Deviation from Maxwell distribution in granular gases with constant restitution coefficient. Nikolai V. .... *URL: http://summa.physik.hu-berlin.de/thorsten/.
PHYSICAL REVIEW E
VOLUME 61, NUMBER 3
MARCH 2000
Deviation from Maxwell distribution in granular gases with constant restitution coefficient Nikolai V. Brilliantov1,2 and Thorsten Po¨schel1,* 1
Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, Invalidenstraße 110, D-10115 Berlin, Germany 2 Physics Department, Moscow State University, Moscow 119899, Russia 共Received 28 June 1999兲
We analyze the velocity distribution function of force-free granular gases in the regime of homogeneous cooling when deviations from the Maxwellian distribution may be accounted only by the leading term in the Sonine polynomial expansion, quantified by the second coefficient a 2 . We go beyond the linear approximation for a 2 and find three different values 共three roots兲 for this coefficient which correspond to a scaling solution of the Boltzmann equation. The stability analysis performed showed, however, that among these three roots only one corresponds to a stable scaling solution. This is very close to a 2 , obtained in previous studies in a linear with respect to a 2 approximation. PACS number共s兲: 81.05.Rm, 36.40.Sx, 51.20.⫹d, 66.30.Hs
Granular gases, i.e., rarefied systems composed of inelastically colliding particles have been of particular interest during the last decade 共e.g., Refs. 关1–4兴兲. Compared to gases of elastically colliding particles, the dissipation of energy at inelastic collisions leads to some novel phenomena in these systems such as clustering 共e.g., Ref. 关1兴兲, formation of vortex patterns 共e.g., Ref. 关2兴兲, etc. Before clustering starts, the granular gas being initially homogeneous, keeps for some time its homogeneity, although its temperature permanently decreases. This regime is called the homogeneous cooling regime 共HC兲. In the present study we address the properties of the velocity distribution of granular particles in the regime of HC, such as the deviation from the Maxwellian distribution and the stability of the distribution function. We assume that the restitution coefficient ⑀ does not depend on the impact velocity, i.e., that ⑀⫽const. The properties of the velocity distribution for the system with impact-velocity dependent coefficient of restitution 共e.g., Ref. 关5兴兲 will be addressed elsewhere 关6兴. It is well known that granular gases in the HC regime do not reveal Maxwellian distribution 共e.g., Refs. 关3,4,7,8兴兲. The high-velocity tail is overpopulated 关3,8兴, while the main part of the distribution is described by the sum of the Maxwellian and correction to it, written in terms of the Sonine polynomial expansion 共e.g., Refs. 关3,4,7兴兲. Usually only the leading, second term, in this expansion is taken into account 关3,4,7兴, moreover in previous studies 关3,4兴 only linear analysis with respect to the coefficient a 2 , which refers to this second term has been performed. Finding that a 2 , obtained within the linear approximation, is small, the authors of Refs. 关4,3兴 conclude a posteriori that the linear approximation is valid. In our approach we also assume that one can restrict oneself to the leading term in the Sonine polynomial expansion. However, we go beyond the linear approximation with respect to the coefficient a 2 and perform complete analysis within this level of the system description. We found three different values of a 2 which correspond to the scaling solution of the Boltzmann equation. The stability analysis for the
*URL: http://summa.physik.hu-berlin.de/⬃thorsten/ 1063-651X/2000/61共3兲/2809共4兲/$15.00
PRE 61
velocity distribution function shows, however, that only one value of a 2 corresponds to a physically acceptable stable scaling solution. The stable solution is close to the result previously obtained within the linear analysis 关3兴. To introduce notations and specify the problem we briefly sketch the derivation of the coefficient a 2 关3,4兴. We introduce the 共time-dependent兲 temperature T(t), and the thermal velocity v 0 (t), which are related to the velocity distribution function f (v,t) for 3D systems as 3 nT 共 t 兲 ⫽ 2
冕
dv
3 v2 f 共 v,t 兲 ⫽ n v 20 共 t 兲 . 2 2
共1兲
Here n is the number density and the particles are assumed to be of unit mass (m⫽1). The inelasticity of collisions is characterized by the coefficient of normal restitution ⑀, which relates the after-collisional velocities v* 1 , v2* to the precollisional ones, v1 , v2 as 1 * ⫽v1/2⫿ 共 1⫹ ⑀ 兲共 v12•e兲 e, v1/2 2
共2兲
where v12⫽v1 ⫺v2 is the relative velocity, the unit vector e ⫽r12 / 兩 r12兩 gives the direction of the vector r12⫽r1 ⫺r2 at the instant of the collision. The time-evolution of the velocity distribution function is subjected to the EnskogBoltzmann equation, which for the force-free case reads 关3,9兴
f 共 v,t 兲 ⫽g 2 共 兲 2 t ⫻
再
1
⑀2
冕 冕 dv2
de⌰ 共 ⫺v12•e兲 兩 v12•e兩
冎
f 共 v1** ,t 兲 f 共 v** 2 ,t 兲 ⫺ f 共 v1 ,t 兲 f 共 v2 ,t 兲 , 共3兲
where is the diameter of particles, g 2 ( )⫽(2⫺ )/2(1 ⫺ ) 3 ( ⫽ 16 n 3 is the packing fraction兲 denotes the contact value of the two-particle correlation function 关10兴, which accounts for the increasing collision frequency due to the excluded volume effects; ⌰(x) is the Heaviside function. and v2** refer to the pre-collisional veThe velocities v** 1 2809
©2000 The American Physical Society
¨ SCHEL NIKOLAI V. BRILLIANTOV AND THORSTEN PO
2810
locities of the so-called inverse collision, which results with v1 and v2 as the after-collisional velocities. The factor 1/⑀ 2 in the gain term appears respectively from the Jacobian of the transformation dv1** dv2** →dv1 dv2 and from the relation ** •e兩 dt between the lengths of the collisional cylinders ⑀ 兩 v12 ⫽ 兩 v12•e兩 dt 关3,9兴. Assuming that the velocity distribution function is of a scaling form f 共 v,t 兲 ⫽
n
共4兲
˜f 共 c兲 v 30 共 t 兲
one can show that the scaling function satisfies the timeindependent equation 关3兴
冉
冊
2 ˜f 共 c兲 ⫽˜I 共˜f ,˜f 兲 3⫹c 1 3 c1
˜I 共˜f ,˜f 兲 ⫽
冕 冕 dc2
de⌰ 共 ⫺c12•e兲 兩 c12•e兩 兵 ⑀ ⫺2˜f 共 c1** 兲˜f 共 c2** 兲
⫺˜f 共 c1 兲˜f 共 c2 兲 其
rately described only by the second term in the expansion 共9兲 with all high-order terms with p⬎2 discarded. Then Eq. 共10兲 is an equation for the coefficient a 2 . Using the above results for 具 c 2 典 and 具 c 4 典 it is easy to show that Eq. 共10兲 converts for p⫽2 into identity, while for p⫽4 it reads 5 2 共 1⫹a 2 兲 ⫺ 4 ⫽0.
p ⫽⫺
dT/dt⫽⫺ 共 2/3兲 BT 2 ,
兺
p⫽1
冎
a pS p共 c 2 兲 ,
共9兲
where (c)⬅ ⫺d/2 exp(⫺c2) is the Maxwellian distribution and the first few Sonine polynomials read S 0 (x)⫽1, S 1 (x) ⫽⫺x 2 ⫹ 32 , S 2 (x)⫽x 2 /2⫺5x/2⫹ 158 , etc. Multiplying both sides of Eq. 共5兲 with c 1p and integrating by parts over dc1 , we obtain 关3兴
2 p 具 c p典 ⫽ p , 3
冉
冕
c p˜f 共 c,t 兲 dc.
冊
3 9 a 2⫹ a2 , 16 1024 2
4 ⫽4 冑2 兵 T 1 ⫹a 2 T 2 ⫹a 22 T 3 其 ,
冉 冊
9 1 T 1 ⫽ 共 1⫺ ⑀ 2 兲 ⫹ ⑀ 2 , 4 2 T 2⫽
共13兲 共14兲
T 3⫽
1 1 共 1⫹ ⑀ 兲 ⫹ 共 1⫺ ⑀ 2 兲共 9⫺30⑀ 2 兲 . 64 8192
The coefficients 2 and 4 were provided in Ref. 关3兴 up to terms of the order of O(a 2 ). One obtains the coefficient a 2 in the Sonine polynomial expansion in this approximation by substituting Eqs. 共13兲, 共14兲 into Eq. 共12兲 and discarding in Eqs. 共13兲, 共14兲 all terms of the order of O(a 22 ): a NE 2 ⫽
共11兲
The odd moments 具 c 2n⫹1 典 are zero, while the even ones, 具 c 2n 典 , may be expressed in terms of a k with 0⭐k⭐n. Calculations show that 具 c 2 典 ⫽ 23 , implying a 1 ⫽0, according to the definition of the temperature 共1兲 共e.g., Ref. 关3兴兲, and that 具 c 4 典 ⫽ 154 (1⫹a 2 ). Now we assume, that the dissipation is not large, so that the deviation from the Maxwellian distribution may be accu-
共15兲
3 1 共 1⫺ ⑀ 2 兲共 69⫹10⑀ 2 兲 ⫹ 共 1⫹ ⑀ 兲 , 128 2
共10兲
where we define
具 c p典 ⬅
de⌰ 共 ⫺c12•e兲 兩 c12•e兩 共 c 1 兲 共 c 2 兲 兵 1
2 ⫽ 冑2 共 1⫺ ⑀ 2 兲 1⫹
共8兲
where B⫽B(t)⬅ v 0 (t)g 2 ( ) 2 n. To proceed we use the Sonine polynomial expansion for the velocity distribution function 关3,4兴
再
dc2
with
while the time-evolution of temperature reads
⬁
dc1
共7兲
dc1 c 1p˜I 共˜f ,˜f 兲 ,
˜f 共 c兲 ⫽ 共 c 兲 1⫹
冕 冕 冕
where ⌬ (ci )⬅ 关 (c* i )⫺ (ci ) 兴 denotes change of some function (ci ) in a direct collision. Calculations, similar to that, described in Ref. 关3兴, yield 共details are given in Ref. 关6兴兲:
and with its moments 关3兴
冕
1 2
⫹a 2 关 S 2 共 c 21 兲 ⫹S 2 共 c 22 兲兴 ⫹a 22 S 2 共 c 21 兲 S 2 共 c 22 兲 其 ⌬ 共 c 1p ⫹c 2p 兲 ,
共6兲
p ⬅⫺
共12兲
The coefficients p may be expressed in terms of a 2 due to the definition 共7兲 and the assumption ˜f ⫽ (c) 关 1 ⫹a 2 S 2 (c 2 ) 兴 . Using the properties of the collision integral one obtains for p 关3兴
共5兲
with the dimensionless collision integral
PRE 61
16共 1⫺ ⑀ 兲共 1⫺2 ⑀ 2 兲 81⫺17⑀ ⫹30⑀ 2 共 1⫺ ⑀ 兲
.
共16兲
Calculations including the next order terms O(a 22 ) in the coefficients 2 and 4 show that Eq. 共12兲 is a cubic equation, which for physical values of ⑀, 0⭐⑀⭐1, has three different real roots, as it shown in Fig. 1. Although the cubic equation may be generally solved, the resultant expressions for the roots are too cumbersome to be written explicitly. However, one of the roots 共the middle one兲 is rather small and close to that given by Eq. 共16兲, obtained within the linear approximation. This suggests the perturbative solution of the cubic equation near this root:
DEVIATION FROM MAXWELL DISTRIBUTION IN . . .
PRE 61
FIG. 1. The left hand side of Eq. 共12兲 over a 2 for ⑀⫽0.8. Obviously Eq. 共12兲 has three real solutions.
冋
1⫺ a 2 ⫽a NE 2
1005共 1⫺ ⑀ 2 兲 ⫺4096T 3 6080共 1⫺ ⑀ 2 兲 ⫺4096T 2
册
a NE 2 ⫹••• , 共17兲
where we do not write explicitly terms of the order NE 3 O( 关 a NE 2 兴 ) and higher. In Fig. 2 the dependence of a 2 and of the corresponding improved value a 2 are shown as a function of the restitution coefficient ⑀. As one can see from Fig. 2 the maximal deviation between these is less than 10% at small ⑀ and decreases as ⑀ tends to 1. The other two roots, shown on Fig. 3 are of the order of 1 or 10, i.e., are not small. Physically, this means that one cannot cut the Sonine polynomial expansion in this case at the second term and next order terms are not negligible. Taking into account the next order terms, i.e., releasing the assumption that a p ⯝0 for p⬎2, breaks down the above analysis, since the coefficients 2 , 4 occur to be dependent not only on a 2 , but on a 3 , a 4 , . . . as well. Thus the occurrence of several roots for the a 2 , found within the above approach, which satisfy the conditions required by the scaling ansatz 共4兲 does not imply the existence of several different scaling solutions. Nevertheless such possibility may not be completely excluded. If one assumes that few scaling distributions of the velocity may realize, depending on the initial conditions at which the HC state has been prepared, a natural question arises: Whether the particular scaling solution is stable with respect to small perturbations, and what is the domain of attraction of this particular scaling solution in some parametric space.
2811
FIG. 3. The other two solutions for second Sonine coefficient a 2 of Eq. 共12兲 over the coefficient of restitution ⑀.
Certainly, the stability problem is very complicated to be solved in general. Therefore, we restrict ourselves to the stability analysis of the scaling distribution 共4兲 where the scaling function ˜f (c) has nonzero value of the coefficient a 2 , while the other coefficients a p with p⬎2 are negligibly small. 共For this scaling solution our above results for the coefficients 2 , 4 are valid兲. Moreover, we assume, that small perturbations of the 共vanishingly small兲 coefficients a p with p⬎2 do not influence the stability of the distribution, and analyze the stability only with respect to variation of the coefficient a 2 . To analyze the stability of the velocity distribution we write it in a more general form f 共 v,t 兲 ⫽
n
˜f 共 c,t 兲
v 30 共 t 兲
共18兲
which leads, as it easy to show, to the following generalization of Eq. 共5兲 关6兴:
冉
冊
2 ˜f 共 c,t 兲 ⫹B ⫺1 ˜f 共 c,t 兲 ⫽˜I 共˜f ,˜f 兲 3⫹c 1 3 c1 t
共19兲
with the collisional integral and coefficients p being now time dependent. The quantities 具 c p 典 also depend now on time, while temperature evolves still according to Eq. 共8兲. Using ˜f ⫽ (c) 关 1⫹a 2 (t)S 2 (c 2 ) 兴 and performing essentially the same manipulations which led before to Eq. 共12兲, we find for the coefficient a 2 (t): a˙ 2 ⫺ 共 4/3兲 B 2 共 1⫹a 2 兲 ⫹ 共 4/15兲 B 4 ⫽0
共20兲
with 2 , 4 still given by Eqs. 共13兲, 共14兲, but with the timedependent coefficient a 2 (t). Writing the above value B(t) as
FIG. 2. The second Sonine coefficient a 2 as a function of the coefficient of restitution ⑀ 共full line兲. The dashed line shows a NE 2 in the first order approximation by van Noije and Ernst 关3兴 according to Eq. 共16兲. The approximation 共17兲 is shown by circles.
B 共 t 兲 ⫽ 共 8 兲 ⫺1/2 c 共 0 兲 ⫺1 u 共 t 兲 1/2,
共21兲
c 共 0 兲 ⫺1 ⬅4 1/2g 2 共 兲 2 nT 1/2 0 ,
共22兲
where c (0) is related to the initial mean-collision time at the initial temperature T 0 , and u(t)⬅T(t)/T 0 is the reduced temperature, we recast Eq. 共20兲 into the form da 2 冑2/ 1/2 ⫽ u F共 a2兲, 15 d ˆt
共23兲
2812
¨ SCHEL NIKOLAI V. BRILLIANTOV AND THORSTEN PO
where ˆt is the reduced time, measured in units of c (0), and where we define a function
PRE 61
The form of the function F(a 2 ) for some particular value of ⑀ is shown on Fig. 1. This form of F(a 2 ) persists for all physical values of the restitution coefficient, 0⭐⑀⭐1. There are three different roots, F(a (i) 2 )⫽0, i⫽1,2,3, which make da 2 /dt vanish yielding the scaling form for the solution of the Enskog-Boltzmann equation. The stability of the scaling solution, corresponding to a (i) 2 requires for the derivative dF/da 2 , taken at a (i) to be negative, since only in this case a 2 (i) from a , corresponding to a scalsmall deviation a 2 ⫺a (i) 2 2 ing solution will decay with time. As one can see from Fig. 1 only the middle root, which corresponds to small values of a 2 , and is close to a NE 2 , predicted by linear theory 关3兴, has negative dF/da 2 , and thus is stable. We also observed that for any 0⭐ ⑀ ⭐1 the point a 2 ⫽0 belongs to the attractive interval of this stable root. Naturally, this means that an initial Maxwellian distribution will relax to the non-Maxwellian with a 2 ⬇a NE 2 . Note that relaxation of any 共small兲 perturbation to this value of a 2 occurs, as it follows from Eq. 共23兲, on the collision time scale, i.e., practically ‘‘immediately’’ on the time scale which describes the evolution of the temperature.
Therefore we conclude, that the scaling solution of the Enskog-Boltzmann equation with a 2 corresponding to the middle root of the function F(a 2 ), given with a high accuracy by Eqs. 共17兲,共16兲, and with negligibly small other coefficients a 3 ,a 4 , . . . , of the Sonine polynomial expansion is a stable one with respect to 共relatively兲 small perturbations. In conclusion, we analyzed the velocity distribution function of a granular gas with constant restitution coefficient in the regime of homogeneous cooling. We assume that the deviations from the Maxwellian distribution may be described using only the leading term in the Sonine polynomial expansion, with all other high-order terms discarded. In this approach the deviations from the Maxwellian distribution are completely characterized by the magnitude of the coefficient a 2 of the leading term. We go beyond previous linear theories and perform a complete analysis 共on the level of the description chosen兲, without discarding any nonlinear with respect to a 2 terms. Performing the stability analysis of the scaling solution of the Enskog-Boltzmann equation we observe that only one value of a 2 , obtained within our nonlinear analysis corresponds to a stable scaling solution. We also report corrections for this value of a 2 with respect to the previous result of the linear theory. These corrections are small 共less than 10%兲 for all values of the restitution coefficient ⑀ and vanishes as ⑀ tends to unity in the elastic limit.
关1兴 I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 共1993兲; S. McNamara and W.R. Young, Phys. Rev. E 50, R28 共1993兲; F. Spahn, U. Schwarz, and J. Kurths, Phys. Rev. Lett. 78, 1596 共1997兲; T. Aspelmeier, G. Giese, and A. Zippelius, Phys. Rev. E 57, 857 共1997兲; P. Deltour and J.L. Barrat, J. Phys. I 7, 137 共1997兲; R. Brito and M.H. Ernst, Europhys. Lett. 43, 497 共1998兲. 关2兴 T.P.C. van Noije and M.H. Ernst, R. Brito, and J.A.G. Orza, Phys. Rev. Lett. 79, 411 共1997兲. 关3兴 T.P.C. van Noije and M.H. Ernst, Granular Matter 1, 57 共1998兲. 关4兴 A. Goldshtein and M. Shapiro, J. Fluid. Mech. 282, 75 共1995兲. 关5兴 N.V. Brilliantov, F. Spahn, J.-M. Hertzsch, and T. Po¨schel,
Phys. Rev. E 53, 5382 共1996兲; T. Schwager and T. Po¨schel, ibid. 57, 650 共1998兲; R. Ramı´rez, T. Po¨schel, N.V. Brilliantov, and T. Schwager, ibid. 60, 4465 共1999兲. N.V. Brilliantov and T. Po¨schel, in Granular Gases, edited by T. Po¨schel and S. Luding 共Springer, Berlin, 2000兲; N.V. Brilliantov and T. Po¨schel, e-print cond-mat/9911212. J.J. Brey, J.W. Dufty, C.S. Kim, and A. Santos, Phys. Rev. E 58, 4638 共1998兲. S.E. Esipov and T. Po¨schel, J. Stat. Phys. 86, 1385 共1997兲. P. Resibois and M. de Leener, Classical Kinetic Theory of Fluids 共Wiley, New York, 1977兲. N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51, 635 共1969兲.
F 共 a 2 兲 ⬅5 2 共 1⫹a 2 兲 ⫺ 4 .
共24兲
关6兴
关7兴 关8兴 关9兴 关10兴
