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Evolution of the entropy for an anisotropic Maxwellian gas
Volume 139, number 1,2
PHYSICS LETTERS A
24 July 1989
EVOLUTION OF THE ENTROPY FOR AN ANISOTROPIC MAXWELLIAN GAS Damián H. ZANETTE’ Centro AtOmico Bari/oche and Inst ituto Balseiro, 8400 S. C. de Bariloche, Rio Negro, Argentina Received 7 January 1989; accepted for publication 18 May 1989 Communicated by A.R. Bishop
The temporal evolution of the entropy for a Maxwellian gas is explicitly written in terms of the Laguerre moments of the distribution function. This approximated form, which is valid when the system is near to the equilibrium state makes evident general features in the relaxation of the entropy.
Let us consider a dilute gas of classical molecules without internal degrees of freedom. Ifthe gas is spatially homogeneous, it can be described by a one-partide distribution function f(v, t), depending on the velocity v and on the time t. The temporal evolution off(v, t) is governed by the Boltzmann equation [1] CI dw CI dngl(g x)Lf(v’ t)f(w’ t) j
j
—f(v,t)f(w t)J
(1)
where I(g, x) is the differential scattering cross section for the binary collision (v, w)—~(v’,w’); g~ p.... w is the initial relative velocity, ii is the direction of scattering and cos(x) =ñg. The symmetries of interchange of identical partides and temporal inversion in each collision ensure the validity of the Boltzmann H-theorem, determining the evolution off(v, t) towards equilibrium [2]. This theorem establishes that the entropy S(t) grows monotonically as time elapses: dS(t) d C d =_~-Jf(v~t)lnf(v,t)dv~O.
sense, S(t) is a relevant function in the analysis of the relaxation process [31. To study the evolution of the entropy for a dilute gas, we begin by specifying a particular model for the interaction between molecules. For Maxwell interaction models, which take a (g, x) =gI(g, ~) independent of g, an exact solution for the Boltzmann equation can be found in terms of a series of orthogonal functions. In the case of two-dimensional systems, we write [4]
f(v, t)=fo(v) ~ ~°
~
Cnq(1)Rnq(V)
,
(3)
‘°
with fo(v)=(2it)
—‘
2
exp(—v/2)
(4)
the equilibrium distribution function, and 2 1! R ~ 1 ~ 2qn1 )/2 F nq~ / ‘~ / 12 k~ q—nJ / j. X (v2/2 ) I 2q~ /2~~~n’~—~2q—n1)/2 ( v2/2) ~
— —
xexp[i(2q—n)i~] (5) with v=v(cos(t9), sin(z9)). The symbol ~~,Y~ri(x) ,
(2)
This inequality is related with a constant loss of information along the evolution of the system: Since two different initial conditions evolve towards the same equilibrium state, there is a destruction of information directly measured by the entropy. In that Consejo Nacional de Investigaciones Cientfficas y Técnicas, Argentina.
denotes the associated Laguerre polynomials. The functions R~~(v) are orthogonal in the following sense
5
dVf
0R~qR~q~ = [~(n + 2q— n )]!
x [~(n I 2q n ) ]! —
—
~
ôqq
,
(6)
and form a complete set in the Hubert space ofsquare integrable functions. The moments CNQ(1) satisfy 39
Volume 139, number 1,2
PHYSICS LETTERS A
(d/dt+ANQ)CA’Q(t) N—I qi
S(t)=S
=
n=I
q~o(—
1 )qpNQ Cna ( I) ~
nQq ( t)
with q0=max(0, Q-.-N+n) and q1 =min(n, Q). The coefficients /A~~and A~Qare given in terms of the scattering cross section as 2zr
C d~a(~) . [cos(X/2)] [sin(X/2)] ,,
0 ,
(8)
an
5
2~
ANQ =
d~a (x) { 1 + ÔNO
0
2)x I cos’V (x/2) exp [i ( Q NI —exp[2iQx+iN(it—x) /2] Sin”~(X/2) }
—
—
(9)
The set of equations (7) can be recursively solved with the condition C 00 ( t) = 1, having a very simple asymptotic form: CNQ (oo) = oN0 O~.Therefore, it constitutes an exact solution for the Boltzmann equation, within the appropriate Hubert space. Near enough to the equilibrium state, eqs. (7) can be linearized to give C\.Q(t) = CNQ(O) exp( —ANQt)
.
(10)
It is easily shown that, for a fixed n, A~qis minimum for q= n/2. From the angular dependence of the functions Rnq(V) in eq. (5), we observe that the isotropic terms evolve slower than the anisotropic ones. The relaxation process approaches an isotropic state before reaching the equilibrium situation. This behaviour can also be understood in terms of the conservation laws of energy and momentum [5]. Furthermore, A~q grows for increasing n. Then, the higher-order moments relax faster than the lower-order ones, Since the moments CNQ(t) determine the distribution function, the entropy can be seen as a function of these moments. Near to the equilibrium situation the logarithm in the definition ofS(t) can be expanded about f0 ( v) to give 40
~ I q=0 ~ a~C~(t)I
(11)
where we have neglected cubic terms in the moments C,~q. The equilibrium entropy is given by S 0= ln(2ite) and ctnq = N (n + I 2q
—
n)]! [~ (n
—
I 2q— fl)]!.
12
Eq. (11) describes the final approach of the entropy towards the equilibrium in terms of the Laguerre . moments for the distribution function. Linear terms in the moments vanish by virtue of the conservation laws of particles, momentum and energy. It is analogous to the expression found in a previous work for the isotropic case [6]. In the moment-space and near to the equilibrium state, the entropy surface has the form of a hyperparaboloid whose maximum corresponds to the equilibrium entropy. The system evolves on this large surface reaching the maximum for asymptotically times. The coefficients a~qin eq. (10) determine the curvature of the hyperparaboloid representing the entropy in the moment-space. From eq. (11) we ob.
J
xexp[i(Q—N/2)x+init/2] d
2,
0—~
(7)
~
24 July 1989
serve that /
a~q= n!
(
~ (n + 2q— nI)
)
.
(13)
For a fixed n, this expression has a minimum when 2q—n~=0, that is, when q=n/2. Furthermore, a~q grows for increasing n. Taking into account the Nand Q-dependence of ANQ discussed before, we ohserve that the greater the curvature a~qis the faster the corresponding Laguerre moment decays. We conclude that “The system climbs the entropy hill looking for the less steep route”. This assertion can be seen to hold also for three-dimensional Maxwell gases, whose distribution function can be expanded in more complex orthogonal functions [4]. The extension of this result to other interaction models requires the generalization of the equations of motion for the corresponding Laguerre moments, which are not recursive in general and then do not enable finding an exact solution.
Volume 139, number 1,2
PHYSICS LETTERS A
24 July 1989
References
[4] E.M. Hendriks and T. Nieuwenhuizen, J. Stat. Phys. 29 (1982) 591.
[1] M.H. Ernst, Phys. Rep. 78 (1981) 1. [2] K. Huang, Statistical mechanics (Wiley, New York, 1965). [3] C. Cercignani, Mathematical methods in kinetic theory (Plenum, New York, 1969).
[5] R.O. Barrachina and C.R. Garibotti, J. Stat. Phys. 45 (1986) 541. [61D.H. Zanette and C.R. Garibotti, Physica A 149 (1988) 638.
41
PHYSICS LETTERS A
24 July 1989
EVOLUTION OF THE ENTROPY FOR AN ANISOTROPIC MAXWELLIAN GAS Damián H. ZANETTE’ Centro AtOmico Bari/oche and Inst ituto Balseiro, 8400 S. C. de Bariloche, Rio Negro, Argentina Received 7 January 1989; accepted for publication 18 May 1989 Communicated by A.R. Bishop
The temporal evolution of the entropy for a Maxwellian gas is explicitly written in terms of the Laguerre moments of the distribution function. This approximated form, which is valid when the system is near to the equilibrium state makes evident general features in the relaxation of the entropy.
Let us consider a dilute gas of classical molecules without internal degrees of freedom. Ifthe gas is spatially homogeneous, it can be described by a one-partide distribution function f(v, t), depending on the velocity v and on the time t. The temporal evolution off(v, t) is governed by the Boltzmann equation [1] CI dw CI dngl(g x)Lf(v’ t)f(w’ t) j
j
—f(v,t)f(w t)J
(1)
where I(g, x) is the differential scattering cross section for the binary collision (v, w)—~(v’,w’); g~ p.... w is the initial relative velocity, ii is the direction of scattering and cos(x) =ñg. The symmetries of interchange of identical partides and temporal inversion in each collision ensure the validity of the Boltzmann H-theorem, determining the evolution off(v, t) towards equilibrium [2]. This theorem establishes that the entropy S(t) grows monotonically as time elapses: dS(t) d C d =_~-Jf(v~t)lnf(v,t)dv~O.
sense, S(t) is a relevant function in the analysis of the relaxation process [31. To study the evolution of the entropy for a dilute gas, we begin by specifying a particular model for the interaction between molecules. For Maxwell interaction models, which take a (g, x) =gI(g, ~) independent of g, an exact solution for the Boltzmann equation can be found in terms of a series of orthogonal functions. In the case of two-dimensional systems, we write [4]
f(v, t)=fo(v) ~ ~°
~
Cnq(1)Rnq(V)
,
(3)
‘°
with fo(v)=(2it)
—‘
2
exp(—v/2)
(4)
the equilibrium distribution function, and 2 1! R ~ 1 ~ 2qn1 )/2 F nq~ / ‘~ / 12 k~ q—nJ / j. X (v2/2 ) I 2q~ /2~~~n’~—~2q—n1)/2 ( v2/2) ~
— —
xexp[i(2q—n)i~] (5) with v=v(cos(t9), sin(z9)). The symbol ~~,Y~ri(x) ,
(2)
This inequality is related with a constant loss of information along the evolution of the system: Since two different initial conditions evolve towards the same equilibrium state, there is a destruction of information directly measured by the entropy. In that Consejo Nacional de Investigaciones Cientfficas y Técnicas, Argentina.
denotes the associated Laguerre polynomials. The functions R~~(v) are orthogonal in the following sense
5
dVf
0R~qR~q~ = [~(n + 2q— n )]!
x [~(n I 2q n ) ]! —
—
~
ôqq
,
(6)
and form a complete set in the Hubert space ofsquare integrable functions. The moments CNQ(1) satisfy 39
Volume 139, number 1,2
PHYSICS LETTERS A
(d/dt+ANQ)CA’Q(t) N—I qi
S(t)=S
=
n=I
q~o(—
1 )qpNQ Cna ( I) ~
nQq ( t)
with q0=max(0, Q-.-N+n) and q1 =min(n, Q). The coefficients /A~~and A~Qare given in terms of the scattering cross section as 2zr
C d~a(~) . [cos(X/2)] [sin(X/2)] ,,
0 ,
(8)
an
5
2~
ANQ =
d~a (x) { 1 + ÔNO
0
2)x I cos’V (x/2) exp [i ( Q NI —exp[2iQx+iN(it—x) /2] Sin”~(X/2) }
—
—
(9)
The set of equations (7) can be recursively solved with the condition C 00 ( t) = 1, having a very simple asymptotic form: CNQ (oo) = oN0 O~.Therefore, it constitutes an exact solution for the Boltzmann equation, within the appropriate Hubert space. Near enough to the equilibrium state, eqs. (7) can be linearized to give C\.Q(t) = CNQ(O) exp( —ANQt)
.
(10)
It is easily shown that, for a fixed n, A~qis minimum for q= n/2. From the angular dependence of the functions Rnq(V) in eq. (5), we observe that the isotropic terms evolve slower than the anisotropic ones. The relaxation process approaches an isotropic state before reaching the equilibrium situation. This behaviour can also be understood in terms of the conservation laws of energy and momentum [5]. Furthermore, A~q grows for increasing n. Then, the higher-order moments relax faster than the lower-order ones, Since the moments CNQ(t) determine the distribution function, the entropy can be seen as a function of these moments. Near to the equilibrium situation the logarithm in the definition ofS(t) can be expanded about f0 ( v) to give 40
~ I q=0 ~ a~C~(t)I
(11)
where we have neglected cubic terms in the moments C,~q. The equilibrium entropy is given by S 0= ln(2ite) and ctnq = N (n + I 2q
—
n)]! [~ (n
—
I 2q— fl)]!.
12
Eq. (11) describes the final approach of the entropy towards the equilibrium in terms of the Laguerre . moments for the distribution function. Linear terms in the moments vanish by virtue of the conservation laws of particles, momentum and energy. It is analogous to the expression found in a previous work for the isotropic case [6]. In the moment-space and near to the equilibrium state, the entropy surface has the form of a hyperparaboloid whose maximum corresponds to the equilibrium entropy. The system evolves on this large surface reaching the maximum for asymptotically times. The coefficients a~qin eq. (10) determine the curvature of the hyperparaboloid representing the entropy in the moment-space. From eq. (11) we ob.
J
xexp[i(Q—N/2)x+init/2] d
2,
0—~
(7)
~
24 July 1989
serve that /
a~q= n!
(
~ (n + 2q— nI)
)
.
(13)
For a fixed n, this expression has a minimum when 2q—n~=0, that is, when q=n/2. Furthermore, a~q grows for increasing n. Taking into account the Nand Q-dependence of ANQ discussed before, we ohserve that the greater the curvature a~qis the faster the corresponding Laguerre moment decays. We conclude that “The system climbs the entropy hill looking for the less steep route”. This assertion can be seen to hold also for three-dimensional Maxwell gases, whose distribution function can be expanded in more complex orthogonal functions [4]. The extension of this result to other interaction models requires the generalization of the equations of motion for the corresponding Laguerre moments, which are not recursive in general and then do not enable finding an exact solution.
Volume 139, number 1,2
PHYSICS LETTERS A
24 July 1989
References
[4] E.M. Hendriks and T. Nieuwenhuizen, J. Stat. Phys. 29 (1982) 591.
[1] M.H. Ernst, Phys. Rep. 78 (1981) 1. [2] K. Huang, Statistical mechanics (Wiley, New York, 1965). [3] C. Cercignani, Mathematical methods in kinetic theory (Plenum, New York, 1969).
[5] R.O. Barrachina and C.R. Garibotti, J. Stat. Phys. 45 (1986) 541. [61D.H. Zanette and C.R. Garibotti, Physica A 149 (1988) 638.
41