We study properties of the consistent Boltzmann algorithm for dense gases, using its limiting kinetic equation. First we derive an H-theorem for this equation. Then, following the classical derivation by Chapman and Cowling, we find approximations to the equations of continuity, momentum and energy. The first order correction terms with respect to the particle diameter turn out to be the same as for the Enskog equation. These results confirm previous derivations, based on the virial, of the corresponding equation of state

Garcia, Alejandro L.

Wagner, Wolfgang

English

We study properties of the consistent Boltzmann algorithm for dense gases, using its limiting kinetic equation. First, we derive an H-theorem for this equation. Then, following the classical derivation by Chapman and Cowling, we find approximations to the equations of continuity, momentum, and energy. The first order correction terms with respect to the particle diameter turn out to be the same as for the Enskog equation. These results confirm previous derivations, based on the virial, of the corresponding equation of state

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Some new properties of the kinetic equation for the consistent Boltzmann algorithm

WeierstraÿInstitutfür Angewandte Analysis und Stochastikim Forschungsverbund Berlin e.V.Preprint ISSN 0946  8633Some new properties of the kinetic equation for theconsistent Boltzmann algorithmAlejandro L. Garcia1, Wolfgang Wagner2submitted: 9th July 20011 Institute for Scientic Computing ResearchLawrence Livermore National LaboratoryLivermore, California 94550, USAPermanent address: Physics DepartmentSan Jose State UniversitySan Jose, California 95192, USAE-Mail: algarcia@algarcia.org2 Weierstrass Institute forApplied Analysis and StochasticsMohrenstrasse 39D-10117 Berlin, GermanyE-Mail: wagner@wias-berlin.dePreprint No. 661Berlin 20011991 Mathematics Subject Classication. 65C05, 76P05, 82C40.Key words and phrases. Kinetic theory, direct simulation Monte Carlo, consistent Boltzmannalgorithm, dense gases, H-theorem.Edited byWeierstraÿInstitut für Angewandte Analysis und Stochastik (WIAS)Mohrenstraÿe 39D  10117 BerlinGermanyFax: + 49 30 2044975E-Mail (X.400): c=de;a=d400-gw;p=WIAS-BERLIN;s=preprintE-Mail (Internet): preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstractWe study properties of the consistent Boltzmann algorithm for dense gases,using its limiting kinetic equation. First we derive an H-theorem for thisequation. Then, following the classical derivation by Chapman and Cowling,we nd approximations to the equations of continuity, momentum and energy.The rst order correction terms with respect to the particle diameter turn outto be the same as for the Enskog equation. These results conrm previousderivations, based on the virial, of the corresponding equation of state.Contents1. Introduction 22. H-theorem 43. Equations of continuity, momentum and energy 63.1. Left-hand side of the kinetic equation . . . . . . . . . . . . . . . . . . 63.2. Right-hand side of the kinetic equation . . . . . . . . . . . . . . . . . 83.3. Comparison of both sides . . . . . . . . . . . . . . . . . . . . . . . . . 10Appendix: Moments of a Gaussian variable 12References 1311. IntroductionDirect Simulation Monte Carlo (DSMC) is presently the most widely used numericalalgorithm in kinetic theory [4]. In this method, a system of simulation particlesxi(t); vi(t); i = 1; : : : ; N ; t  0 ;is used to approximate the behaviour of the real gas. Independent motion (freeow) of the particles and their pairwise interactions (collisions) are separated usinga splitting procedure. During the free ow step, particles are moved according totheir velocities,xi(t+t) = xi(t) +Z t+ttvi(s) ds ;and boundary conditions are taken into account. During the collision step, particlepairs (x; v); (y;w) are randomly chosen in small cells of the position space, accord-ing to the collision probability for the interparticle potential. The post-collisionvelocitiesv = v + e (e;w   v) ; w = w   e (e;w  v) (1.1)are determined by randomly selecting a direction vector e from the unit sphereS2R3 : Here ( :; : ) and k : k denote the scalar product and the Euclidean norm inR3 ; respectively. The number of collisions at each time step t is computed fromthe local collision frequency.Recently, the Consistent Boltzmann Algorithm (CBA) was introduced as a sim-ple variant of DSMC for dense gases [2]. Besides the standard problems in kinetictheory, CBA has proved useful in the study of granular material [9] and nuclearphysics [10]. Although CBA can be generalized to other potentials [1] here we willonly consider the hard sphere gas with particle diameter  : In CBA the collisionprocess is as in DSMC with two modications. First, when a pair collides eachparticle is displaced a distance  into the direction e or  e ; i.e. (cf. (1.1))x = x+ (v   w)   (v   w)k(v   w)   (v   w)k; y = y   (v   w)  (v   w)k(v   w)  (v   w)k: (1.2)Second, the dense hard sphere collision frequency is used.The limiting equation of CBA (as N !1) has been established in [8],@@tp(t; x; v) + (v;rx) p(t; x; v) =ZR3dwZS2deB(v;w; e) (1.3)h(%(t; x)) p(t; x; v) p(t; x; w)  (%(t; x)) p(t; x; v) p(t; x; w)i:HereB(v;w; e) = const j(e;w  v)j (1.4)2is the hard sphere collision kernel, and%(t; x) =ZR3p(t; x; v) dvdenotes the density. The function  is equal to unity for a rareed gas, and increaseswith increasing density, becoming innity as the gas approaches the state of close-packing. With the notationsS2+ = S2+(v;w) = fe : (e;w  v) > 0g ; S2 = fe : (e;w  v) < 0g ; (1.5)equation (1.3) takes the form@@tp(t; x; v) + (v;rx) p(t; x; v) = 2ZR3dwZS2+deB(v;w; e) (1.6)h(%(t; x+  e)) p(t; x+  e; v) p(t; x+  e;w)  (%(t; x)) p(t; x; v) p(t; x; w)i:Compare this equation with the Enskog equation (cf. [7, Ch.16.3])@@tp(t; x; v) + (v;rx) p(t; x; v) = (1.7)2ZR3dwZS2+deB(v;w; e)h(%(t; x+12 e)) p(t; x; v) p(t; x+  e;w) (%(t; x 12 e)) p(t; x; v) p(t; x   e;w)i:Note that, in the case  1 ;  = 0 ; both equations (1.6) and (1.7) reduce to theBoltzmann equation@@tp(t; x; v) + (v;rx) p(t; x; v) =ZR3dwZS2deB(v;w; e)hp(t; x; v) p(t; x; w)  p(t; x; v) p(t; x; w)i:Here we study some properties of equation (1.6). In Section 2 we derive an H-theorem. In Section 3, following the classical derivation by Chapman and Cowling[6], [7], we nd approximations to the equations of continuity,momentumand energy.The rst order correction terms with respect to the particle diameter turn out tobe the same as for the Enskog equation. These results conrm previous derivations,based on the virial, of the corresponding equation of state [2].32. H-theoremAccording to (1.1), the displacements (1.2) take the formx = x+  (v;w; e) ; y = y    (v;w; e) ;where the notation (v;w; e) =  e sign(e;w   v)is used. Note that (v; w; e) =   (v;w; e) =  (w; v; e) (2.1)and (cf. (1.4))B(v;w; e) = B(v; w; e) = B(w; v; e) = B(v;w; e) : (2.2)Using (2.1), (2.2), one obtainsZR3dxZR3dvZR3dwZS2de'(x; v)B(v;w; e)(%(t; x)) p(t; x; v) p(t; x; w)=ZR3dxZR3dvZR3dwZS2de'(x   (v;w; e); v)B(v;w; e)(%(t; x)) p(t; x; v) p(t; x; w)=ZR3dxZR3dvZR3dwZS2de'(x; v)B(v;w; e)(%(t; x)) p(t; x; v) p(t; x;w) :Thus, the weak form of equation (1.3) isddtZR3R3'(x; v) p(t; x; v) dx dv =ZR3R3(v; (rx')(x; v))p(t; x; v) dx dv+ZR3ZR3ZR3ZS2(%(t; x))B(v;w; e)h'(x; v)  '(x; v)ip(t; x; v) p(t; x; w) de dv dw dx ;or, equivalently,ddtZR3R3'(x; v) p(t; x; v) dx dv =ZR3R3(v; (rx')(x; v))p(t; x; v) dx dv+ (2.3)12ZR3dxZR3dvZR3dwZS2de(%(t; x))B(v;w; e)h'(x+  (v;w; e); v) + '(x   (v;w; e); w)  '(x; v)  '(x;w)ip(t; x; v) p(t; x; w) :The form (2.3) is convenient for deriving an H-theorem. We consider'(x; v) = log p(t; x; v)andH(t) =ZR3ZR3p(t; x; v) log p(t; x; v) dv dx :4Note that ZR3R3(v;rx log p(t; x; v))p(t; x; v) dx dv =ZR3R3(v;rxp(t; x; v))p(t; x; v)p(t; x; v) dx dv = 0 :Using the elementary inequalitya (log b  log a)  b  a ; a; b > 0 ;one obtainsddtH(t) =12ZR3dxZR3dvZR3dwZS2de(%(t; x))B(v;w; e)nloghp(t; x+  (v;w; e); v) p(t; x   (v;w; e); w)i loghp(t; x; v) p(t; x; w)iop(t; x; v) p(t; x; w)12ZR3dxZR3dvZR3dwZS2de(%(t; x))B(v;w; e)hp(t; x+  (v;w; e); v) p(t; x   (v;w; e); w)  p(t; x; v) p(t; x; w)i=12ZR3dxZR3dvZR3dwZS2de(%(t; x))B(v;w; e)hp(t; x   (v;w; e); v) p(t; x+  (v;w; e); w)  p(t; x; v) p(t; x; w)i=: I(t) :With the notations (1.5), the correction functional takes the formI(t) =ZR3dxZR3dvZR3dwZS2+(v;w)de(%(t; x))B(v;w; e)hp(t; x   (v;w; e); v) p(t; x+  (v;w; e); w)  p(t; x; v) p(t; x; w)i=ZR3dxZR3dvZR3dwZS2 (v;w)de(%(t; x))B(v;w; e)hp(t; x   (v;w; e); v) p(t; x+  (v;w; e); w)  p(t; x; v) p(t; x; w)i=ZR3dxZR3dvZR3dwZS2+(v;w)de(%(t; x))B(v;w; e)hp(t; x   e; v) p(t; x+  e;w)  p(t; x; v) p(t; x; w)i:In analogy with [3], one may introduce the functional~H(t) = H(t) Z t0I(s) ds ;for whichddt~H(t)  0 :Note that, in the Boltzmann case  = 0 ; one obtains I(t) = 0 :53. Equations of continuity, momentum and energyWe follow [7, Ch. 16] (cf. also [6, Ch. 16], [5, Ch.V.6]). Adapted to the notations of[7], equation (1.6) takes the form"@@t+ c @@r#f =Z Zdk dc1 2 (c1   c)  khf 0(r +  k) f 01(r +  k)  f(r) f1(r)i; (3.1)where f = n p : Here n denotes the number density, and integration dk is overS2+(c; c1) (cf. (1.5)). For simplicity we set   1 :The uniform steady state isf (0) = nm2k T 32exp m kc  c0k22k T: (3.2)A rst approximation to the solution of equation (3.1) is f = f (0) ; a second approx-imation isf (1) = f (0) (1 + (1)) ; (3.3)where (1) is a linear function of the rst derivatives of n ; T and the mass velocityc0 : In the following derivations we neglect all products of derivatives and derivativesof higher order.3.1. Left-hand side of the kinetic equationConsider the left-hand side of equation (3.1):"@@t+ c @@r#f (1) ="@@t+ c @@r#f (0) = f (0)"@@t+ c @@r#log f (0) :Note that@@tlog f (0) =1n@@tn 32T@@tT +m2k T 2@@tT kc  c0k2 +mkT(c  c0) @@tc0and@@rlog f (0) =1n@@rn 32T@@rT +m2k T 2@@rT kc  c0k2 +mkT @@rc0!(c  c0) :Multiplying with  = 1 and integrating with respect to c ; one obtainsZdc f (0)@@tlog f (0) =@@tn  3n2T@@tT +3n2T@@tTand Zdc f (0) c @@rlog f (0) =c0 @@rn  3n2Tc0 @@rT +mn2k T 2c0 @@rT3kTm+ ndiv(c0)= c0 @@rn + ndiv(c0) :6Finally,Zdc f (0)"@@t+ c @@r#log f (0) =DDtn+ ndiv(c0) (3.4)whereDDt=@@t+ c0 @@r: (3.5)Multiplying with  = c   c0 and integrating with respect to c ; one obtains (cf.(A.2))Zdc (c   c0) f(0) @@tlog f (0) = n@@tc0and (cf. (A.2), (A.4), (A.3))Zdc (c   c0) f(0) c @@rlog f (0) =kTm@@rn 3kn2m@@rT+m2k T 2Zdc (c   c0) f(0) (c  c0) @@rT kc  c0k2 +mkTZdc (c  c0) f(0) (c  c0)  @@rc0!(c  c0) +mkTZdc (c   c0) f(0) c0 @@rc0!(c  c0)=kTm@@rn 3kn2m@@rT +m2k T 25n kTm!2@@rT +mkTkTnmc0 @@rc0!=kTm@@rn+nkm@@rT + n c0 @@r!c0 :Note that @@rc0 is a matrix,@@ric0;j3i;j=1and"c0 @@rc0!#i= c0;1@@r1c0;i + c0;2@@r2c0;i + c0;3@@r3c0;i = c0 @@r!c0 :Finally one obtains (cf. (3.5))Zdc (c  c0) f(0)"@@t+ c @@r#log f (0) = (3.6)n@@tc0 + n c0 @@r!c0 +kTm@@rn+nkm@@rT = nDDtc0 +1m@@r(k nT ) :Multiplying with  = kc   c0k2 and integrating with respect to c ; one obtains(cf. (A.5))Zdc kc   c0k2 f (0)@@tlog f (0) =@@tn3kTm 32T@@tT n3kTm+m2k T 2@@tT 15n kTm!2=3kTm@@tn 9kn2m@@tT +15kn2m@@tT =3kTm@@tn+3knm@@tT7and (cf. (A.5), (A.6))Zdc kc   c0k2 f (0) c @@rlog f (0) =3kTmc0 @@rn 9kn2mc0 @@rT +m2k T 2c0 @@rT 15n kTm!2+mkTZdc kc  c0k2 f (0) (c  c0)  @@rc0!(c  c0)=3kTmc0 @@rn 9kn2mc0 @@rT +15nk2mc0 @@rT +mkT5n kTm!2div(c0)=3kTmc0 @@rn+3knmc0 @@rT +5nkTmdiv(c0) :Finally one obtains (cf. (3.5))Zdc kc   c0k2 f (0)"@@t+ c @@r#log f (0) =3kTmDDtn +3knmDDtT +5nkTmdiv(c0) : (3.7)3.2. Right-hand side of the kinetic equationConsider the term in brackets at the right-hand side of equation (3.1). Expandingf1 ; f01 by Taylor's theorem, and retaining only the rst derivatives, gives(f 0 f 01   f f1) +  k  f 0@@rf 01 + f01@@rf 0!: (3.8)Substituting from (3.3) into the rst term on the right of (3.8) (neglecting termsas before) givesf (0) f(0)1(1)0+ (1)01   (1)   (1)1; (3.9)sincef (0)0f(0)01 = f(0) f(0)1 :The second term on the right of (3.8) involves space-derivatives. Thus we maywrite f (0) in place of f (1) and obtainf (0)0f(0)01@@rlog f(0)01 + f(0)01 f(0)0 @@rlog f (0)0= f (0) f(0)1@@rloghf(0)01 f(0)0iand@@rloghf(0)01 f(0)0i=2n@@rn 3T@@rT+m2k T 2@@rTkc01   c0k2 + kc0   c0k2+mk T @@rc0!(c01   c0) + (c0   c0)= : : : +m2k T 2@@rTkc1   c0k2 + kc  c0k2+mk T @@rc0! (c1   c0) + (c  c0):8The integral on the right-hand side of the equation givesI =Z Z k  f (0) f(0)1@@rloghf(0)01 f(0)0i!2 (c1   c)  k dk dc1=2n3 f (0)Zf(0)1Zk @@rn (c1   c)  k dk dc1 3T3 f (0)Zf(0)1Zk @@rT (c1   c)  k dk dc1+m2k T 23 f (0)Zf(0)1Zk @@rTkc1   c0k2 + kc  c0k2(c1   c)  k dk dc1+mk T3 f (0)Zf(0)1Zk  @@rc0!(c1   c0) + (c  c0)(c1   c)  k dk dc1 :(3.10)According to [6, formula 16.8,2] we haveZk (c1   c)  k dk =23(c1   c) :Thus, (3.10) impliesI =232n3 f (0)Zf(0)1 (c1   c) @@rn dc1  233T3 f (0)Zf(0)1 (c1   c) @@rT dc1+23m2k T 23 f (0)Zf(0)1 (c1   c) @@rTkc1   c0k2 + kc  c0k2dc1+23mk T3 f (0)Zf(0)1 (c1   c)  @@rc0!(c1   c0) + (c  c0)dc1 : (3.11)Note that (cf. (3.2))Zf(0)1 (c1   c0) @@rTkc1   c0k2 + kc  c0k2dc1 = 0 ;Zf(0)1 (c  c0) @@rTkc1   c0k2 + kc  c0k2dc1= (c  c0) @@rT n"3kTm+ kc  c0k2#;(cf. (A.1))Zf(0)1 (c1   c0)  @@rc0!(c1   c0) + (c  c0)dc1=Zf(0)1 (c1   c0)  @@rc0!(c1   c0) dc1 = nkTmdiv(c0)andZf(0)1 (c  c0)  @@rc0!(c1   c0) + (c  c0)dc1 = n (c  c0)  @@rc0!(c  c0)9Thus, (3.11) impliesI =  232n3 f (0) n (c   c0) @@rn+233T3 f (0) n (c  c0) @@rT 23m2k T 23 f (0)(c  c0) @@rT n"3kTm+ kc  c0k2#+23mk T3 f (0)"nkTmdiv(c0)  n (c   c0)  @@rc0!(c  c0)#=  23n3 f (0)2n(c  c0) @@rn 23n3 f (0)(c  c0) @@rT 32T+m2k T 2kc  c0k2+23n3 f (0)"div(c0) mk T(c  c0)  @@rc0!(c  c0)#:When multiplying with  = 1 ; c  c0 ; kc  c0k2 and integrating with respect toc ; many terms vanish. One obtains (cf. (A.1))Zdc I =23n3"ndiv(c0) mk Tk Tnmdiv(c0)#= 0 ; (3.12)(cf. (A.2), (A.4), (A.3))Zdc (c   c0) I =  23n32nkTnm@@rn+23n332TkTnm@@rT  23n3m2k T 25n kTm!2@@rT=  23n32kTm@@rn+23n3@@rT"3kn2m 5kn2m#=  23n32kTm@@rn 23n3knm@@rT =  23m3@@r(k n2 T ) (3.13)and (cf. (A.6))Zdc kc  c0k2 I =23n3 div(c0)3kTnm 23n3mk T5n kTm!2div(c0)=23n3 div(c0)"3kTnm 5kTnm#=  23n3 div(c0)2kTnm: (3.14)Note that the corresponding integrals of the term (3.9) are zero.3.3. Comparison of both sidesFrom (3.4), (3.12) one obtainsDDtn+ ndiv(c0) = 0 : (3.15)10This equation is identical with [7, (16.33,3)].From (3.6), (3.13) one obtainsnDDtc0 +1m@@r(k nT ) +23m3@@r(k n2 T ) = 0 ;ornDDtc0 +1m@@rk nTh1 +233 ni= 0 : (3.16)Introducing (cf. [7, (16.33,2)])p0 = k nTh1 +233 ni; (3.17)and up to some notations, equation (3.16) is identical with [7, formula 16.33,4]. Thisequation of state is in agreement with that obtained from the virial [2].From (3.7), (3.14) one obtains3kTmDDtn+3knmDDtT +5nkTmdiv(c0) +23n3 div(c0)2kTnm= 0or, using (3.15),3knmDDtT + k T div(c0)2nm1 +23n3= 0 ;i.e.DDtT +23T div(c0)1 +23n3= 0 : (3.18)Taking into account (3.17), this equation is identical with [7, formula 16.33,5].Equations (3.15), (3.16), (3.18) are the rst order approximations to the equa-tions of continuity, momentum and energy. These are the Euler equations with thehydrostatic pressure given by (3.17) and they are identical to those obtained for theEnskog equation (recall that for simplicity  was taken as unity). For future work,the Chapman-Enskog analysis may be continued to evaluate the transport coe-cients by computing the collisional transfer of momentum, energy, and for CBA,mass. We anticipate that, as with the Enskog equation, the resulting viscosity, ther-mal conductivity, and self-diusion coecient will be in good agreement with theresults already obtained by Green-Kubo analysis (cf. [2] and [11]).Acknowledgments. The authors want to thank B. Alder, F. Alexander, andM. Malek Mansour for useful discussions. This work was supported, in part, by agrant from the European Commision DG 12 (PSS*1045) and was performed, in part,at Lawrence Livermore National Laboratory under the auspices of the Departmentof Energy under Contract No. W-7405-Eng-48.11Appendix: Moments of a Gaussian variableLet  = c  c0 : ThenZdc f (0)  A =Zdc f (0)Xj;kj aj;k k =kTnm[a1;1+ a2;2 + a3;3] ; (A.1)Zdc f (0) i   b =Zdc f (0)2i bi =kTnmbi ; (A.2)Zdc f (0) i  A =Zdc f (0) iXj;kj aj;k k = 0 ; (A.3)Zdc f (0) i b A =Zdc f (0) iXj;kbj aj;k k = 0 =kTnmXjbj aj;i ;Zdc f (0) i   b kk2 =Zdc f (0)2i bi kk2 = n bi kTm!2[1 + 1 + 3] ; (A.4)Zdc f (0) kk4 =Zdc f (0)(21 + 22 + 23)2 = 15n kTm!2(A.5)and Zdc f (0) kk2  A =Zdc f (0) kk2Xj;kj aj;k k =Zdc f (0)Xi2iXj2j aj;j= n kTm!2[3 a1;1 + a2;2 + a3;3 + a1;1 + 3 a2;2 + a3;3 + a1;1 + a2;2 + 3 a3;3]= 5n kTm!2[a1;1+ a2;2 + a3;3] : (A.6)These formulas follow from elementary properties of one-dimensional Gaussianrandom variables, in particular, E4 = 3 (E2)2 ; i.e.1nZdc f (0) kik4 = 31nZdc f (0) kik22= 3 kTm!2:12References[1] F. J. Alexander and A. L. Garcia. The direct simulation Monte Carlo method.Computers in Physics, 11(6):588593, 1997.[2] F. J. Alexander, A. L. Garcia, and B. J. Alder. A consistent Boltzmann algo-rithm. Phys. Rev. Lett., 74(26):52125215, 1995.[3] N. Bellomo and M. Lachowicz. On the asymptotic theory of the Boltzmannand Enskog equations: a rigorous H-theorem for the Enskog equation. InMathematical aspects of uid and plasma dynamics (Salice Terme, 1988), pages1530. Springer, Berlin, 1991.[4] G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows.Clarendon Press, Oxford, 1994.[5] C. Cercignani. The Boltzmann Equation and its Applications. Springer, NewYork, 1988.[6] S. Chapman and T. G. Cowling. The Mathematical Theory of NonUniformGases. Cambridge Univ. Press, 1952.[7] S. Chapman and T. G. Cowling. The mathematical theory of non-uniform gases.An account of the kinetic theory of viscosity, thermal conduction and diusionin gases. Cambridge University Press, London, 1970.[8] A. L. Garcia and W. Wagner. The limiting kinetic equation of the consistentBoltzmann algorithm for dense gases. J. Statist. Phys., 101(5-6):10651086,2000.[9] H. J. Herrmann and S. Luding. Modeling granular media on the computer.Contin. Mech. Thermodyn., 10(4):189231, 1998.[10] G. Kortemeyer, F. Dan, and W. Bauer. Nuclear ow in consistent Boltzmannalgorithm models. Phys. Lett. B, 374:2530, 1996.[11] T. E. Wainwright. Calculation of hard-sphere viscosity by means of correlationfunctions. J. Chem. Phys., 40(10):29322937, 1964.13

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Some new properties of the kinetic equation for the consistent Boltzmann algorithm

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