A GENERALIZED ENTROPY OPTIMIZATION AND MAXWELL-BOLTZMANN DISTRIBUTION A.M.Mathai Emeritus Professor of Mathematics and Statistics, McGill University, Canada
[email protected] ;
[email protected] and Hans J. Haubold Office of Outer Space Affairs, United Nations Vienna, Austria,
[email protected] Abstract A generalized entropy introduced earlier by the first author is optimized under various conditions and it is shown that Maxwell-Boltzmann distribution, Raleigh distribution and other distributions can be obtained through such optimization procedures. Some properties of the entropy measure are examined and then MaxwellBoltzmann and Raleigh densities are extended to multivariate cases. Connections to geometrical probability problems, isotropic random points, and spherically symmetric and elliptically contoured statistical distributions are pointed out. Keywords Generalized entropy, Maxwell-Boltzmann density, Raleigh density, geometrical probability, isotropic random points, multivariate analogues 1. Introduction Consider the following entropy measure ∫ δ−α [f (x)]1+ η dx − 1 x Mα (f ) = , α ̸= δ, η > 0, α < η + δ α−δ
(1.1)
where δ is a real number or anchoring point, α is a varying parameter, η > 0 is the measuring unit of the distance δ − α, x is a real scalar or vector or matrix variable. We will use small x to denote a scalar variable and capital X to denote a vector or matrix variable. dX stands for the wedge product of differentials. f (x) is a real∫ valued scalar function of x such that f (x) ≥ 0 for all x and x f (x)dx = 1 or f (x) is a statistical density. If X is a p × 1 vector with X ′ = (x1 , ..., xp ) denoting its transpose, then dX = dX ′ = dx1 ∧ ... ∧ dxp . If X = (xij ) is a m × n matrix with n distinct real scalar variable elements xij ’s then dX = ∧m i=1 ∧j=1 dxij . Here δ is an 1
anchoring point, any real number including zero. When α → δ then 1 + δ−α →1 η ∫ δ−α and x [f (x)]1+ η dx − 1 → 0. Hence {∫ } ∫ 1+ δ−α η dx − 1 [f (x)] 1 x lim Mα (f ) = lim =− f (x) ln f (x)dx α→δ α→δ α−δ η x which is Shannon’s measure of entropy when x is a real scalar variable. Hence (1.1) is a generalization of Shannon’s entropy measure in the real scalar variable case as well as extended to vector and matrix-variate cases. Observe that ∫ ∫ δ−α δ−α 1+ δ−α η [f (x)] dx = [f (x)] η f (x)dx = E[f (x)] η x
x
where E denotes statistical expectation. When α → δ this expected value is E(1) = 1. Here δ is a fixed point on the real line and the entropy is anchored at the point x = δ when x is a scalar variable. When α = δ, Mα (f ) goes to Shannon’s entropy and hence Mα (f ) can be taken as a measure of departure from Shannon’s entropy, departure measured in terms of δ−α units. If α = δ gives a stable stage in a physical η situation then when α moves away from δ then Mα (f ) will measure the entropy in the neighborhood of the stable stage. Later it will be shown that α can describe a pathway for the movement from a stable situation to the unstable neighborhoods. We will start with our discussion when x is a real scalar variable. First we will optimize (1.1) and obtain the Maxwell-Boltzmann density. 1.1. Entropy Optimization for the Maxwell-Boltzmann Density One form of the Maxwell-Boltzmann velocity density is the following: { 3 2 √4 β 2 v 2 e−βv , 0 ≤ v < ∞, β = m 2kT π f (v) = (1.2) 0, elsewhere . ∫∞ Note that f (v) ≥ 0 for all v and 0 f (v)dv = 1. It is a statistical density. Consider an arbitrary density for a real scalar positive variable x, denoted by f (x), and consider the following moments for the real scalar positive variable x: ∫ ∞ δ−α 2( δ−α ) η µ1 = E[x ]= x2( η ) f (x)dx, η > 0, (1.3) 0
∫
and 2( δ−α )+2 η
µ2 = E[x
∞
]= 0
2
x2(
δ−α )+2 η
f (x)dx, η > 0.
(1.4)
Observe that when α = δ, (1.3) says E(1) = 1 with respect to any density f (x), and (1.4) says about the second moment of the arbitrary density f (x). Let us assume that µ1 and µ2 are fixed or given in the class of all densities f (x) or∫ in the set of all real-valued scalar functions f (x) such that f (x) ≥ 0 for all x and x f (x)dx = 1. Let us optimize the entropy in (1.1) under the constraints that µ1 and µ2 are given for fixed δ, α, η with η > 0. If we use calculus of variation to optimize (1.1) then the Euler equation is the following: δ−α δ−α δ−α ∂ {f 1+ η − λ1 x2( η ) f + λ2 x2( η )+2 f } = 0 ∂f where λ1 and λ2 are Lagrangian multipliers. Then (i) gives
(1 + That is,
δ−α δ−α δ − α δ−α )f η − λ1 x2( η ) + λ2 x2( η )+2 = 0. η
[
1 f= 1 + δ−α η By taking
λ2 λ1
η ] δ−α
η λ2 2 δ−α x] . λ1 η [ ] δ−α
x2 [1 −
= a(δ − α), α < δ, a > 0 and c1 =
1 1+ δ−α η
(i)
(ii)
(iii)
we have
η
f1 (x) = c1 x2 [1 − a(δ − α)x2 ] δ−α , a > 0, α < δ
(1.5)
where 1 − a(δ − α)x2 > 0, c1 can act as the normalizing constant when f1 (x) is a statistical density, and f1 (x) = 0 elsewhere. If α > δ then write δ −α = −(α−δ), α > δ and then (1.5) is transformed to η
f2 (x) = c2 x2 [1 + a(α − δ)x2 ]− α−δ , α > δ, η > 0, a > 0, x ≥ 0
(1.6)
and zero elsewhere, where c2 can act as the normalizing constant. Observe that c2 is different from c1 , and in fact that the two functions f1 (x) and f2 (x) are structurally different, one belonging to the generalized type-1 beta family of densities and the other belonging to the generalized type-2 beta family of densities, where the support 1 of f1 (x) is finite 0 ≤ x ≤ [a(δ − α)]− 2 , and f2 (x) has the support 0 ≤ x < ∞. When α → δ then both f1 (x) and f2 (x) go to f3 (x) = c3 x2 e−aηx , η > 0, a > 0, x ≥ 0 2
(1.7)
and zero elsewhere, where c3 is the normalizing constant. Note that (1.7) is a form of the Maxwell-Botzmann velocity density. Observe that from (1.5) one can go to (1.6) 3
and (1.7). Also one can go from (1.6) to (1.5) and (1.7). Hence (1.5) or (1.6) is called a pathway version of the Maxwell-Boltzmann density, where δ is a fixed point such as δ = 1 and α is the pathway parameter and the departure from the point x = δ is measured in terms of η units, η > 0. If we want to incorporate the parameters m m in the Maxwell-Boltzmann density then we may take aη = 2kT or a = η2kT . Then (1.5),(1.6) and (1.7) reduce to the following: f1∗ (x) = c∗1 x2 [1 − (δ − α)(
η m )x2 ] δ−α , α < δ, η > 0 η2kT
m )x2 > 0; η2kT η m ∗ ∗ 2 f2 (x) = c2 x [1 + (α − δ)( )x2 ]− α−δ , α > δ, x ≥ 0 η2kT m ∗ ∗ 2 − 2kT x2 f3 (x) = c3 x e , x ≥ 0.
(1.8)
1 − (δ − α)(
(1.9) (1.10)
We can evaluate c∗1 by using a type-1 beta integral, c∗2 by using a type-2 beta integral and c∗3 by using a gamma integral. The results are the following: 3
c∗1
3
η m + 5 )(δ − α) 2 ( η2kT )2 4Γ( δ−α √ 2 η = , α < δ, η > 0, πΓ( δ−α + 1) 3
(1.11)
3
η m η 4 Γ( α−δ )(α − δ) 2 ( η2kT ) 3 3 , α > δ, =√ − >0 η 3 α−δ 2 π Γ( α−δ − 2 ) 4 m 3 c∗3 = √ ( )2 . π η2kT
c∗2
(1.12) (1.13)
We will call (1.8) and (1.9) as the pathway generalized Maxwell-Boltzmann density. 2. Raleigh Density and Optimization of the Generalized Entropy One form of Raleigh density is the following: g(x) =
1 − 2γx22 xe , 0 ≤ x < ∞, γ > 0 γ2
(2.1)
and zero elsewhere. If (2.1) is to be obtained from the generalized entropy (1.1) then consider the following constraints: ∫ ∞ δ−α δ−α η ν1 = E[x ] = x η g(x)dx, η > 0 (2.2) 0
4
∫
and ν2 = E[x
δ−α +2 η
∞
]=
x
δ−α +2 η
g(x)dx, η > 0.
(2.3)
0
Assuming that ν1 and ν2 are fixed for an arbitrary density g(x), for fixed α, δ, η and proceeding as in section 1.1 we have the following densities corresponding to f1 , f2 , f3 in section 1.1: η
g1 (x) = d1 x[1 − a(δ − α)x2 ] δ−α , a > 0, α < δ, η > 0, 1 − (δ − α)x2 > 0 η
g2 (x) = d2 x[1 + a(α − δ)x2 ]− α−δ , a > 0, α > δ, η > 0, x ≥ 0 −aηx2
g3 (x) = d3 xe
, a > 0, η > 0, x ≥ 0
(2.4) (2.5) (2.6)
where g1 , g2 , g3 are zero outside the support indicated above, and d1 , d2 , d3 are the respective normalizing constants. Comparing (2.6) with the Raleigh density in (2.1) 1 we may take aη = 2γ12 or a = η2γ 2 . Then g1 , g2 transform to the following: g1∗ (x) = d∗1 x[1 − (δ − α)(
η 1 2 δ−α )x ] , α < δ, η > 0, η2γ 2
1 )x2 > 0 η2γ 2 η 1 2 − α−δ g2∗ (x) = d∗2 x[1 + (α − δ)( )x ] , α > δ, η > 0, x ≥ 0 η2γ 2
(2.7)
1 − (δ − α)(
(2.8)
where the normalizing constants d∗1 and d∗2 can be evaluated by using a type-1 beta inegral and a type-2 beta integral respectively. Then the resulting densities are the following: η (δ − α) 2 δ−α η+δ−α x[1 − x ] , α < δ, η > 0, 2 2 ηγ 2ηγ eta η+δ−α (α − δ) 2 − α−δ g2∗ (x) = x[1 + x] , α > δ, η > 0, x ≥ 0, 2 2 ηγ 2ηγ 1 − x2 g3∗ (x) = 2 xe 2γ 2 , x ≥ 0, γ > 0. γ
g1∗ (x) =
(2.9) (2.10) (2.11)
If we take the anchoring point δ as δ = 1 and the parameter η = 1 then the normalizing constants d∗1 = d∗2 = 2−α . Then or α < 1 one has type-1 beta case in (2.9), for γ2 α > 1 one has the type-2 beta case in (2.10) and for α → 1 one has the gamma case in (2.11). 5
3. Some General Observations Consider the generalized entropy in (1.1). This can be written as ∫ ∫ ∫ δ−α δ−α [f (x)]1+ η dx − 1 [f (x)]1+ η dx − x f (x)dx x x Mα (f ) = = α−δ [ δ−α α − ]δ δ−α ∫ f η −1 [f (x) η − 1] = f (x) dx = E α−δ α−δ x
(3.1)
δ−α
where E denotes the expected value. When α → δ we have f η = 1. Thus, depending upon the departure of α from the fixed anchoring point x = δ we have a δ−α very small or larger departure from 1 in f η . Also lim
α→δ
f
δ−α η
δ−α
−1 e η ln f − 1 = lim α→δ α−δ α−δ δ−α
∂ limα→δ ∂α [e η ln f − 1] 1 = = − ln f. ∂ η limα→δ ∂α [α − δ]
That is, 1 1 lim Mα (f ) = − E[ln f ] = − α→δ η η where S denotes Shannon’s entropy. Hence
(3.2)
∫ f (x) ln f (x)dx = S x
1 1 S = − E[ln f ] = E[− ln f ] = lim Mα (f ). α→δ η η δ−α η
Hence, what is done in Mα (f ) is to approximate − η1 ln f by f α−δ−1 , where δ is a fixed anchoring point, η is fixed and positive and α can vary, where α < δ, α > δ, α → δ. Consider a simple example. Let x = energy generated in a physical system. Then the physical law of conservation of energy can be stated as an expected value E in statistical terms, that is, E(x) is fixed in the density f (x) of x. That is, ∫ ∞ E(x) = xf (x)dx = fixed. 0
For example, if f (x) is the exponential density, f (x) = ce−cx , c > 0, x ≥ 0 then E(x) = 1c . Instead of E(x) = fixed, let us consider a slight disturbance and consider the following constraints: E[x
δ−α η
] = fixed and E[x 6
δ−α +1 η
] = fixed.
(3.3)
When α → δ then the two restrictions above are E(1) = 1 and E(x) is fixed or the law of conservation of energy. Hence in (3.3) we consider only a slight disturbance to the law of conservation of energy. Let us consider an arbitrary density f (x) and let us optimize Mα (f ) of (1.1) under the constraints in (3.3). Then proceeding as in the derivation from (2.1) to (2.6) we see that the Euler equation, if we use calculus of variations for the optimization, as the following: δ−α δ−α δ−α ∂ {f 1+ η − λ1 x η f + λ2 x1+ η f } = 0 ∂f
(3.4)
where λ1 and λ2 are the Lagrangian multipliers. Then (3.4) gives η
f1 (x) = c1 x[1 − a(δ − α)x] δ−α , α < δ, η > 0
(3.5)
for 1 − a(δ − α)x > 0, a > 0 and zero elsewhere, where λλ21 is taken as a(δ − α), a > 0 and c1 can act as the normalizing constant. For α > δ, (3.5) changes to the following: η
f2 (x) = c2 x[1 + a(α − δ)x]− α−δ , α > δ, a > 0, x ≥ 0
(3.6)
and zero elsewhere. When α → δ, both (3.5) and (3.6) go to f3 (x) = c3 xe−aηx , a > 0, η > 0, x ≥ 0
(3.7)
and zero elsewhere. This is Maxwell-Boltzmann’s energy density. Note that (3.5) to (3.7) give the pathway form of the Maxwell-Boltzmann density which is available by optimizing E[ δ−α
f
δ−α η −1
α−δ
] which is an approximation to E[− η1 ln f ], under the constraints δ−α
E[x η ] is fixed and E[x1+ η ] is fixed, which correspond to a slight disturbance 1 from the law of conservation of energy. Note that if in (3.7), aη = kT where k is Boltzmann’s constant and T is the temperature, then the densities in (3.5) to (3.7) change to the following: η 1 δ − α δ−α ( )x] , α < δ, η > 0, kT η ηkT 0≤x≤ δ−α η 1 α − δ − α−δ f21 (x) = c2 x[1 + ( )x] , α > δ, η > 0, x ≥ 0 kT η x f31 (x) = c3 xe− kT , x ≥ 0.
f11 (x) = c1 x[1 −
7
(3.8)
(3.9) (3.10)
Note that if a in (3.5) to (3.7) is taken as 1 then η = to (3.10) change to the following:
1 . kT
Then the densities in (3.8)
1
f12 (x) = c1 x[1 − (δ − α)x] kT (δ−α) , α < δ, 1 − (δ − α)x > 0 1 − kT (α−δ)
f22 (x) = c2 x[1 + (α − δ)x] x − kT
f32 (x) = c3 xe
, α > δ, x ≥ 0
, x ≥ 0.
(3.11) (3.12) (3.13)
In this case the restrictions can be stated as the following: E[xkT (δ−α) ] = fixed and E[xkT (δ−α)+1 ] = fixed, α < δ. These are only slight deviations from E(1) = 1 and E(x) = fixed. As before, the normalizing constants can be evaluated with the help of type-1 beta, type-2 beta and gamma integrals and they are the following: c1 = c2 = c3 =
1 (δ − α)2 Γ( kT (δ−α) + 3) 1 + 1) Γ( kT (δ−α) 1 (α − δ)2 Γ( kT (α−δ) ) 1 Γ( kT (α−δ)
− 2)
,α < δ
, α > δ,
1 −2>0 kT (α − δ)
1 . (kT )2
(3.14) (3.15) (3.16)
3.1. Differential pathway d ′ f12 (x) = f12 Consider the differential equation for (3.11). Denoting dx (x) we have the following: 1 1 ′ f12 (x) = f12 (x)[ − ], α < δ. (3.17) x kT [1 − (δ − α)x]
Similar differential equations corresponding to (3.15) and (3.16) can be derived. This will provide a differential pathway. If Shannon’s entropy is optimized subject to the constraint that the first moment is fixed then we automatically arrive at (3.1) or (3.13). 3.2. Evaluation of δ − α for α < δ From (3.11) or (3.14) the first moment E(x) is the following: ] [ 2 2kT − E(x) 1 E(x) = ⇒ (δ − α) = , α < δ. 1 δ − α kT (δ−α) + 3 3kT E(x) 8
(3.19)
But, through the constraint, E(x) is fixed. Then (δ − α) is evaluated in terms of E(x). Note that in the stable situation α → δ we have E(x) = kT . For α > δ, α−δ =
E(x) − 2kT . 3kT E(x)
(3.20)
4. Generalized Energy Density If Mα (f ) of (1.1) is optimized under the conditions E[xγ(
δ−α ) η
] = fixed and E[xγ(
δ−α )+ρ η
] = fixed
(4.1)
for some γ > 0, ρ > 0, then proceeding as in (3.4) to (3.7) one gets the following models: η (δ − α) ρ δ−α x ] , α < δ, a > 0, ρ > 0, γ > 0, η 1 η 0≤x≤[ ]ρ , a(δ − α) η (α − δ) ρ − α−δ , α > δ, a > 0, ρ > 0, γ > 0, x ≥ 0, g2 (x) = cˆ2 xγ [1 + a x ] η ρ g3 (x) = cˆ3 xγ e−ax , a > 0, ρ0, x ≥ 0.
g1 (x) = cˆ1 xγ [1 − a
(4.2)
(4.3) (4.4)
The models in (4.2) or (4.3) is the pathway model of Mathai (2005) for the real scalar positive variable case. For γ = ρ − 1 one has the power transformed version of the energy densities in (3.5) to (3.7). Then for γ = ρ − 1 we can call (4.2) to (4.4) as the Weibull forms of the energy densities. If γ = 1, ρ = 2 then we have extended Raleigh density in (4.1) and (4.2) and when α → δ it is the Raleigh density. If γ = 2, ρ = 2 then we have extended Maxwell-Botzmann density in (4.1) and (4.2), and when α → δ it is the Maxwell-Boltzmann density. 5. Multicomponent Energy Generation x1 .. Consider the matrix X = . . Let the total energy produced by X be a norm xp 1 of X, say ∥X∥. Taking the Euclidean norm ∥X∥ = (x21 + ... + x2p ) 2 , where x1 , ..., xp are the real components of X, we can look at the density of u = ∥X∥2 = x21 + ... + x2p . 9
Let dX = dx1 ∧ ... ∧ dxp the wedge product of the differentials dxj ’s. Then Mathai’s entropy (1.1) in this case is the following: ∫
δ−α
[f (X)]1+ η dX − 1 Mα (f ) = , α ̸= δ, η > 0, dX = dx1 ∧ ... ∧ dxp (5.1) α−δ ∫ and f is a density, that is f (X) ≥ 0 for all X and X f (X)dX = 1. Let us optimize (1.1) under the conditions X
E[uγ
(δ−α) η
] = fixed and E[uγ
(δ−α) +ρ η
] = fixed.
(5.2)
Observe that when α = δ the first condition is E(1) = 1 and the second condition is that the ρ-th moment is fixed. Going through the steps as in earlier sections the density f (X) is the following for the three cases α < δ, α > δ, α → δ, denoted by g1 (X), g2 (X), g3 (X) respectively: η δ − α ρ δ−α )u ] , α < δ, a > 0, η > 0, ρ > 0, γ > 0 η η δ−α 2 = C1 (x21 + ... + x2p )γ [1 − a( )(x1 + ... + x2p )ρ ] δ−α , α < δ, η 1 η ]ρ , 0 ≤ ∥X∥ ≤ [ a(δ − α) η α−δ 2 g2 (X) = C2 (x21 + ... + x2p )γ [1 + a( )(x1 + ... + x2p )ρ ]− α−δ , η α > δ, η > 0, a > 0, ∥X∥ ≥ 0,
g1 (X) = C1 uγ [1 − a(
g3 (X) = C3 (x21 + ... + x2p )γ e−aη(x1 +...+xp ) , a > 0, η > 0, ∥X∥ ≥ 0 2 ρ
2
(5.3)
(5.4)
(5.4)
where C1 , C2 , C3 are the normalizing constants. The densities in (5.3) to (5.5) are also connected with isotropic random points, type-1 beta, type-2 beta and gamma distributed, in geometrical probability problems, see for example Mathai (1999). Observe that ∥X∥2 is invariant under orthonormal transformations on X. That is, if Y = QX with QQ′ = I, Q′ Q = I where a prime denotes the transpose and I is the identity matrix, then Y ′ Y = X ′ X = ∥X∥2 = ∥Y ∥2 = y12 + ... + yp2 . In statistical problems, the components of X, namely xj ’s, may be correlated and one may want to make the components noncorrelated. Then we consider the trans1 formation Y = V − 2 X where the p × p matrix V > O (positive definite) is the 10
1
covariance matrix and V 2 denotes the positive definite square root to V . In this case 1 the norm ∥V − 2 X∥ = X ′ V −1 X = ∥Y ∥2 , or it is a positive definite quadratic form of the type X ′ AX, A = A′ > O where A = V −1 in the correlated case. Observe that X ′ AX = c > 0 is an ellipsoid in the p-space. In this situation, if we consider an orthonormal transformation X = Q′ Y, QQ′ = I, Q′ Q = I then X ′ AX = Y ′ QAQ′ Y = Y ′ DY = λ1 y12 + ... + λp yp2
(5.6)
where λj > 0, j = 1, ..., p are the eigenvalues of A > O. Then, instead of the spherically symmetric densities in (5.3),(5.4),(5.5) we end up with the elliptically contoured densities. If Mathai’s entropy Mα (f ) is optimized under the conditions δ−α δ−α E[v γ( η ) ] is fixed and E[v γ( η )+ρ ] is fixed, where v = X ′ AX then we have the following densities: η δ−α )(X ′ AX)ρ ] δ−α , α < δ, η 1 η η > 0, a > 0, γ > 0, ρ > 0, 0 ≤ X ′ AX ≤ [ ]ρ , a(δ − α) η α−δ g5 (X) = C5 (X ′ AX)γ [1 + a( )(X ′ AX)ρ ]− α−δ , α > δ η ′ a > 0, γ > 0, ρ > 0, X AX > 0,
g4 (X) = C4 (X ′ AX)γ [1 − a(
′
g6 (X) = C6 (X ′ AX)γ e−a(X AX) , a > 0, X ′ AX ≥ 0 ρ
(5.7)
(5.8)
(5.9)
where C4 , C5 , C6 are the normalizing constants. Let us evaluate the normalizing constant C1 in (5.7). This procedure will also hold for∫ the evaluation of C5 and C6 . From (5.7) the total probability is 1. Hence 1 = X g4 (X)dX. Consider the p+1 1 transformation A 2 X = Y then dX = |A|− 2 dY , see Mathai (1997) where |(·)| denotes the determinant of (·). Then − p+1 2
∫
1 = C4 |A|
(Y ′ Y )γ [1 − a(
Y
η δ−α )(Y ′ Y )ρ ] δ−α dY. η
Put Y ′ Y = s = r2 . Then p
p
π2 p π2 dY = p s 2 −1 ds = 2 p rp−1 dr, Γ( 2 ) Γ( 2 ) 11
see Mathai (1997). Then − p+1 2
∫
1 = C4 |A|
∞
r2γ [1 − a(
r=0
η δ − α 2ρ δ−α )r ] η
p 2
2π p−1 r dr, α < δ. Γ( p2 )
× Put
z = a(
1 1 δ − α 2ρ η )r ⇒ r = z 2ρ [ ] 2ρ , η a(δ − α)
and dr =
1 1 1 η [ ] 2ρ z 2ρ −1 dz. 2ρ a(δ − α) γ
r
2γ+p−1
p
γ p z ρ + 2ρ −1 η dr = [ ] ρ + 2ρ . 2ρ a(δ − α)
Integration over r gives
p η )Γ( δ−α + 1) 2ρ η γ p . Γ( δ−α + 1 + ρ + 2ρ )
Γ( γρ +
Therefore C4 =
|A|
p+1 2
ρπ
η Γ( p2 )Γ( δ−α +1+ p 2
Γ( γρ
+
p η )Γ( δ−α 2ρ
γ ρ
p ) 2ρ
+
+ 1)
[
a(δ − α) γρ + 2ρp , α < δ. ] η
(5.10)
For α > δ, follow through the same steps as above and then evaluate the integral by using a type-2 beta integral, then one has the following: C5 = η for α > δ, α−δ −
γ ρ
−
p 2ρ
|A|
p+1 2
η Γ( p2 )Γ( α−δ )
p
ρπ 2 Γ( γρ +
p ) 2ρ
p
γ
[a (α−δ) ] ρ + 2ρ η η Γ( α−δ −
γ ρ
−
p ) 2ρ
(5.11)
> 0 and
C6 =
|A|
p+1 2 p
γ
p
Γ( p2 )(a) ρ + 2ρ
ρπ 2 Γ( γρ +
p ) 2ρ
, a > 0, γ > 0, ρ > 0.
(5.12)
In all the above cases, A > O, γ > 0, ρ > 0, a > 0, η > 0. Note that the energy density is the case for γ = 1, p = 1. Recently, Mathai and Princy (2017) constructed matrix-variate analogues of the Maxwell-Boltzmann and Raleigh densities. 12
References A.M. Mathai (1997): Jacobians of Matrix Transformations and Functions of Matrix Argument, World Scientific Publishing, New York. A.M. Mathai (1999): An Introduction to Geometrical Probability: Distributions Aspects with Applications, Gordon and Breach, Amsterdam. A.M. Mathai (2005): A pathway to matrix-variate gamma and normal densities, Linear Algebra and its Applications, 396, 317-328. A.M. Mathai and T. Princy (2017): Multivariate and matrix-variate analogues of Maxwell-Boltzmann and Raleigh densities, Physica A, 468, 668-676.
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