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 IJAA  Vol.10 No.3 , September 2020
New Probability Distributions in Astrophysics: III. The Truncated Maxwell-Boltzmann Distribution
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Abstract: 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.

1. Introduction

The Maxwell-Boltzmann (MB) distribution, see [1] [2], is a powerful tool to explain the kinetic theory of gases. The range in velocity of this distribution spans the interval [ 0, ] , which produces several problems:

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 [3]; propagation of longitudinal electron waves in a collisionless, homogeneous, isotropic plasma, whose velocity distribution function is a truncated MB [4]; fast ion production in laser plasma [5]; the release of a dust particle from a plasma-facing wall [6]; an explanation of an anomaly in the Dark Matter (DAMA) experiment [7]; a distorted MB distribution of epithermal ions observed associated with the collapse of energetic ions [8]; and deviations to MB distribution that could have observable effects which can be measured trough the vapor spectroscopy at an interface [9]. However, these approaches do not clearly cover the effect of introducing a lower and an upper boundary in the MB distribution, which is the subject that will be analyzed in this paper.

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.

2. The Maxwell-Boltzmann Distribution

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 [1] [2]. Conversion to the physics is done by introducing the variable a, which is defined as

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 [10]. The root-mean-square speed , v r m s , can be obtained from this formula by inserting r = 2

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 [11]. This equation allows us to derive the temperature once the root-mean-square speed is measured

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] for more details. The coefficient of skewness is

( 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 [12], a random number generation can be obtained via inverse transform sampling when the distribution function or cumulative distribution function, F ( x ) , is known: 1) a pseudo number generator gives a random number R between zero and one; 2) the inverse function x = F 1 ( R ) is evaluated; and 3) the procedure is repeated for different values of R. In our case, the inverse function should be evaluated in a numerical way by solving for v the following nonlinear equation

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 .

3. The Double Truncated Maxwell-Boltzmann Distribution

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 [10]. The physical meaning of a is still represented by Equation (2); however, due to the tendency to obtain complicated expressions, we will omit the double notation. The DF, F t ( v ; a , v l , v u ) , is

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 [10]. The root-mean-square speed, v r m s , t ( a , v l , v u ) , can be obtained from this formula by inserting r = 2 , and is

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.

4. A Laboratory Application

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 [13], where a v r m s , m = 388 m / s at 400˚C was found for a metallic vapor. In the truncated MB distribution, there are three parameters that can be measured in the laboratory from a kinematical point of view, as follows: the lowest velocity, v l ; the highest velocity, v u ; and the root-mean-square speed, v r m s , m . Setting for simplicity v l = 0 , we will now explore the effect of the variation of v u on the root-mean-square speed; see Figure 1. The first example of the influence of the upper limit in velocity on the temperature is given by potassium gas [14] [15], in which molecular mass is 6.492429890 × 10−26 kg. In Figure 2, we evaluate in a numerical way the temperature when v l = 0 and v u is variable in the case of a measured value of v r m s , m .

The second example is given by diatomic nitrogen, N2, in which molecular mass is 4.651737684 × 10−26 kg. In Figure 3, we evaluate the temperature when v l = 0 and v u is a variable in the case of a measured value of v r m s , m .

5. The Jeans Escape

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.

5.1. The Standard Case

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

Figure 1. The theoretical root-mean-square speed as a function of the upper limit in velocity (continuous line) and standard value of the temperature (dotted line) when a = 340 and v l = 0 .

Figure 2. Temperature as a function of the upper limit in velocity for Potassium (continuous line) and standard value of the temperature (dotted line) when v l = 0 and v r m s , m = 589.111511 m / s .

Figure 3. Temperature as a function of the upper limit in velocity for diatomic nitrogen, N2, (continuous line) and standard value of the temperature (dotted line) when v l = 0 and v r m s , m = 695.9756308 m / s .

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 [16] [17] [18] [19]. On adopting the parameters of Table 1 the Jeans escape flux for hydrogen is

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 [20] or Figure 1 in [21]. and Φ j 4 × 10 11 molecules m 2 s 1 , see [22]. Therefore, our choice of parameters is compatible with the suggested interval in flux.

5.2. The Truncated Case

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 Figure 4. In other words, an increase in λ l produces an increase in the flux of the molecules. The dependence of the flux when two parameters are variable, λ l and λ u , is reported in Figure 5.

Table 1. Adopted physical parameters for the exosphere.

Figure 4. The flux of molecules as a function of λ l with parameters as in Table 1, λ e = 7.78 and λ u = 1000 λ e .

Figure 5. The flux of molecules as a function of λ l and λ u with parameters as in Table 1.

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

6. Conclusion

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.

Cite 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. doi: 10.4236/ijaa.2020.103010.
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