generating Function

The generating function [math]\displaystyle{ G_\mu(s,t) }[/math] of the fractional Poisson probability distribution function is defined as (see, Ref.[1]).

[math]\displaystyle{ G_\mu (s,t)=\sum\limits_{n=0}^\infty s^nP_\mu (n,t). }[/math]

The generating function of the fractional Poisson probability distribution was obtained for the first time by Nick Laskin in Ref.[1].

[math]\displaystyle{ G_\mu (s,t)=E_\mu (\nu t^\mu (s-1)), }[/math]

where [math]\displaystyle{ E_\mu (z) }[/math] is the Mittag-Leffler function given by its series representation

[math]\displaystyle{ E_\mu (z)=\sum\limits_{m=0}^\infty \frac{z^m}{\Gamma (\mu m+1)}. }[/math]

8. Moment Generating Function

The equation for the moment of any integer order of the fractional Poisson can be easily found by means of the moment generating function [math]\displaystyle{ H_\mu (s,t) }[/math] which is defined as

[math]\displaystyle{ H_\mu (s,t)=\sum\limits_{n=0}^\infty e^{-sn}P_\mu (n,t). }[/math]

For example, for the moment of [math]\displaystyle{ k^{{\rm th}} }[/math] order we have

[math]\displaystyle{ \overline{n_\mu ^k}=(-1)^k\frac{\partial ^kH_\mu (s,t)}{\partial s^k}|_{s=0}. }[/math]

The moment generating function [math]\displaystyle{ H_\mu (s,t) }[/math] is (see, Ref.[1])

[math]\displaystyle{ H_\mu (s,t)=E_\mu (\nu t^\mu (e^{-s}-1)), }[/math]

or in a series form

[math]\displaystyle{ H_\mu (s,t)=\sum\limits_{m=0}^\infty \frac 1{\Gamma (m\mu +1)}\left( \nu t^\mu (e^{-s}-1)\right) ^m, }[/math]

with the help of the Mittag-Leffler function series representation.

9. Waiting Time Distribution Function

A time between two successive arrivals is called as waiting time and it is a random variable. The waiting time probability distribution function is an important attribute of any arrival or counting random process.

Waiting time probability distribution function [math]\displaystyle{ \psi _\mu (\tau) }[/math] of the fractional Poisson process is defined as (see, Refs.[1,3])

[math]\displaystyle{ \psi _\mu (\tau )=-\frac d{d\tau }P_\mu (\tau ), }[/math]

where [math]\displaystyle{ P_\mu (\tau ) }[/math] is the probability that a given interarrival time is greater or equal to [math]\displaystyle{ \tau }[/math]

[math]\displaystyle{ P_\mu (\tau )=1-\sum\limits_{n=1}^\infty P_\mu (n,\tau )=E_\mu (-\nu \tau^\mu ), }[/math]

and [math]\displaystyle{ P_\mu (n,\tau ) }[/math] is the fractional Poisson probability distribution function.

The waiting time probability distribution function [math]\displaystyle{ \psi _\mu (\tau) }[/math] of the fractional Poisson process was found for first time by Nick Laskin in Ref.[1],

[math]\displaystyle{ \psi _\mu (\tau )=\nu \tau ^{\mu -1}E_{\mu ,\mu }(-\nu \tau ^\mu ),\qquad t\geq 0,\qquad 0\lt \mu \leq 1, }[/math]

here [math]\displaystyle{ E_{\alpha ,\beta }(z) }[/math] is the generalized two-parameter Mittag-Leffler function

[math]\displaystyle{ E_{\alpha ,\beta }(z)=\sum\limits_{m=0}^\infty \frac{z^m}{\Gamma (\alpha m+\beta )},\qquad E_{\alpha ,1}(z)=E_\alpha (z). }[/math]

Waiting time probability distribution function [math]\displaystyle{ \psi _\mu (\tau ) }[/math] has the following asymptotic behavior (see, Ref.[1])

[math]\displaystyle{ \psi _{\mu }(\tau )\simeq 1/\nu \tau ^{\mu +1},\qquad \tau \rightarrow \infty , }[/math]

and

[math]\displaystyle{ \psi _{\mu }(\tau )\simeq \nu \tau ^{\mu -1},\qquad \tau \rightarrow 0. }[/math]

10. Fractional Compound Poisson Process

Fractional compound Poisson process has been introduced and developed for the first time by Nick Laskin (see, Ref.[1]). The fractional compound Poisson process [math]\displaystyle{ \{X(t) }[/math], [math]\displaystyle{ t\geq 0\} }[/math] is represented by

[math]\displaystyle{ X(t)=\sum\limits_{i=1}^{N(t)}Y_i, }[/math]

where [math]\displaystyle{ \{N(t) }[/math], [math]\displaystyle{ t\geq 0\} }[/math] is a fractional Poisson process, and [math]\displaystyle{ \{Y_i }[/math], [math]\displaystyle{ i=1,2,\ldots\} }[/math] is a family of independent and identically distributed random variables with probability distribution function [math]\displaystyle{ p(Y) }[/math] for each [math]\displaystyle{ Y_i }[/math]. The process [math]\displaystyle{ \{N(t) }[/math], [math]\displaystyle{ t\geq 0\} }[/math] and the sequence [math]\displaystyle{ \{Y_i }[/math], [math]\displaystyle{ i=1,2,\ldots\} }[/math] are assumed to be independent.

The fractional compound Poisson process is natural generalization of the compound Poisson process.

11. Applications of Fractional Poisson Probability Distribution

The fractional Poisson probability distribution has physical and mathematical applications. Physical application is in the field of quantum optics. Mathematical applications are in the field of combinatorial numbers (see, Ref.[4]).

12. Physical Application: New Coherent States

A new family of quantum coherent states [math]\displaystyle{ |\varsigma \gt }[/math] has been introduced as^[3]

[math]\displaystyle{ |\varsigma \rangle=\sum_{n=0}^{\infty }\frac{(\sqrt{\mu }\varsigma ^{\mu })^{n}}{\sqrt{n!}}(E_{\mu }^{(n)}(-\mu |\varsigma |^{2\mu }))^{1/2}|n\rangle, }[/math]

where [math]\displaystyle{ |n\rangle }[/math] is an eigenvector of the photon number operator, complex number [math]\displaystyle{ \varsigma }[/math] stands for labeling the new coherent states,

[math]\displaystyle{ E_\mu ^{(n)}(-\mu |\varsigma |^{2\mu }) = \left. \frac{d^n}{dz^n}E_\mu (z) \right|_{z=-\mu |\varsigma |^{2\mu}} }[/math]

and [math]\displaystyle{ E_\mu(x) }[/math] is the Mittag-Leffler function.

Then the probability [math]\displaystyle{ P_\mu(n) }[/math] of detecting n photons is:

[math]\displaystyle{ P_\mu(n)=|\langle n\mid\varsigma \rangle|^2 =\frac{(\mu |\varsigma |^{2\mu})^n}{n!} \left( E_\mu^{(n)}(-\mu |\varsigma |^{2\mu})\right), }[/math]

which is recognized as fractional Poisson probability distribution.

In terms of photon field creation and annihilation operators [math]\displaystyle{ a^{+} }[/math] and [math]\displaystyle{ a }[/math] that satisfy the canonical commutation relation [math]\displaystyle{ [a,a^{+}]=aa^{+}-a^{+}a=1 }[/math], the average number of photons [math]\displaystyle{ \bar n }[/math] in a coherent state [math]\displaystyle{ |\varsigma \rangle }[/math] can be presented as (see, Ref.[4])

[math]\displaystyle{ \bar n=\langle \varsigma |a^{+}a|\varsigma \rangle =\sum_{n=0}^\infty nP_\mu(n) =(\mu|\varsigma |^{2\mu})/\Gamma (\mu +1). }[/math]

13. Mathematical Applications: New Polynomials and Numbers

The fractional generalization of Bell polynomials, Bell numbers, Dobinski's formula and Stirling numbers of the second kind have been introduced and developed by Nick Laskin (see, Ref.[4]). The appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been applied to evaluate the skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of the Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been discovered (see, Ref.[4]).

In the limit case μ = 1 when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed applications turn into the well-known results of the quantum optics and the enumerative combinatorics.

14. Statistical Application and Inference

The point and interval estimators for the model parameters are developed by Cahoy et. al, (2010) (see, Ref.[5]). ^[4]

The content is sourced from: https://handwiki.org/wiki/Fractional_Poisson_process