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arXiv:0906.0243v1 [cond-mat.stat-mech] 1 Jun 2009
Functions of Mittag-Leffler and Fox:
The Pathway Model to Tsallis Statistics and Beck-Cohen Superstatistics
A.M. Mathai
Centre for Mathematical Sciences Pala Campus
Arunapuram P.O., Pala, Kerala-686574, India
and Department of Mathematics and Statistics
McGill University, Canada
mathai@math.mcgill.ca; cmspala@gmail.com
and
H.J. Haubold
Office for Outer Space Affairs, United Nations
Vienna International Centre, P.O. Box 500,
A-1400 Vienna, Austria
and Centre for Mathematical Sciences Pala Campus
Arunapuram P.O., Pala, Kerala-686574, India
hans.haubold@unvienna.org; cmspala@gmail.com
Keywords: generalized hypergeometric functions, Mathai’s pathway model, Tsallis statistics, Beck-Cohen
superstatistics, Mellin-Barnes integrals, fractional integrals, fractional differential equations.
AMS Subject classification: 33C60, 82C31, 62E15
Abstract
In reaction rate theory, in production-destruction type models and in reaction-diffusion problems when
the total derivatives are replaced by fractional derivatives the solutions are obtained in terms of Mittag-
Leffler functions and their generalizations. When fractional calculus enters into the picture the solutions
of these problems, usually available in terms of generalized hypergeometric functions, switch to Mittag-
Leffler functions and their generalizations into Wright functions and subsequently into Fox functions. In this
paper, connections are established among generalized Mittag-Leffler functions, Mathai’s pathway model,
Tsallis statistics, Beck-Cohen superstatistics, and among corresponding entropic measures. The Mittag-
Leffler function, for large values of the parameter, approaches a power-law. For values of the parameter
close to zero, the Mittag-Leffler function behaves like a stretched exponential. The Mittag-Leffler function
is a generalization of the exponential function and represents a deviation from the exponential paradigm
whenever it shows up in solution of physical problems. The paper elucidates the relation between analytic
representations of the q-exponential function that is fundamental to Tsallis statistics, Mittag-Leffler, Wright,
and Fox functions, respectively, utilizing Mellin-Barnes integral representations.
1. Introduction
Fundamental laws of physics are written as equations for the time evolution of a quantity x(t),
dx(t)
dt
= Ax(t),
(1.1)
where if A is limited to a linear operator we have Maxwell’s equation or Schroedinger equation, or it could
be Newton’s law of motion or Einstein’s equations for geodesics if A may also be a nonlinear operator [8].
When A is linear then the mathematical solution is
x(t) = x0e−At
(1.2)
where x0 is the initial value at t = 0. Taking the simplest case of (1.1), as used in reaction rate theory
([8],[4]), if the number density at time t of the i-th particle is Ni(t) and if the number of particles produced
or the production rate is proportional to Ni(t) then the reaction equation is
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dN(t)
dt
= k1N(t), k1> 0
deleting i for convenience. If the decay rate is also proportional to N(t) then the corresponding equation is
dN(t)
dt
= −k2N(t), k2> 0.
Then the residual effect in a production-destruction mechanism is of the form
dN(t)
dt
= −cN(t),c > 0 ⇒ N(t) − N0= −c
?
N(t)dt
(1.3)
if the destruction rate dominates so that the production-destruction model is a decaying model.
production-destruction models abound in scientific disciplines. If the total integral or the total derivative in
(1.3) is replaced by a fractional integral then we have
Such
N(t) − N0= −cν
0D−ν
t N(t),
(1.4)
where0D−ν
t
is the Riemann-Liouville fractional integral operator ([5],[7])defined by
0D−ν
t f(t) =
1
Γ(ν)
?t
0
(t − u)ν−1f(u)du
(1.5)
for ℜ(ν) > 0 where ℜ(·) denotes the real part of (·), c is replaced by cνfor convenience, and then the solution
of (1.4) goes into the category of a Mittag-Leffler function [4], namely,
N(t) = N0
∞
?
k=0
(−cktk)ν
Γ(1 + kν)= N0Eν[−(ct)ν] (1.6)
where Eν(·) is the Mittag-Leffler function. The generalized Mittag-Leffler function is defined as
∞
?
and when Γ(γ) is defined, it has the following Mellin-Barnes representation
Eγ
α,β(z) =
k=0
(γ)k
k!Γ(β + αk), ℜ(α) > 0,ℜ(β) > 0,
(1.7)
Eγ
α,β(z) =
1
Γ(γ)
∞
?
1
2πi
k=0
Γ(γ + k)
k!Γ(β + αk)
(1.8)
=
1
Γ(γ)
?c+i∞
c−i∞
Γ(s)Γ(γ − s)
Γ(β − αs)
(−z)−sds
(1.9)
for 0 < c < ℜ(γ),ℜ(γ) > 0,i =√−1. Some special cases of the generalized Mittag-Leffler function are the
following:
E1
α,β(z) = Eα,β(z) =
∞
?
k=0
1
Γ(β + kα)zk
(1.10)
E1
α,1= Eα,1(z) = Eα(z) =
∞
?
k=0
zk
Γ(1 + kα)
(1.11)
and when α = 1 we have
Eα(z) = E1(z) = ez.
(1.12)
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Thus the Mittag-Leffler function can be looked upon as an extension of the exponential function. The
Mellin-Barnes representation in (1.9) is a special case of the Mellin-Barnes representation of the Wright’s
function [3] which is defined as
pψq(z) =pψq
?
?p
?c+i∞
z
????
j=1Γ(bj+ βjk)
(aj,αj),j=1,...,p
(bj,βj),j=1,...,q
?
=
∞
?
1
2πi
k=0
j=1Γ(aj+ αjk)
?q
Γ(s)
zk
k!
(1.13)
=
c−i∞
??p
j=1Γ(aj− αjs)
?q
j=1Γ(bj− βjs)
?
(−z)−sds
(1.14)
where 0 < c < min1≤j≤pℜ
αj> 0,j = 1,...,p and βj> 0,j = 1,...,q being real quantities. Observe from (1.14) that Wright’s function
is a special case of the H-function [7] and the H-function is defined as the following Mellin-Barnes integral
?aj
αj
?
with aj,j = 1,...,p and bj,j = 1,...,q being complex quantities and
Hm,n
p,q(z) = Hm,n
p,q
?
z
????
(a1,α1),...,(ap,αp)
(b1,β1),...,(bq,βq)
?
=
1
2πi
?c+i∞
c−i∞
φ(s)z−sds
(1.15)
where
φ(s) =
??m
j=m+1Γ(1 − bj− βjs)
j=1Γ(bj+ βjs)
???n
j=1Γ(1 − aj− αjs)
???p
?
??q
j=n+1Γ(aj+ αjs)
?,
(1.16)
for max1≤j≤m
αj> 0,j = 1,...,p and βj> 0,j = 1,...,q re real quantities. Existence conditions and various contours may
be seen from books on the H-function, for example, ([5],[7]). Equations (1.8), (1.14), and (1.15) establish
the relations between Mittag-Leffler, Wright, and Fox functions to be used in the following for q-exponential
function in Tsallis statistics, Beck-Cohen superstatistics, and their link to Mathai’s pathway model.
ℜ(−bj)
βj
< c < min1≤j≤n
ℜ(1−aj)
αj
where aj,j = 1,...,p and bj,j = 1,...,q are complex quantities,
1.1. Extension of the reaction rate model and Mittag-Leffler function
The reaction rate model in (1.3) can be extended in various directions ([8],[4]). For example, if N0is
replaced by N0f(t) where f(t) is a general integrable function on the finite interval [0,b] then it is easy to
see that for the solution of the equation
N(t) − N0f(t) = −cν
0D−ν
t N(t) (1.17)
there holds the formula
N(t) = cN0
?t
0
H1,1
1,2
?
cν(t − x)ν
????
(−1
ν,1)
(−1
ν,1),(0,ν)
?
f(x)dx.
(1.18)
Some special cases are the following: Let ν > 0,ρ > 0,c > 0. Then for the solution of the fractional equation
N(t) − N0tρ−1= −cν
0D−ν
t N(t) (1.19)
there holds the formula
N(t) = N0Γ(ρ)tρ−1Eν,ρ(−(ct)ν).
(1.20)
Let c > 0,ν > 0,µ > 0. Then for the solution of the fractional equation
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N(t) − N0tµ−1Eγ
ν,µ[−(ct)ν] = −cν
0D−ν
t N(t) (1.21)
there holds the formula
N(t) = N0tµ−1Eγ+1
ν,µ[−(ct)ν].
(1.22)
1.2. Fractional partial differential equations and Mittag-Leffler functions
In a series of papers, see for example, ([2],[4]) it is illustrated that the solutions of certain fractional
partial differential equations, resulting from fractional diffusion problems, are available in terms of Mittag-
Leffler functions. For example, consider the equation
0Dν
tN(x,t) −
t−ν
Γ(1 − ν)δ(x) = −cν∂2
∂x2N(x,t) (1.23)
with initial conditions
0Dν−k
t
N(x,t)|t=0= 0,k = 1,...,n
where n = [ℜ(ν)] +1,cνis the diffusion constant, δ(x) is the Dirac’s delta function and [ℜ(ν)] is the integer
part of ℜ(ν). The solution of (1.23), by taking Laplace transform with respect to t nd Fourier transform
with respect to x and then inverting, can be shown to be of the form
N(x,t) =
1
(4πcnutν)
1
2H2,0
1,2
?
|x|2
4cνtν
????
(1−ν
2,ν)
(0,1),(1
2,1)
?
(1.24)
which in special cases reduce to Mittag-Leffler functions.
This paper is organized as follows: Section 2 gives the classical special function technique of eliminating
the upper or lower parameters from a general hypergeometric series. Section 3 establishes the pathways of
going from Mittag-Leffler function to pathway models to Tsallis statistics and Beck-Cohen superstatistics
through the parameter elimination technique. Section 4 gives representations of Mittag-Leffler functions and
pathway model in terms of H-functions and then gives a pathway to go from a Mittag-Leffler function to the
pathway model through H-function.
2. A classical special function technique
The well established technique of eliminating a numerator or denominator parameter from a general
hypergeometric function in the classical theory of special functions, is the following: For convenience, we
will illustrate it on a confluent hypergeometric function.
1F1(a;b;z) =
∞
?
k=0
(a)k
(b)k
zk
k!,(a)m= a(a + 1)...(a + m − 1),a ?= 0,(a)0= 1.
(2.1)
Observe that
(a)k
ak
=a
a
(a + 1)
a
...(a + m − 1)
a
= 1(1 +1
a)(1 +2
a)...(1 +m − 1
a
) → 1 as a → ∞(2.2)
for all finite k. Similarly
bk
(b)k
→ 1 when b → ∞.
(2.3)
Therefore
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lim
a→∞1F1(a;b;z
a) =0F1( ;b;z) (2.4)
which is a Bessel function. Thus we can go from a confluent hypergeometric function to a Bessel function
through this process. Similarly
lim
b→∞1F1(a;b;bz) =1F0(a; : z) = (1 − z)−a, |z| < 1.
(2.5)
Thus, from a confluent hypergeometric function we can go to a binomial function. Further,
lim
b→∞0F1( ;b;bz) =0F0( ; ;z) = ez
(2.6)
and
lim
a→∞1F0(a; ;z
a) =0F0( ; ;z) = ez.
(2.7)
Thus, we can go from a Bessel function as well as from a binomial function to an exponential function. These
two results can be stated in a slightly different form as follows:
lim
q→10F1(;
1
q − 1;−
z
q − 1) = e−z
(2.8)
and
lim
q→11F0(
1
q − 1; ;−(q − 1)z) = lim
q→1[1 + (q − 1)z]−
1
q−1= e−z.
(2.9)
Equation (2.9) is the starting point of Tsallis statistics [9]. The left side in (2.9) is the q-exponential function
of Tsallis, namely,
[1 + (q − 1)z]−
1
q−1= expq(−z)
= e−zwhen q → 1.
(2.10)
We consider a more general form of (2.9) given by
lim
q→1c|x|γ1F0(
η
q − 1; ;−a(q − 1)|x|δ) = lim
q→1c|x|γ[1 + a(q − 1)|x|δ]−
= c|x|γe−aη|x|δ,a > 0,η > 0,−∞ < x < ∞.
η
q−1
(2.11)
Thus, we obtain Mathai’s pathway model [6], namely,
f(x) = c|x|γ[1 + a(q − 1)|x|δ]−
η
q−1,a > 0,η > 0(2.12)
where c is the normalizing constant. If η = 1,γ = 0,a = 1,δ = 1,x > 0 then (2.12) gives Tsallis statistics
[9]. For q > 1,a = 1,η = 1,x > 0 in (2.12) we get Beck-Cohen superstatistics [1].
It may be observed that (2.7), which is the binomial form or1F0going to exponential, is exploited to
produce the pathway model, Tsallis statistics and Beck-Cohen superstatistics whereas (2.6), which is the
Bessel function form going to exponential is not yet exploited. This produces a rich variety of applicable
functions as will be shown in a forthcoming paper.
3. Connection of Mittag-Leffler function to the pathway model
In order to see the connection, let us recall the Mellin-Barnes representation of the generalized Mittag-
Leffler function.
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Γ(β)Eγ
α,β(zδ) =Γ(β)
Γ(γ)
1
2πi
?c+i∞
c−i∞
Γ(γ − s)Γ(s)
Γ(β − αs)
(−zδ)−sds
(3.1)
for ℜ(γ) > 0,i =√−1,ℜ(β) > 0. Let us examine the situation when |β| → ∞. We will state two basic
results as lemmas.
Lemma 3.1 When |β| → ∞ and α > 0 is finite then
lim
|β|→∞
Γ(β)
Γ(β − αs)[−b(zβ
α
δ)δ]−s= (−bzδ)−s.
(3.2)
Proof. The Stirling’s formula is given by
Γ(z + a) ≈
√2πzz+a−1
2e−z, for |z| → ∞(3.3)
and a is a bounded quantity. Now, applying Stirling’s formula we have
Γ(β)
Γ(β − αs)[−b(xβ
α
δ)δ]−s≈
√2πββ−1
√2πββ−1
= βαs(−bzδ)−sβ−αs= (−bzδ)−s.
2e−β
2−αse−β[−b(xβ
α
δ)δ]−s
(3.4)
Lemma 3.2. For ℜ(γ) > 0,ℜ(β) > 0,
lim
|β|→∞Γ(β)Eγ
α,β(−bzδ) =
1
Γ(γ)
∞
?
1
2πi
?c+i∞
c−i∞
Γ(γ − s)Γ(s)[bzδ]−sds
(3.5)
=
k=0
(γ)k
k!
(−bzδ)k= [1 + bzδ]−γfor |bzδ| < 1.
(3.6)
Proof. After applying Lemma 3.1 we can evaluate the contour integral in (3.5) by using the residue theorem
at the poles of Γ(s) to obtain the series form in (3.6). For |bzδ| > 1 we obtain the series form as analytic
continuation or by evaluating the integral in (3.5) as the sum of the residues at the poles of Γ(γ − s). This
will be equal to the following:
lim
|β|→∞Γ(β)Eγ
α,β(−bzδ) = (bzδ)−γ[1 + (bzδ)−1]−γ
(3.7)
for |bzδ| > 1 which will be of the same form as in (3.6). Replacing b by a(q − 1) and γ by
following:
η
q−1we have the
lim
|β|→∞c|z|γΓ(β)Eη/(q−1)
α,β
(−a(q − 1)|z|δ)
= c|z|γ[1 + a(q − 1)|z|δ]−
η
q−1.
(3.8)
Observe that (3.8) is nothing but Mathai’s pathway model [6] for a > 0,δ > 0,η > 0 with c being the
normalizing constant, for both q > 1 and q < 1. Then when q → 1 (3.8) will reduce to the exponential form.
That is,
lim
q→1
lim
|β|→∞c|z|γΓ(β)Eη/(q−1)
α,β
[−a(q − 1)|z|δ] = c|z|γe−aη|z|δ.
(3.9)
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Note from (3.8) that when q > 1 the functional form in (3.8) remains the same whether δ > 0 or δ < 0 with
−∞ < z < ∞. When q < 1 the support in (3.8) will be different for δ > 0 and δ < 0 if f(x) is to remain as
a statistical density.
For γ = 0,a = 1,δ = 1,η = 1,z > 0 we have Tsallis statistics [9] coming from (3.8) for both the cases
q > 1 and q < 1. For q > 1,a = 1,δ = 1,η = 1,z > 0 we have Beck-Cohen superstatistics [1] coming from
(3.8). Observe that the limiting process from (3.5) to (3.6) holds for γ = 1 or γ = 1 and α = 1 also. Thus
the special cases of Mittag-Leffler function are also covered.
Thus, through β a pathway is created to go from a generalized Mittag-Leffler function to Mathai’s
pathway model and then to Tsallis statistics and Beck-Cohen superstatistics. If β is real then as β becomes
larger and larger then
c|z|γΓ(β)Eη/(q−1)
α,β
(−a(q − 1)|z|δ)
goes closer and closer to the pathway model in (3.8). In other words, a pathway is created through β to
go from a Mittag-Leffler function to the pathway model to Tsallis statistics and Beck-Cohen superstatistics.
Thus for large real value of β or for large value of |β|,
c|z|γΓ(β)Eη/(q−1)
α,β
[−a(q − 1)|z|δ] ≈ c|z|γ[1 + a(q − 1)|z|δ]−
η
q−1.
(3.10)
4. Connections through the H-function
Recalling the generalized Mittag-Leffler function from (3.1) and representing it in terms of a H-function
we have the following:
Γ(β)Eγ
α,β(zβ
α
δ)δ=Γ(β)
Γ(γ)
1
2πi
?c+i∞
?
c−i∞
Γ(s)Γ(γ − s)
Γ(β − αs)
????
[−(zβ
?
α
δ)δ]−sds
(4.1)
=Γ(β)
Γ(γ)H1,1
1,2
−(zβ
α
δ)δ
(1−γ,1)
(0,1),(1−β,α)
.
(4.2)
From the limiting process discussed in (3.2) we have the following result:
Lemma 4.1 For ℜ(β) > 0,ℜ(γ) > 0,
lim
|β|→∞
Γ(β)
Γ(γ)H1,1
1,2
?
−(zβ
α
δ)δ??(1−γ,1)
−zδ??(1−γ,1)
?c+i∞
(0,1),(1−β,α)
?
=
1
Γ(γ)H1,1
1,1
?
(0,1)
?
(4.3)
=
1
Γ(γ)
1
2πi
c−i∞
Γ(s)Γ(γ − s)(−zδ)−sds
(4.4)
= [1 − zδ]−γ.
(4.5)
Therefore we have the following theorem:
Theorem 4.1 For ℜ(β) > 0,ℜ(γ) > 0,x > 0,a > 0,q > 1,c > 0
lim
|β|→∞c
γ(β)
?
= cxγ[1 + a(q − 1)xδ]−
Γ
η
q−1
?xγEη/(q−1)
α,β
[−a(q − 1)β
α
δx]δ
η
q−1.
(4.6)
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Observe that the right side in (4.6) is the pathway model [6] for x > 0 from where one has Tsallis statistics,
Beck-Cohen superstatistics and power law where the constant c can act as the normalizing constant to create
a statistical density in the right side of (4.6).
We will write the right side in (4.6) as a H-function and then establish a connection between Mittag-
Leffler function and the pathway model through the H-function. To this end, the practical procedure is
to look at the Mellin transform of the right side of (4.6) and then write it as a H-function by taking the
inverse Mellin transform. The Mellin transform of the right side of (4.6), denoted by Mf(s), is given by the
following:
Mf(s) =
?∞
0
c xγ+s−1[1 + a(q − 1)xδ]−
?
η
q−1dx,a > 0,q > 1,η > 0,δ > 0
=c
δ
Γ?γ+s
[a(q − 1)]
δ
?
γ+s
δ
Γ
η
q−1−γ+s)
Γ
?
δ
?
η
q−1
?
(4.7)
for ℜ
by putting s = 1 the right side of (4.7) must be 1. Therefore,
?
γ+s)
δ
?
> 0,ℜ
?
η
q−1−γ+s)
δ
?
> 0. Note that if c is the normalizing constant for the density f(x) then
Mf(s) =
[a(q − 1)]
?Γ
1
δ
Γ?γ+1
δ
?
η
q−1−γ+1
δ
?Γ
?γ + s
δ
?
Γ
?
η
q − 1−γ + s
δ
?
[a(q − 1)]−s
δ.
(4.9)
Hence the right side of (4.6) is available as the inverse Mellin transform of (4.9). That is,
f(x) = cxγ[1 + a(q − 1)xδ]−
for a > 0,η > 0,q > 1,δ > 0,x > 0
η
q−1,
(4.10)
=
[a(q − 1)]
?Γ
?c+i∞
[a(q − 1)]
Γ?γ+1
= lim
|β|→∞c
1
δ
Γ?γ+1
1
2πi
δ
?
η
q−1−γ+1
?γ + s
1
δ
δ
?
Γ
×
c−i∞
Γ
δ
??
η
q − 1−γ + s
?
δ
?
[x(a(q − 1))
1
δ]−sds
(4.11)
=
δ
?Γ
?
η
q−1−γ+1
Γ(β)
?
δ
?H1,1
1,1
x(a(q − 1))
1
δ
????
(1−
η
q−1+γ
δ,1
δ)
(γ
δ,1
δ)
?
(4.12)
Γ
η
q−1
?xγEη/(q−1)
?
q−1−γ+1
α,β
[−x(a(q − 1))
1
δβ
α
δ]δ
(4.13)
where
c =
δ[a(q − 1)]
Γ?γ+1
γ+1
δ Γ
η
q−1
?
?
δ
?Γ
?
η
δ
(4.14)
for q > 1,δ > 0,η > 0,a > 0,ℜ(γ + 1) > 0,ℜ
?
η
q−1−γ+1
δ
?
> 0.
Remark 4.1. From (4.13) and (4.10) it can be noted that as the parameter β becomes larger and larger the
Mittag-Leffler function in (4.13) comes closer and closer to the pathway model and eventually in the limiting
situation both become identical. In a physical situation, if (4.10) represents the stable situation then the
unstable neighborhoods are given by (4.13). When q → 1 then (4.10) goes to the exponential form. Thus
if the exponential form is the stable situation then the pathway model of (4.1) itself models the unstable
neighborhoods. This unstable neighborhood is farther extended by the Mittag-Leffler form in (4.13).
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Remark 4.2. Observe that the functional form in (4.10) remains the same whether q > 1 or q < 1. But
for q < 1 or when q → 1 the normalizing constant c will be different.
Remark 4.3. When dealing with problems such as reaction-diffusion situations or a general production-
destruction model one goes to fractional differential equations to get a better picture of the solution. Then
we usually end up in Mittag-Leffler functions and their generalizations into Wright’s function or Fox function.
Comparison of (4.13) and (4.10) reveals that as β gets larger and larger the effect of fractional derivative
becomes less and less and finally when |β| → ∞ the effect of taking fractional derivatives, instead of total
derivatives, gets nullified.
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