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Behavior of the H-function at a local Maxwellian state
Physica 43 (1969) 29-32. © North-Holland Publishing Co., Amsterdam
BEHAVIOR
OF THE H-FUNCTION
AT A LOCAL MAXWELLIAN
STATE
J. H. F E R Z I G E R * Mathematisch Instituut, Rijksuniversiteit te Groningen, Groningen, Nederland Received 26 July 1968
Synopsis It is shown that when a gas in a finite container passes through a local Maxwellian state the H-function for the system takes on a horizontal inflection point. A corollary to this is that dH/dt = 0 is not a sufficient condition for equilibrium. I n the p r o o f of the H - t h e o r e m 1) for a gas in a finite container, one finds t h a t sufficient conditions for d H / d t ---- 0 are t h a t the d i s t r i b u t i o n function / be a local Maxwellian a n d t h a t no mass, m o m e n t u m , or e n e r g y is t r a n s ferred f r o m the gas to the b o u n d a r y . F u r t h e r m o r e , w h e n the d i s t r i b u t i o n f u n c t i o n is locally Maxwellian, H reduces to - - ( l / k ) t i m e s the t h e r m o d y n a m i c e n t r o p y of the s y s t e m H = S n(r)
[log n(r)
-
~ log T(r)] d3r
+ const.
(1)
Now, it is well k n o w n f r o m t h e r m o d y n a m i c s t h a t the e n t r o p y is m a x i m i z e d (for fixed t o t a l n u m b e r of particles in the s y s t e m a n d fixed t o t a l energy) w h e n n(r) a n d T(r) are c o n s t a n t s . T h e question t h u s arises as to w h y the H - t h e o r e m does not yield this result. G r a d 2) has a t t e m p t e d to resolve this question b y showing t h a t a gas will n e v e r r e a c h a local Maxwellian state. This, however, begs the question. I n this n o t e we will a t t e m p t to answer w h a t h a p p e n s if a gas does r e a c h a local Maxwellian. I t will be s h o w n t h a t w h e n a s y s t e m passes t h r o u g h such a state, H , r e g a r d e d as a function of time, t a k e s on a horizontal inflection point. An interesting corollary to this result is t h a t dH/dt = 0 is not a sufficient condition for t h e r m a l equilibrium, a l t h o u g h it is clearly necessary. To p r o v e the a b o v e result it is sufficient to show t h a t w h e n a s y s t e m passes t h r o u g h a local Maxwellian s t a t e : (a) dH/dt = 0, (b) dZH/dt 2 = O, * On leave during the academic year 1967-'68 from Department of Mechanical Engineering, Stanford University, Stanford, California, USA. 29
30
J . H . FERZIGER
and (c) d3H/dt 3 < 0. The first of these has, of course, been proven b y other authors but is very simply shown again. We start with the Boltzmann equation:
~t
-
c.V/+
Y(/,/)
(2)
and the definition of the H-function n ( t ) ---- ~ dSr d3c ](r, c, t) log/(r, c, t).
(3)
Since our purpose is merely to construct an example, we shall assume, for simplicity, that the hydrodynamic velocity is zero in the local Maxwellian state. To obtain (a) we differentiate eq. (3) with respect to t to obtain the wellknown result: dH dt
dar dac (1 + log ]) ~t "
(4)
Again, substituting ~]/8t = - - c . 17] which is valid for the local Maxwellian, we can, with the aid of Green's theorem, write eq. (4) as dt --
n . c ] log I d2S dac,
(5)
which is zero b y virtue of ] being MaxweUian; n is a unit normal at the surface. Next, to prove (b) we differentiate eq. (4) again d2Hdt 2 --
d3rd3c (1 + l o g ] ) ~ - ~ - +
] \~t]i
(6)
1,~-
(7)
and we differentiate eq. (2) :
~9/
-
c.
V
+j
,
/ +j
Substituting eq. (7) in eq. (6) and once again using 8]/bt : - - c . 17], we find that the terms not involving J can be written as a divergence, and again using Green's theorem and the fact that ] is Maxwellian, one finds that these terms give a contribution ~ ( A n . V n + B n . V T ) dS, where A and B are constants. Then, if we assume that n . V n and n . [TT are zero at the surface, which is consistent with Fick's and Fourier's laws and the assumptions of no mass or energy transfer to the boundary, these terms yield a zero contribution. That the terms involving J are likewise zero follows from the fact that 1 + log ] is a summational invariant.
BEHAVIOR
OF THE H-FUNCTION
AT A LOCAL MAXWELLIAN
STATE
3
Differentiating eq. (6) once more, we obtain:
daH-- f dardac [ (1 + l o g / ) ~ -aa/+
a/ 73 ~/
a2/
and we shall also need 8t3 --
c.r~-
+Jkst
2 ,l
q-J
/, at2] ÷ 2 J
st'
~
"
(9)
When the expressions for the various derivatives are substituted into eq. (8), there results a rather complicated expression which m a y be simplified by treating various parts separately. First, we find that the terms which do not contain any J operators can again be reduced to a divergence and, subsequently, by Gauss's theorem, to
n. ccc: V( 1 + log/) [7] d2S d3c.
(10)
The integrand is odd in all components of c and is therefore zero. Next there are terms which yield f(1 + log/)[j
\(a~/at2 , ] ) + J (], a2j2lst ,1 + 2.] (\ ats]' ~ ) ] dar d3c" (11)
This is also zero, because 1 + l o g / is a summational invariant and the term in brackets is symmetric in Cl and c, and antisymmetric under an interchange of the primed and unprimed variables. Finally, there is the term
--3f c'Vlog/[J (~t , /) + J (/, [~t)l dard3c.
(12)
Now, using b//Ot : --v.V/ and v.V/ = [c.Vn/n + v(c2 -- 4)'VT/T~ ], we find that the terms in lTn vanish because c is a summational invariant. The remaining terms m a y be written, in the notation of Chapman and Cowling 3) :
3 f cS~1)(c2) VT X
VT ~
-- 4
d3cd~Cl d3r :
~2~2,2)(T) dSr < 0,
32
BEHAVIOR OF THE H-FUNCTION AT A LOCAL MAXWELLIAN STATE
where ~2(2,2) is the Q-integral related to the t h e r m a l c o n d u c t i v i t y . This completes the proof t h a t H has a h o r i z o n t a l inflection point. T h u s it has been shown t h a t dH/dt = 0 is not a sufficient condition for equilibrium. T o assure thatTan equilibrium solution is obtained, one m u s t f u r t h e r require t h a t H take its m i n i m u m value with respect to restricted v a r i a t i o n s of n(r) and T(r) or t h a t ~]/~t also be zero. Acknowledgments. The a u t h o r wishes to t h a n k the M a t h e m a t i s c h Instituut, Rijksuniversiteit te Groningen for its hospitality during the y e a r 1967-8 and, in particular, Dr. H. G. K a p e r , in discussion with w h o m the p r o b l e m arose. The a u t h o r wishes also to t h a n k the U n i t e d States E d u cational F o u n d a t i o n in the N e t h e r l a n d s for a F u l b r i g h t grant t h a t m a d e the s t a y possible.
REFERENCES
I) DarrozSs, J. and Guiraud, J. P., Comptes Rendus Acad. Sci. 262 (1966) 1368. 2) Grad, H., J. Soc. Ind. Appl. Math. 13 (1965) 259. 3) Chapman, S. and Cowling, T. G., Mathematical Theory of Non-Uniform Gases, Second ed., Cambridge Univ. Press, 1952.
BEHAVIOR
OF THE H-FUNCTION
AT A LOCAL MAXWELLIAN
STATE
J. H. F E R Z I G E R * Mathematisch Instituut, Rijksuniversiteit te Groningen, Groningen, Nederland Received 26 July 1968
Synopsis It is shown that when a gas in a finite container passes through a local Maxwellian state the H-function for the system takes on a horizontal inflection point. A corollary to this is that dH/dt = 0 is not a sufficient condition for equilibrium. I n the p r o o f of the H - t h e o r e m 1) for a gas in a finite container, one finds t h a t sufficient conditions for d H / d t ---- 0 are t h a t the d i s t r i b u t i o n function / be a local Maxwellian a n d t h a t no mass, m o m e n t u m , or e n e r g y is t r a n s ferred f r o m the gas to the b o u n d a r y . F u r t h e r m o r e , w h e n the d i s t r i b u t i o n f u n c t i o n is locally Maxwellian, H reduces to - - ( l / k ) t i m e s the t h e r m o d y n a m i c e n t r o p y of the s y s t e m H = S n(r)
[log n(r)
-
~ log T(r)] d3r
+ const.
(1)
Now, it is well k n o w n f r o m t h e r m o d y n a m i c s t h a t the e n t r o p y is m a x i m i z e d (for fixed t o t a l n u m b e r of particles in the s y s t e m a n d fixed t o t a l energy) w h e n n(r) a n d T(r) are c o n s t a n t s . T h e question t h u s arises as to w h y the H - t h e o r e m does not yield this result. G r a d 2) has a t t e m p t e d to resolve this question b y showing t h a t a gas will n e v e r r e a c h a local Maxwellian state. This, however, begs the question. I n this n o t e we will a t t e m p t to answer w h a t h a p p e n s if a gas does r e a c h a local Maxwellian. I t will be s h o w n t h a t w h e n a s y s t e m passes t h r o u g h such a state, H , r e g a r d e d as a function of time, t a k e s on a horizontal inflection point. An interesting corollary to this result is t h a t dH/dt = 0 is not a sufficient condition for t h e r m a l equilibrium, a l t h o u g h it is clearly necessary. To p r o v e the a b o v e result it is sufficient to show t h a t w h e n a s y s t e m passes t h r o u g h a local Maxwellian s t a t e : (a) dH/dt = 0, (b) dZH/dt 2 = O, * On leave during the academic year 1967-'68 from Department of Mechanical Engineering, Stanford University, Stanford, California, USA. 29
30
J . H . FERZIGER
and (c) d3H/dt 3 < 0. The first of these has, of course, been proven b y other authors but is very simply shown again. We start with the Boltzmann equation:
~t
-
c.V/+
Y(/,/)
(2)
and the definition of the H-function n ( t ) ---- ~ dSr d3c ](r, c, t) log/(r, c, t).
(3)
Since our purpose is merely to construct an example, we shall assume, for simplicity, that the hydrodynamic velocity is zero in the local Maxwellian state. To obtain (a) we differentiate eq. (3) with respect to t to obtain the wellknown result: dH dt
dar dac (1 + log ]) ~t "
(4)
Again, substituting ~]/8t = - - c . 17] which is valid for the local Maxwellian, we can, with the aid of Green's theorem, write eq. (4) as dt --
n . c ] log I d2S dac,
(5)
which is zero b y virtue of ] being MaxweUian; n is a unit normal at the surface. Next, to prove (b) we differentiate eq. (4) again d2Hdt 2 --
d3rd3c (1 + l o g ] ) ~ - ~ - +
] \~t]i
(6)
1,~-
(7)
and we differentiate eq. (2) :
~9/
-
c.
V
+j
,
/ +j
Substituting eq. (7) in eq. (6) and once again using 8]/bt : - - c . 17], we find that the terms not involving J can be written as a divergence, and again using Green's theorem and the fact that ] is Maxwellian, one finds that these terms give a contribution ~ ( A n . V n + B n . V T ) dS, where A and B are constants. Then, if we assume that n . V n and n . [TT are zero at the surface, which is consistent with Fick's and Fourier's laws and the assumptions of no mass or energy transfer to the boundary, these terms yield a zero contribution. That the terms involving J are likewise zero follows from the fact that 1 + log ] is a summational invariant.
BEHAVIOR
OF THE H-FUNCTION
AT A LOCAL MAXWELLIAN
STATE
3
Differentiating eq. (6) once more, we obtain:
daH-- f dardac [ (1 + l o g / ) ~ -aa/+
a/ 73 ~/
a2/
and we shall also need 8t3 --
c.r~-
+Jkst
2 ,l
q-J
/, at2] ÷ 2 J
st'
~
"
(9)
When the expressions for the various derivatives are substituted into eq. (8), there results a rather complicated expression which m a y be simplified by treating various parts separately. First, we find that the terms which do not contain any J operators can again be reduced to a divergence and, subsequently, by Gauss's theorem, to
n. ccc: V( 1 + log/) [7] d2S d3c.
(10)
The integrand is odd in all components of c and is therefore zero. Next there are terms which yield f(1 + log/)[j
\(a~/at2 , ] ) + J (], a2j2lst ,1 + 2.] (\ ats]' ~ ) ] dar d3c" (11)
This is also zero, because 1 + l o g / is a summational invariant and the term in brackets is symmetric in Cl and c, and antisymmetric under an interchange of the primed and unprimed variables. Finally, there is the term
--3f c'Vlog/[J (~t , /) + J (/, [~t)l dard3c.
(12)
Now, using b//Ot : --v.V/ and v.V/ = [c.Vn/n + v(c2 -- 4)'VT/T~ ], we find that the terms in lTn vanish because c is a summational invariant. The remaining terms m a y be written, in the notation of Chapman and Cowling 3) :
3 f cS~1)(c2) VT X
VT ~
-- 4
d3cd~Cl d3r :
~2~2,2)(T) dSr < 0,
32
BEHAVIOR OF THE H-FUNCTION AT A LOCAL MAXWELLIAN STATE
where ~2(2,2) is the Q-integral related to the t h e r m a l c o n d u c t i v i t y . This completes the proof t h a t H has a h o r i z o n t a l inflection point. T h u s it has been shown t h a t dH/dt = 0 is not a sufficient condition for equilibrium. T o assure thatTan equilibrium solution is obtained, one m u s t f u r t h e r require t h a t H take its m i n i m u m value with respect to restricted v a r i a t i o n s of n(r) and T(r) or t h a t ~]/~t also be zero. Acknowledgments. The a u t h o r wishes to t h a n k the M a t h e m a t i s c h Instituut, Rijksuniversiteit te Groningen for its hospitality during the y e a r 1967-8 and, in particular, Dr. H. G. K a p e r , in discussion with w h o m the p r o b l e m arose. The a u t h o r wishes also to t h a n k the U n i t e d States E d u cational F o u n d a t i o n in the N e t h e r l a n d s for a F u l b r i g h t grant t h a t m a d e the s t a y possible.
REFERENCES
I) DarrozSs, J. and Guiraud, J. P., Comptes Rendus Acad. Sci. 262 (1966) 1368. 2) Grad, H., J. Soc. Ind. Appl. Math. 13 (1965) 259. 3) Chapman, S. and Cowling, T. G., Mathematical Theory of Non-Uniform Gases, Second ed., Cambridge Univ. Press, 1952.