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The Maxwell-Boltzmann (MB) distribution for velocities in ideal gases is usually defined between zero and infinity. A double truncated MB distribution is here introduced and the probability density function, the distribution function, the average value, the rth moment about the origin, the root-mean-square speed and the variance are evaluated. Two applications are presented: 1) a numerical relationship between root-mean-square speed and temperature, and 2) a modification of the formula for the Jeans escape flux of molecules from an atmosphere.

The Maxwell-Boltzmann (MB) distribution, see [

1) The maximum velocity of a gas cannot be greater than the velocity of light, c.

2) The kinetic theory is developed in a classical environment, which means that the involved velocities should be smaller than ≈1/10c.

These items point toward the hypothesis of an upper bound in velocity for the MB. We will now report some approaches, including an upper bound in velocity: the ion velocities parallel to the magnetic field in a low density surface of a ionized plasma [

This paper is structured as follows. Section 2 reviews the basic statistics of the MB distribution and it derives a new approximate expression for the median. Section 3 introduces the double truncated MB and it derives the connected statistics. Section 4 derives the relationship for root-mean-square speed versus temperature in the double truncated MB. Finally, Section 5.2 derives a new formula for Jeans flux in the exosphere.

Let V be a random variable defined in [ 0, ∞ ] ; the MB probability density function (PDF), f ( v ; a ) , is

f ( v ; a ) = 2 v 2 e − 1 2 v 2 a 2 π a 3 , (1)

where a is a parameter and v denotes the velocity, see [

a = k T m , (2)

where m is the mass of the gas molecules, k is the Boltzmann constant and T is the thermodynamic temperature. With this change of variable, the MB PDF is

f p ( v ; m , k , T ) = 2 v 2 e − 1 2 v 2 m k T π ( k T m ) 3 2 , (3)

where the index p stands for physics. The distribution function (DF), F ( x ; a ) , is

F ( v ; a ) = 2 a 2 ( a π 2 erf ( 1 2 2 v a ) − 2 v e − 1 2 v 2 a 2 ) 2 π a 3 (4)

F p ( v ) = 2 ( ( k T m ) 3 2 π 2 erf ( 1 2 2 v 1 k T m ) m − 2 v e − 1 2 v 2 m k T k T ) 2 π ( k T m ) 3 2 m . (5)

The average value or mean, μ , is

μ ( a ) = 2 2 a π , (6)

μ ( m , k , T ) p = 2 2 π k T m , (7)

the variance, σ 2 , is

σ 2 ( a ) = a 2 ( − 8 + 3 π ) π (8)

σ 2 ( m , k , T ) p = k T ( − 8 + 3 π ) m π . (9)

The rth moment about the origin for the MB distribution is, μ ′ r , is

μ ′ r ( a ) = 2 r / 2 + 1 a r Γ ( r / 2 + 3 2 ) π (10)

μ ′ r ( m , k , T ) p = 2 r / 2 + 1 ( k T m ) r Γ ( r / 2 + 3 2 ) π , (11)

where

Γ ( z ) = ∫ 0 ∞ e − t t z − 1 d t , (12)

is the gamma function, see [

v r m s ( a ) = 3 a (13)

v r m s ( m , k , T ) p = 3 k T m , (14)

see Equations (7-10-16) in [

T = 1 3 v r m s 2 m k . (15)

The coefficient of variation (CV) is

C V = σ ( a ) μ ( a ) = 3 8 π − 1 , (16)

which is constant. The first three rth moments about the mean for the MB distribution, μ r ( a ) , are

μ 2 ( a ) = a 2 ( − 8 + 3 π ) π (17)

μ 3 ( a ) = − 2 a 3 2 ( 5 π − 16 ) π 3 / 2 (18)

μ 4 ( a ) = a 4 ( 15 π 2 + 16 π − 192 ) π 2 . (19)

The mode is at

v ( a ) = 2 a (20)

v ( m , k , T ) p = 2 k T m . (21)

An approximate expression for the median can be obtained by a Taylor series of the DF around the mode. The approximation formula is

v ( a ) = − 1 4 a ( − 6 + e ( erf ( 1 ) − 1 2 ) π ) 2 , (22)

v ( m , k , T ) p = − 1 4 k T m ( − 6 + e ( erf ( 1 ) − 1 2 ) π ) 2 , (23)

which has a percent error, δ , of δ ≈ 0.04 % in respect to the numerical value. The entropy is

l n ( 2 π a ) − 1 2 + γ , (24)

l n ( 2 π k T m ) − 1 2 + γ , (25)

where γ is the Euler-Mascheroni constant, which is defined as

γ = l i m n → ∞ ( 1 + 1 2 + 1 3 + ⋯ + 1 n − l n n ) = 0.57721 ⋯ , (26)

see [

( − 10 π + 32 ) 2 ( − 8 + 3 π ) 3 2 ≈ 0.48569, (27)

and the coefficient of kurtosis is

15 π 2 + 16 π − 192 ( − 8 + 3 π ) 2 ≈ 3.10816. (28)

According to [

F ( v ; a ) − R = 0, (29)

F ( v ; m , k , T ) p − R = 0, (30)

where F ( v ) and F p ( v ) are the two DF represented by Equations (4) and (5). As a practical example, by inserting in Equation (29) a = 1 and R = 0.5 , we obtain in a numerical way v = 1.538 .

Let V be a random variable that is defined in [ v l , v u ] ; the double truncated version of the Maxwell-Boltzmann PDF, f t ( v ; a , v l , v u ) , is

f t ( v ; a , v l , v u ) = v 2 e − 1 2 v 2 a 2 , (31)

where

C = − 2 C D , (32)

where

C D = a 2 ( − a π 2 erf ( 1 2 2 v u a ) + a π 2 erf ( 1 2 2 v l a ) + 2 v u e − 1 2 v u 2 a 2 − 2 v l e − 1 2 v l 2 a 2 ) , (33)

and erf ( x ) is the error function, which is defined as

erf ( x ) = 2 π ∫ 0 x e − t 2 d t , (34)

see [

F t ( v ; a , v l , v u ) = C a 2 ( π 2 a erf ( 1 2 2 v a ) − 2 v e − 1 2 v 2 a 2 ) 2 . (35)

The average value μ t ( a , v l , v u ) , is

μ t ( a , v l , v u ) = C a 2 ( 2 e − 1 2 v l 2 a 2 a 2 − 2 e − 1 2 v u 2 a 2 a 2 + e − 1 2 v l 2 a 2 v l 2 − e − 1 2 v u 2 a 2 v u 2 ) . (36)

The rth moment about the origin for the double truncated MB distribution is, μ ′ r , t ( a , v l , v u ) ,

μ ′ r , t ( a , v l , v u ) = M N r + 3 (37)

where

M N = C 2 r 4 + 5 4 a 2 × ( ( v u 2 a 2 ) − r 4 − 1 4 v u r + 1 e − 1 4 v u 2 a 2 M r 4 + 1 4 , r 4 + 3 4 ( 1 2 v u 2 a 2 ) − v l r + 1 e − 1 4 v l 2 a 2 M r 4 + 1 4 , r 4 + 3 4 ( 1 2 v l 2 a 2 ) ( v l 2 a 2 ) − r 4 − 1 4 ) (38)

where M μ , ν ( z ) is the Whittaker M function, see [

v r m s , t ( a , v l , v u ) = N V 5 ( v u 2 a 2 ) 3 / 4 ( v l 2 a 2 ) 3 / 4 , (39)

where

N V = 2 C 2 3 / 4 a 2 ( v u 3 e − 1 / 4 v u 2 a 2 M 3 / 4 , 5 / 4 ( 1 / 2 v u 2 a 2 ) ( v l 2 a 2 ) 3 / 4 − v l 3 e − 1 / 4 v l 2 a 2 M 3 / 4 , 5 / 4 ( 1 / 2 v l 2 a 2 ) ( v u 2 a 2 ) 3 / 4 ) . (40)

The variance σ t 2 ( a , v l , v u ) is defined as

σ t 2 ( a , v l , v u ) = μ ′ 2, t ( a , v l , v u ) − ( μ ′ 1, t ( a , v l , v u ) ) 2 (41)

and has the following explicit form

σ t 2 ( a , v l , v u ) = 4 ( ( ( v l + 2 v u ) a 2 + v l v u ( v l + 1 2 v u ) ) ( a 2 + 1 / 2 v u 2 ) C 2 a 4 e − 1 2 v l 2 + 2 v u 2 a 2 − 2 ( ( v l + 1 2 v u ) a 2 + 1 4 v l v u ( v l + 2 v u ) ) C 2 a 4 ( a 2 + 1 2 v l 2 ) e − 1 2 2 v l 2 + v u 2 a 2 + ( a 2 + 1 2 v u 2 ) ( C erf ( 1 2 2 v l a ) a 3 2 π − C erf ( 1 2 2 v u a ) a 3 2 π + 4 ) C a 2 ( a 2 + 1 2 v l 2 ) e − 1 2 v l 2 + v u 2 a 2

+ C 2 a 4 ( a 2 + 1 2 v l 2 ) 2 v l e − 3 2 v l 2 a 2 − ( a 2 + 1 2 v u 2 ) 2 C 2 a 4 v u e − 3 2 v u 2 a 2 + ( 3 4 a 2 v l + 1 4 v l 3 ) e − 1 2 v l 2 a 2 + ( − 3 4 a 2 v u − 1 4 v u 3 ) e − 1 2 v u 2 a 2 − 1 2 ( ( C erf ( 1 2 2 v l a ) a 3 2 π − C erf ( 1 2 2 v u a ) a 3 2 π + 4 ) C ( a 2 + 1 2 v l 2 ) 2 e − v l 2 a 2

+ ( a 2 + 1 2 v u 2 ) 2 ( C erf ( 1 2 2 v l a ) a 3 2 π − C erf ( 1 2 2 v u a ) a 3 2 π + 4 ) C e − v u 2 a 2 + 3 4 π 2 ( − erf ( 1 2 2 v u a ) + erf ( 1 2 2 v l a ) ) a ) a 2 ) C a 2 . (42)

Although the coefficients of skewness and kurtosis for the truncated MB exist, they have a complicated expression.

The temperature as a function of root-mean-square speed for the MB is given by Equation (15). In the truncated MB distribution, the temperature can be found by solving the following nonlinear equation

v r m s , t ( k , m , T , v l , v u ) = v r m s , m , (43)

where v r m s , m is not a theoretical variable but is the root-mean-square speed measured in the laboratory and v r m s , t is given by Equation (39). The laboratory measures of v r m s , m started with [^{−26} kg. In

The second example is given by diatomic nitrogen, N_{2}, in which molecular mass is 4.651737684 × 10^{−26} kg. In

The standard formula for the escape of molecules from the exosphere is reviewed in the framework of the MB distribution. A new formula for the Jeans escape is derived in the framework of the truncated MB.

In the exosphere, a molecule of mass m and velocity v e is free to escape when

1 2 m v e 2 − G M m R e x = 0, (44)

where G is the Newtonian gravitational constant, M is the mass of the Earth, R e x = R + H is the radius of the exosphere, R is the radius of the Earth and H is the altitude of the exosphere. The flux of the molecules that are living in the exosphere Φ j is

Φ j = 1 4 N e x μ e , (45)

where N e x is the number of molecules per unit volume and μ e is the average velocity of escape. In the presence of a given number of molecules per unit volume, the standard MB distribution in velocities in a unit volume, f m , is

f m ( v ; m , k , T , N e x ) = N e x 2 v 2 e − 1 2 v 2 m k T π ( k T m ) 3 2 . (46)

The average value of escape is defined as

μ e = ∫ v e ∞ v f m ( v ; m , k , T , N e x ) d v ∫ 0 ∞ f m ( v ; m , k , T , N e x ) d v . (47)

In this integral, the following changes are made to the variables

λ = 1 2 m v 2 k T . (48)

Therefore,

μ e = 2 ( λ e + 1 ) e − λ e 2 k T π m , (49)

with

λ e = 2 G M R e x v 0 2 , (50)

where v 0 is the mode as represented by Equation (21). The flux is now

Φ j = N e x ( λ e + 1 ) e − λ e v 0 2 π . (51)

For more details see [

Φ j = 3.98 × 10 11 molecules ⋅ m − 2 ⋅ s − 1 , (52)

and

λ e = 7.78. (53)

The Jeans escape flux for Earth at T = 900 K varies between Φ j ≈ 2.7 × 10 11 molecules ⋅ m − 2 ⋅ s − 1 ; see [

The average value of escape for a truncated MB distribution, μ e , t , is

μ e , t = ∫ v e ∞ v f t ( v ; m , k , T , N e x , v l , v u ) d v ∫ 0 ∞ f m ( v ; m , k , T , N e x , v l , v u ) d v . (54)

This integral can be solved by introducing the change of variable as given by Equation (48)

μ e , t = − 2 ( ( λ u + 1 ) e − λ u − e − λ e ( λ e + 1 ) ) 2 2 λ l e − λ l − 2 λ u e − λ u − π erf ( λ l ) + π erf ( λ u ) k T m , (55)

where λ l is the lower value of λ and λ u is the upper value of λ . The flux of the molecules that are living the exosphere in the truncated MB, Φ j , t , is

Φ j , t = N e x ( ( λ u + 1 ) e − λ u − e − λ e ( λ e + 1 ) ) 2 4 λ u e − λ u + 2 π erf ( λ l ) − 2 π erf ( λ u ) − 4 λ l e − λ l k T m . (56)

The increasing flux of molecules is outlined when one parameter, λ l , is variable; see

Parameter | Value |
---|---|

R_{ex} | 6900 km |

T | 900 K |

N_{ex} | 10^{11} m^{−3} |

These Jeans escape fluxes for Earth are compatible with the observed values that were reported in Section 5.1.

This paper derived analytical formulae for the following quantities for a double truncated MB distribution: the PDF, the DF, the average value, the rth moment about the origin, the root-mean-square speed and the variance. The traditional correspondence between root-mean-square speed and temperature is replaced by the nonlinear Equation (43). The new formula (56) for the Jeans escape flux of molecules from an atmosphere is now a function of the lower and upper boundary in velocity.

The author declares no conflicts of interest regarding the publication of this paper.

Zaninetti, L. (2020) New Probability Distributions in Astrophysics: III. The Truncated Maxwell-Boltzmann Distribution. International Journal of Astronomy and Astrophysics, 10, 191-202. https://doi.org/10.4236/ijaa.2020.103010