Title: Non-Maxwellian molecular velocity distribution at large

Title: Non-Maxwellian Molecular Velocity Distribution at Large Knudsen Numbers
Author: Jae Wan Shim
Affiliation: KIST and University of Science and Technology 136-791 Seoul Korea
Abstract: We have derived a non-Maxwellian molecular velocity distribution at large Knudsen numbers
for ideal gas. This distribution approaches Maxwellian molecular velocity distribution as the Knudsen
number approaches zero. We have found that the expectation value of the square of velocity is the same in
the non-Maxwellian molecular velocity distribution as it is in the Maxwellian distribution; however, the
expectation value of the speed is not the same.
Keywords: molecular velocity distribution, large Knudsen number, non-Maxwellian distribution
Main Text:
A molecular velocity distribution at large Knudsen numbers is fundamental but still uncertain.
The regime of large Knudsen numbers includes molecular dynamics in micro chambers and interplanetary
space as well as in ultrahigh vacuum chambers [1, 2]. A molecular velocity distribution function for ideal
gas in an equilibrium state is commonly assumed to be the Maxwellian distribution at large Knudsen
numbers even though disagreement between the Maxwellian and experimental data exists [3, 4]. We have
rigorously derived a molecular velocity distribution function with respect to a number of molecules, N ,
which is applicable for molecules at large Knudsen numbers, and we have obtained the expectation values
of the square of velocity and of the speed. The non-Maxwellian molecular velocity distribution is
1
precisely not identical to the Maxwellian molecular velocity distribution, but it rapidly approaches it if
N increases; the non-Maxwellian molecular velocity distribution exactly matches the Maxwellian
molecular velocity distribution when N approaches infinity. The expectation value of the square of
velocity is the same in the non-Maxwellian molecular velocity distribution as it is in the Maxwellian;
however, the expectation value of the speed is not the same.
We will follow Cercignani’s derivation process [5]. Let us start our derivation by defining a
physical space, S , and a region, R  S , containing N molecules bounded by impermeable walls
against which the molecule's energy is preserved but its momentum is not after a collision. We can, then,
write the molecular velocity distribution function as
f ( v1 ,
m

, v N )  c  i vi  vi  E 
2


(1)
where v i is a velocity of the i-th molecule, c is a normalizing constant,  is the Dirac delta function,
m is the molecular mass, and E is the given energy of the system in R . Note that a relationship,
E
m
D
v  vi  NkT , exists where D is the dimension of the physical space, k is the

i i
2
2
Boltzmann constant, and T is the temperature.
For simplicity, let us consider a one-dimensional space. Then, Equation (1) becomes
f (v1 ,
, vN )  c
 v
i
i
2
E

(2)
where E  NkT / m . The normalizing constant c satisfies
  c   v
i
i
2

 E dv1
dvN  1 .
(3)
2
Let us use the polar coordinate system for the integration as vi  r j 1 sin  j cos i for
i 1
i  1,2,
, N  1 and vN  r j 1 sin  j , where r  [0, ] , i  [0, ] for i  1,2,
N 1
N 1  [0,2 ] . Then, we have
space can be written as dv1
v
2
i
i
, N  2 , and
 r 2 and the infinitesimal volume of the N -dimensional velocity
dvN  r N 1drdA N where A N is the surface area of the N -dimensional
unit ball and its value is A N  2 N /2 / ( N / 2) where  is the gamma function defined by
( N / 2)  2( N 1)/2 (n  2)!!  . Then, Equation (3) becomes



c   r 2  E r N 1dr
0
Therefore, the normalizing constant is c  2 E

N 2
2
 dA N  cA N E
N 2
2
/ 2 1.
(4)
/ AN .
The molecular velocity distribution function is symmetric with respect to the index changes of
the molecular velocities. This is reasonable because the molecules are identical. Hence, when we integrate
Equation (2) with respect to vi for all i except for i  N , then v N can be a representative velocity
describing the velocity of any molecule among N identical molecules. If we integrate Equation (2) by
F (vN )  
 f (v , , v ) dv dv
 cA    r  ( E  v )  r
1

N 1
0
N
N 1
1
2
2
N 2
N
dr
N 3
 A N 1  N2 2
E
( E  vN 2 ) 2 for v N 2  E

  AN

0 for v N 2  E

(5)

  N / 2 m
mv 2 N 3
(1  N ) 2 for v N 2  E

    ( N  1) / 2   NkT
NkT

0 for v N 2  E

or, for vN 2  E ,
F (v ) 
  N / 2 m
  ( N  1) / 2   NkT
(1 
mv 2 N23
)
NkT
(6)
3
by defining the representative velocity v N by v for simplicity. We can define g (v )  (1 
mv 2 N23
)
NkT
and, then, we have
N 3
mv 2
mv 2
ln(1 
)
N 
2
NkT
2kT
lim ln g (v)  lim
N 
or lim g (v )  e
N 

mv 2
2 kT
(7)
. We also have
lim
N 
  N / 2 m
  ( N  1) / 2   NkT

m
.
2 kT
(8)
Therefore, we can obtain a molecular velocity distribution when N approaches infinity by
1/2
2
mv
 m   2 kT
lim F (v )  
e

N 
 2 kT 
(9)
which is Maxwellian and we will call it FM (v ) .
Let us define a dimensionless velocity v  v
F (v ) 
m
. Then, we have
kT
  N / 2
F (v )
v 2 N 3

(1  ) 2
N
m / (kT )   ( N  1) / 2   N
(10)
and
2
FM (v ) 
FM (v )
1  v2

e .
m / (kT )
2
(11)
We draw F (v ) for N {3,4,5,10} and FM (v ) as Figure 1. By increasing N , F (v )
approaches FM (v ) .
For the case of the Maxwellian molecular velocity distribution, the expectation values of the
square of velocity and the square of the speed are obtained by EM (v 2 )   v 2 FM (v )dv 
EM ( v )   v FM (v )dv 
kT
and
m
2 kT
, respectively.
 m
Meanwhile, for the case of F (v ) , the expectations values are E(v 2 )   v 2 F (v )dv 
kT
and
m
4
E( v )   v F (v )dv 
N
( N / 2)
EM ( v ) , respectively.
2 (( N  1) / 2)
We define  ( N ) 
N
( N / 2)
and plot  ( N ) with respect to N . The ratio between
2 (( N  1) / 2)
E( v ) and EM ( v ) rapidly approaches 1 as N increases in Figure 2.
In conclusion, we confirm that molecular velocity distribution can be approximated by the
Maxwellian molecular velocity distribution in conventional vacuum chambers. Because they normally
contain more than 1000 molecules, and in these cases, E( v ) very closely approaches EM ( v ) .
Otherwise, a system having a small number of molecules should be considered using the molecular
velocity distribution derived in this paper, which can be considered as an equilibrium distribution at large
Knudsen numbers in discrete kinetic theory [6-8]. When comparing the non-Maxwellian molecular
velocity distribution to the Maxwellian molecular velocity distribution, the expectation value of the
square of velocity is identical; however, the expectation value of the speed of the non-Maxwellian
molecular velocity distribution can be calculated by multiplying a factor
N / 2 ( N / 2) (( N  1) / 2)
to the expectation value of the speed of the Maxwellian.
References:
[1] M. Knudsen, The Kinetic Theory of Gases, Metheun, London, 1950
[2] A. Chambers, Modern Vacuum Physics, CRC Press, Boca Raton. 2004
5
[3] W. Jitschin, G. Reich, J. Vac. Sci. Technol. A 9 (1991) 2752.
[4] J. Denur, Found. Phys. 37 (2007) 1685
[5] C. Cercignani, Mathematical Methods in Kinetic Theory, second ed., Plenum Press, New York, 1990.
[6] J. W. Shim, R. Gatignol, Z. Angew. Math. Phys. (2012) DOI 10.1007/s00033-012-0265-1
[7] J. W. Shim, R. Gatignol, Phys. Rev. E 83 (2011) 046710
[8] J. W. Shim, R. Gatignol, Phys. Rev. E 81 (2010) 046703
Figure Captions:
Figure 1. Graphs of F (v ) for N {3,4,5,10} and FM (v ) . The linear line (blue) is the graph of N  3
and the curved lines (blue) are the graph of N {4,5,10} . The dashed line (red) is the graph of FM (v ) .
Note that F (v ) approaches FM (v ) by increasing N .
Figure 2. Log-linear plot of  ( N ) with respect to N . The ratio  ( N ) rapidly approaches 1 as N
increases.
Figure 1.
6
Figure 2.
7

Download Report